I hope you enjoyed the wild goose chase through Numberland. If you'd like to learn more about infinity, check out Brilliant's intro to infinity course brilliant.org/upandatom/
Yes, it's a very approachable video on a complex mathematical issue. I am passing it around. I really love mathematics, but it's sometimes difficult to explain this concept to people that are not into math but are curious about many things and will love to know.
This number was described using language even before it was measured. It's the amount of time it took that ball to hit the ground when dropped from that bridge.
We could create a new set of numbers I would call the "useful numbers" which would be a countably finite set of numbers containing every number any human will ever need for any purpose. In that sense just describing or even thinking about a number would add it to that set, but that set would never be infinite.
Well if you have to measure it exactly you end up with uncountably infinite number of words to describe it considering the gravitational pull of all the objects with mass in universe along with quantum interactions and air resistance it experienced , so impossible....😂
In fact, wouldn't it be describable by relating the time (t) to the mass of the ball (m), acceleration due to gravity (g) and height of the bridge (h)?
… with a certain set of conditions that we could describe here in detail if we had an infinite amount of room but one of the conditions will be the time at which it was dropped so we’ll need to get the release time and then hope it’s an unfathomable miracle that the exact time of day she let go of the ball isn’t a transcendental, so here we go again…
The physicist in me wants to say you only need 44 decimal places. But the mathematician in me really appreciated how you presented this. I really like your content :)
Now I wonder the likelyhood of it being an undescribable trancendental if restricted to 44 decimals instead of infinite. Is it now 0 instead of 100 percent?
@@backwashjoe7864 A unit of Planck time is 5.39×10−44 seconds. It's the smallest unit of time that makes physical sense, at least according to current theories. There could be smaller segments of time, but we currently can't describe them through physics. That wasn't the point of this video though :)
Excellent abstract math, but for physical events like a ball drop, you have to consider the physics, especially "plank time" which is 10^-43 seconds. Plank time is "the length of time at which no smaller meaningful length can be validly measured". Since the drop time is a finite number of states, all of times can be enumerated.
This is by far the best video I have seen on the relationship between real, rational, irrational, algebraic and transcendental numbers. So well layed out and tidily expalined!
The only video I've seen. Early on in the video, my thought, if you can define two points, there are always infinite number of points between them no matter how close they are together.
I recommend you check out the Numberphile videos on the same subject. They are so much fun: - *Transcendental Numbers - Numberphile* - *All the Numbers - Numberphile*
Don’t worry! Heisenberg uncertainty principle says that the time it takes for the ball to drop has a little slop to it, so you can almost certainly find a rational number that could plausibly have described the “exact” time to drop for any physical definition of exact :)
I'd also argue that, given that we only have physics to describe a finite temporal resolution(thanks, Planck), you can one hundo find a rational number to describe the exact time, at least within our known physics.
Год назад+3
The funny thing is, if you could measure time exactly, you would know nothing about the energy of the ball.
@ check out Alain Connes 2015 talk to physicists on noncommutativity as the origin of time and entropy. Fascinating stuff! See Professor Basil J. Hiley for followup.
@@PunnamarajVinayakTejas My review of math professor Joseph Mazur's book "The Motion Paradox" - reissued under a different title. Professor Mazur does an expert job of giving the behind-the-scenes wrangling of conceptual philosophy which gave rise to applied science. What is the difference between time and motion exactly? If that question seems too abstract, this book proves the opposite. Most college graduates assume that Zeno's paradoxes of motion were solved by calculus with its continuous functions. Mazur puts the calculus at the heart of the book, from Descartes and Cavalieri to Galileo, Newton and last but not least Mazur's favorite: Gabrielle-Emilie de Breteuil. In fact, upon investigation, one finds many top scientists still studying and learning from the anomalies in infinite measurement. Regarding relativity Mazur states the wonder of absolute motion is that it "conspires with our measuring instruments to prevent any possibility of detection." As Mazur points out "we don't measure with infinitesmial instruments" and so the perceptual illusion of time continuity remains despite the reliance of science on discrete symbols. With attempts at a unification of quantum mechanics and relativity Zeno's paradoxes reemerge with full-force in the "Calabi-Yau manifold." Mazur writes that the original concept of dimension still holds but now means measuring more by abstract reason than by sight. Although each scientist featured by Mazur appears to have increasingly solved the paradox of motion in the end I think Zeno will be avenged and science will return to right back where it started. There seems to be a deadlocked struggle between discreteness (particle) and continuity (wave) in science and Mazur argues that indeed Nature "makes jumps" despite seeming continuous. But Mazur admits we are left with "splitting operations that can take place only in the mind."
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
This was really interesting, but I'm not sure, if I understood correctly. Are the following two statements correct? 1) "The natural numbers are able to assign a unique label to all algebraic numbers." 2) "A single transcendental number contains all natural numbers infinitely many times as intervals of its digits."
That last part where you explained how even language, when pushed to the extreme, is still a countable infinite and so cannot be used to accurately describe real numbers (and beyond) was really great!
Mathematically speaking, you are correct. Physically speaking, I believe that there is a finite smallest length as well as time, so that technically speaking, the amount of time required for the ball to drop is rational.
We don't know if there is a finite smallest length. We just know that there is a smallest (Planck) length that it would even theoretically be possible to measure.
@@John_Fx It would be correct to say that it would be the smallest length *physically possible* which would apply to matter and energy within our universe. Anything smaller than that wouldn't make sense from a Physics perspective, which would include things like dropping a ball. Therefore, the amount of time, as well as the distance that the ball dropped, would be rational.
There is a set of numbers between the agebratic nubers and the transcendent: the computable nunbers. These are numbers, that can be calculated to a given accuracy by a coumputer (or a touring machine).. In this set there are numbers like e and pi. This set is cpuntable infinit too, because the set of comtuter programms is countable infinit.
@@GarryDumblowski Computable numbers are described by their Turing machine, so they are describable. The converse is not true. I will describe a number that is not computable. Start by enumerating all Turing machines T_1, T_2, T_3,. ... Set a_i = 0 if T_i halts and set a_i = 1 if T_i does not halt. Now let alpha = sum_i a_i/2^i. The number alpha is describable, because I just described it, but it is not computable.
I recently rewatched Matt Parker's Numberphile video "all the numbers" where he talked about the computable numbers, and the normal numbers(numbers which contain any arbitrary string of numbers)
@@ronald3836how is that number not computeable? If you chose to enumerate the machines by alternating between on that does halt and one that does not then your answer is a geometric series that converges and can be computed to any arbitrary level of precision. It's 0.0101010... in binary which is 1/3?
The rule of two -- Darth Bane, Sith lord. Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Jade, you are such a captivating educator. The point about the number of ways a number could possibly be described being countably infinite was particularly interesting to me. Great video!
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
this is the first time i am watching a video from this channel, and it blew me away! i did not expect such a crystal clear, nicely paced, logically flowing and informative video... absolutely gripping stuff, thoroughly enjoyed it.. thank you creator!
Infinity is a concept that you can make tons of videos about with completely different angles of approach. Like, a lot of videos. There should be a number to describe that.
1:30 Computer engineer here, going crazy, looking at what appears to be a common laptop, which has a cpu that runs on the order of few GHz, which means any digits below 9th are physically impossible and just some randomized garbage
I'm just spit-balling here -- don't know if this is actually workable, but . . . . I think we are all comfortable with pretending that her set-up (minus the limitations of the laptop) is capable of perfect precision, right? What if she has a measuring system with multiple known time delay lines. So for one event, we could get hundreds of measurements of the time. Those delays could be calibrated to fall within certain known fractions of a CPU cycle. By comparing the measurements at the different "beats" we could get precision to many more decimal places (I think). So, I'm certain that is the setup that she used :-)
That was sort of like what I was thinking, Quantum Mechanics tells us that there are fundamental limits to how precise any physical property can be defined, thus there will ALWAYS be a rational value that uniquely specifies a given possible value from all other possible values. Mathematics might have infintes and infintesimals, but the Physical Universe doesn't
@@RichardDamon " thus there will ALWAYS be a rational value that uniquely specifies a given possible value from all other possible values." Nope, not even close. All it means is that there is no 1 true value as it is always a range of possibilities. But for timescales that can still be shorter than a planck-time, it is just fundamentally impossible to measure.
@@ABaumstumpf No, it isn't a "Measurement" phenomenon, it is that time is actually indeterminate at that scale, so time doesn't exist finer than that. You could say that the idea of a "Precise Time" doesn't exist. Just as no integer exists between 1 and 2, no time exists between one time quanta and the next.
@@RichardDamon "it is that time is actually indeterminate at that scale, so time doesn't exist finer than that." Nice claim, but again - that is not what is says. Also "No, it isn't a "Measurement" phenomenon" I never claimed that - so why the strawmen? "Just as no integer exists between 1 and 2, no time exists between one time quanta and the next." Which is 100% wrong.
Injective is dual to surjective synthesizes bijective or isomorphism (duality). Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
YEY! Finally a video that's not one of those terrible shorts. We like normal videos. ☺ EDIT: Let me clarify that by "terrible" I mean the format not the content. 🙃 If I wouldn't like Jade's videos I wouldn't watch and subscribe to the chanel. I just don't like the shorts format and I always skip those.
There is an argument to be made that the existence of the planck measurements (time, length, etc.) proves that irrational numbers do not exist in the real world. The planck measurements are the smallest meaningful measurements in our current framework of physics, and since everything is made up from integer multiples of these, you can't actually have an irrational distance in the real world. Edit: This would only be true under the condition that the planck measurements are actual limits of time and space, and not simply limits of the ability to measure them. This is still an unsolved problem however.
I thought Planck units were just what can be, in theory, meaningfully measured. As in, it doesn't mean there's nothing smaller, it's just that we would never be able to measure it.
@@tomshieff You're right, it is currently unknown if the planck measurements are actually real or just limits of measurement. They can be calculated using the uncertainty principle and our current understanding of gravity (relativity), so until someone finds an accurate model of quantum gravity, no one knows if the planck measurements are actually real.
The limits that define the Planck units are fundamental, but it's definitely not known that everything is integer multiples of them. This is similar to the idea behind loop quantum gravity, which experiments have found no evidence of (and good, albeit not 100% definitive, evidence against).
Fascinating video! The end bit about counting descriptions got me thinking about cellular automata and language modeling, which I've never cut through before. It's really great to see, think about, experience, how branches of knowledge coalesce. Thanks, Jade!
Synthetic a priori knowledge -- Immanuel Kant. Knowledge is dual according to Immanuel Kant. Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Assuming zero delay in starting and stopping a timer. You could use a numberless analogue stopwatch to measure the drop of your ball. The arc travelled by the stopwatch pointer is precisely how long it took your ball to drop. For the measurement to remain precise, it can not be described using numbers because numbers introduce divisions.
I've always been dissatisfied with the explanations of transcendental numbers, but this is the best so far. Thank you Jade. Fun to think about the time it takes for the ball to fall being an actual value but never being able to write it down.
Penso que não é bem assim, entendemos muitas coisas, porém somos limitados e essa limitação não se dá basicamente em nossa capacidade mental, ela se define mais pelo nosso tamanho em proporção ao universo ao redor, não entendo que haja algo que a mente humana esteja impossibilitada de entender para sempre, mas certamente há muito que não podemos alcançar, medir, ver etc, ou seja, coisas que são necessárias para que possamos chegar no entendimento. Não é uma questão de inteligência, mas de nossas limitações em relação ao contexto onde estamos inseridos. Pior que, uma das coisas que entendemos, é que tinha que ser assim, pelo menos nesse universo nessa vida, não haveria como ser diferente.
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
I really liked how you aligned the things we must know (like the definition of countable infinity and transcendental numbers etc.) in an ordered track.
These definitions feel like a schoolyard argument. "Ofc i can count your set! I have infinite numbers to count with!" "Nu-uh! My set is an uncountable infinity! I win!"
You described that infinitesimal just fine tho.. it's the gravitational distance in time from the top of that bridge to the earth plus or minus human and computational error!
The problem is you haven't described it precisely enough to make it reproducible. The ratio of the diameter of a circle to its circumference is reproducible because it's based on theoretically perfect geometric objects. If you were to reproduce her experiment, the probability of measuring the exact same number is essentially zero because there's no way for you to perfectly replicate the experiment to infinite precision. All of the subtle forces of gravity from the Earth, Sun, Moon, etc... would all be different and would result in a different measurement. So for you to describe her number, your description would also have to specify the exact initial conditions of her experiment including for example the starting height, time it was dropped, etc.., but those numbers when measured to infinite precision are ALSO likely to be indescribable. So there actually isn't any way to perfectly describe her experiment to reproduce her exact number. The way you described it doesn't specify her specific number, it specifies an infinite set of numbers.
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Which means it actually is rational. Though I guess then the question becomes, what exactly counts as the start and the end. At the level of Planck times, the concept of "the moment it touches the floor" probably isn't so clear cut.
@1vader I think that the instant that either the acceleration is first zero, or the velocity is first zero after release, would be two good candidates for defining "touching the floor". It depends on whether you want to define "touching the floor" to be defined as when it exerts enough force to have altered the object's velocity, or to define it as when the objectsvdownward velocity is cancelled. But then again, air resistance would affect those two points, so maybe it needs to be specified as occuring in a vacuum? 🤷🏻♂️
@@AMcAFaves That's still thinking on a way too macro level. Not all of the atoms of the ball will come to a stop at the same time. And at the scale of plank times, we could differentiate stuff on an even smaller scale where the particles might not even have something like a defined location. Actually, because of Heisenberg's uncertainty principle, I guess it wouldn't even be possible to know.
The funniest, the most pleasant, kind and warm person that makes a very arid topic/domain which is math to be likeable by people that usually are away of math (NOT mt case...). Anyway: it is my youtube favorite channel, it does not matter what type, above everything. Pur and simple I adore the way she is and how much she enjoys what she does and of course I like her. When I want to have a nice mood, I watch one of her math videos which always give deep knowledge expressed simply, funny, enjoyable, so that many can get the point...
as an educator, I learned that its not the teaching method (which yours is great, BTW), its the emotions and enthusiasm of the educator that really inspires students. And in this department - you are 100%
If you choose the right unit of measure, the ball-drop-number can be algebraic or rational or a natural number or even just 1. It might be hard to choose the right unit, though.
Sure, you could declare that the ball takes exactly 1 Bleep to reach the ground. But would that be usefull ? As soon as you want to translate your Bleep unit into another unit like the second, you'll be back to square one. Unless you decide your Bleep unit is the new standard, and any time measurement must be expressed in Bleeps. However if your goal is to avoid dealing with strange numbers, then you must create a new unit fit for any new measurement you make, and forget about comparing results between experiments. Maybe that's not such a good idea after all. :p
@@theslay66 - Did you fail to understand my last sentence? I tried to make it succinct so the reader would have a little "aha" moment. Was it effective for you?
Rational is dual to irrational -- numbers. Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant. Subgroups are dual to subfields -- the Galois Correspondence. Homology (convergence, syntropy) is dual to co-homology (divergence, entropy). The integers are self dual as they are their own conjugates -- the conjugate root theorem. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron. "Always two there are" -- Yoda. Concept are dual to percepts" -- the mind duality of Immanuel Kant. Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
What a beautifully deliveredl lesson. I feel there were several times where it could have become very confusing. You kept everything crystal clear. Thank you very much. 😊
2:55 Reminds me of doing drawing octagons in Paint. Trying to make the sides equally long, having the straight sides 14 pixels long would make the skewed sides about 10 pixels (14 / √2). As I can only draw whole pixels, at that scale it's as precise as it gets. Making a pattern of nested octagons, it doesn't scale well while drawing larger octagons around the first one and trying to keep everything nice and centered. So this vid tells me/confirms that, even if I start with an octagon with straight sides of 14,000,000,000 pixels or even larger, I can only approach the length of the skewed sides and scaling up or down from that will never work out perfectly.
This was an awesome video! Im glad i found your channel. You related all of these concepts together and answered some questions ive had for a while now. Dont stop making videos!
Just a pleasure as always to watch all your exuberant content on anything that captures your attention, Jade! Your enthusiasm and passion for learning are just but more than contagious! It's been my immense pleasure and honor to have been your audience for almost more than three years now, and it is with utmost admiration that I admit that I am very proud to sponsor your content to all my friends and family members, and bug them constantly with all my ramblings about science and its wonders in our Nature :) I hope you keep spreading your contagious energies here forever, and I am thrilled to be a member of our little community here! Best, Sai
Dear Jade, great video. I love how passionate you are about describing mathematical and physical nuances. As for the time measurement description (or any other measurement), there is a handy tool - measurement uncertainty (and the way to express it and measurement results). It would be great to watch you explaining probability issues connected to them.
Great explanation on infinities and types of real numbers. My only slight nitpick, based on the title I thought this was going to have something to do with the fundamental properties of time… like I was like “yeah, can we measure time? I feel like all we can really do is find very regular sources of a periodic signal and call that time… let’s learn more on this interesting topic”… but the concept of this video really applies to any exact physical measurement, not just time (which as you correctly alluded to, the question of what constitutes a measurement and how to minimize error is itself a endlessly complex subject). Anyways, still really enjoyed the great explanations, and loved your personality. Thanks for sharing!
At one moment I thought the Stern-Brocot Tree would make an appearance… a clean way to show every positive rational number (well, by clean I mean there are no duplicates)… and you can do a binary search on it with your irrational number and each further depth on your search leads you to a better approximation, already in reduced form.
i learned more in this video than in my 3 years of high school. my math teachers never talked about real numbers this deep (maybe because its not required in exams)
It feels interesting to rediscover this channel after having seen her explain knot theory related to the painting on Tom Scott's channel over 4 years ago. This was a great video and it seems like there is a whole lot of material/videos I should catch up on here.
There's no reason to prefer cardinality over ordinality for measuring size. Indeed, I would say ordinality is far more accurate. When you do that you see the rationals are roughly omega^2, whereas the integers are roughly omega. As for Cantor's "proof" of the uncountability of the reals, that turns out to be wrong. I published a proof that his "proof" was wrong on RUclips a while back, but it had an error in it. Shortly thereafter I fixed my proof, but I have not republished yet due to lack of time. It's fairly simple to show that Cantor's idea of "cardinality" is flawed and only ordinallity exists as a measure of size (or order). The basic idea is to establish there must be an ordinal number of digits for real numbers. We can then show that no matter what ordinal number of digits you choose, when you try to make a list it will always be longer than wide. This shows the diagonal does not cross all elements of the list, and subsequently, Cantor's proof falls.
An uncomputable number is Chaitin’s constant, or more precisely, family of constants. It is related to the probability that some Turing machine will halt, something that there exists no Turing machine for determining. Another one is constructed from a base-2 representation of a number between 0 and 1, where each trailing digit gets its value from whether or not a Turing machine associated with it will halt; each digit has its own Turing machine.
But... we can describe that number, can't we? That number is equal to the number generated on a computer by a man attempting to measure the time it takes for a ball dropped by Jade from a bridge of X height plus the height from the bridge to Jade's hands, with the ball falling at the rate of acceleration 32.17 ft/s^2, and your man with a reaction time of Y to press the start and stop button. I don't know what number that is exactly but isn't whatever that number is equal to the description above?
How precisely can you define that velocity you mentioned however? You will run in the same problem eventually, since you are just shifting on what part of the equation of the event needs to be exactly defined. You will also run into the whole issue of uncertainty at a certain level of precision where it becomes involved in quantum dynamics and that is just opening another can of worms outside of just the mathematical principles of it.
But wouldn't that be describing an approximation of the number? Like, if you try to reproduce the number based in this description, you would get very similar results, sure, but you can't guarantee the same number to pop up? I'm not a mathematician tho, so idk
I have a similar question. The problem only arises because of the units you select. You could just say, the time it took is 1 Ballbridge, which is the amount of time it takes for this ball to fall from this bridge under these conditions. Then the problematic number becomes how many seconds there are in a Ballbridge, which you can just describe as "the number of seconds in a Ballbridge." What is the number you can't do this for? If you talk about it, you've described it.
Each of the other numbers you mention (height) has the same problem of being hard (impossible) to know exactly. Theoretically. In a physical world made up of quantum elements, "infinitely exact" isn't a meaningful concept.
In theory, technically, the number you get by the result of a given experimental process can be described as "the result of (description of the experimental process)", that that description does not actually tell us anything meaningful - the whole point of describing a number that is the answer to some question is to be able to understand that answer in some greater, more meaningful context. There is also the problem that a given description may not actually describe a specific number, or may require as part of its description other 'merely describable' numbers - for example, your suggestion involves things like the acceleration due to gravity at Earth's surface (itself only an approximation, and a number which varies from moment to moment, with latitude and longitude, and with height above the surface), and also involves descriptions of individuals who are not constant with time, performing actions with initial conditions that are not specifiable with exactness except to describe them 'as what they are' in the same ultimately unhelpful way. Which is all to say that, in a very technical sense, the number is technically describable as the result of a very specific description of events which happened, described unambiguously (though only descriptively) in their time and location, the actual value of that number becomes utterly inaccessible - Describable numbers need not be computable. We then reach an interesting philosophical possibility - we are assuming that all mathematically possible numbers are physically possible. We know that, for example, pi (square units) is the area of a circle that is one unit in radius... but *can we physically make a circle*? Does it physically matter if we cannot exactly express pi if we cannot *make* pi in the physical world? After all, there is some evidence that space is, in a sense, 'pixelated', quantized with some minimum possible meaningful distance, meaning any attempt at any physical circle will only be 'approximately' a circle. Likewise, even with something as simple as sqrt(2), can we say with certainty that a triangle with unit legs can physically exist to the point that its hypotenuse is an irrational value, given that the length of that hypotenuse must be some whole number multiple of that minimum possible length (so must therefore be rational). Perhaps it is not so surprising that we cannot meaningfully describe most of the numbers that can mathematically exist, since, even as big as the observable universe is, it is finite, and thus can only contain a finite number of combinations of things... If the entire universe, and all 10^80 particles it contained were a mechanism for storing binary data, it could only represent 2^(10^80) different unique states - a very very large number, but still finite.
This was brilliant. Accessible yet profound. Like an episode of James Burke's Connections. I'm convinced that Jade is among the world's best STEM communicators. But . . . did you get your ball back?
Ironically, My daughter who is getting a math minor with her bio degree and I were just talking about the sizes of infinity the other day related to the infinite hotel experiment. Because I am that nerdy. When I saw your video I had to check it out. I have to say you brilliantly explained this concept so well. I remember in engineering school in Calculus, that we just got 'infinity' but never considered different types of infinity. I never thought about it until years later. Thank you!
Well, for calculus, you don't need the different types of infinity. All you need to know is if a quantity is bounded or grows unbounded. That's true for science, engineering, and many other fields of math as well. Even within math, the different infinities arise only when you get deep into Axiomatic Set Theory. Which-however fascinating it is in its own right-isn't useful in science and engineering. After all, human knowledge is so vast, it's impossible to learn all of it in one lifetime, let alone a mere 4 years of college. If you want to graduate in a reasonable amount of time-so that you can leave academia, come out into the real world, solve real-world problems, and earn a living-it shouldn't come as a surprise that you were taught very specific subjects that were relevant to your major, while vastly more subjects were left untaught. Luckily, we all have the option to learn more if we are so inclined.
I know you probably know this already (I just like talking about math), but there is a way to write 1/3 in (repeating) decimal notation: 1/3 = 0.(3) = \lim_{x \to \infty} \sum_{i=0}^{x} {3 \over {10^i}}. (And, as a quick random mention, it has some funky properties like 0.(9) = 1)
There is a Planck constant that limits the accuracy of distance measurments, when you measure the time - you measure the distance too, as each object have some size. So the exatct time will be limited by Planck constant. The minimum amount of time that it is possible to measure - is when a photon passes a one Planck distance. And you can mesure time relative to the number of Planck distances that photon passes at the same period. In this units time measurment will always be an integer.
I am quite glad that this lady has gotten to fulfill her lifelong ambition of dropping a soccer ball off a very tall bridge! She did quite a good job of at least trying to explain something that cannot be made sense of, at least at this time. The number of unexplainable in our universe is, quite literally, infinite.
Time is a measurement of motion. Just like inches are a measurement of length. Measuring time with a clock is like measuring inches with a ruler. How precise do you want to be? Currently, clocks measure length down to the frequency/ wavelength of the microwave beam. What happens when you boost the frequency up to the X-ray/Gamma ray frequencies? You shorten the wavelength and the distance the photon has to travel to complete a cycle. Precision goes from 10 to 10th to 10 to 20th/25th.
Certainly you can eliminate the mortal reaction factor. They do it all the time with sports competition, etc. For example, this thing could be based on a computer and cameras.
Isn't time considered (according to our current knowledge) quantized with the plank-time (the time in which light travels one plank-length) being the shortest amount of time being able to pass?
Hey this is great as a maths tutorial but the premise may not be true here due to some underlying physics. The smallest distance that has any physical meaning is the planck length, and the smallest grain of time that has any meaning is the length of time causality (speed of light c) takes to travel the minimum meaningful distance. This means we can likely break time down into finite, or quantised, grains. Using this minimum length of time as a unit, we can then count how long the ball takes to reach the ground. This of course depends on your interpretation of physics and whether you think plancks length is truly the smallest distance and granular, but in this case it would be genuinely possible :)
for measuring length or time it's sufficient to use a natural number, as they are always multiplies of the Planck length and Planck time. I thought you are going to take it to a different problem: the gravitation of the start point is slightly less than the end point down there, so the clocks run at a different speed. Of course, the own clock of the ball runs at yet another speed. And it is not clear what means to measure the time difference between two distant events at the points at which clocks cannot be synchronized
I had this exact realization during a measure theory course recently. We were discussing something about how you can approximate any measurable function within epsilon with a continuous function and something in my brain clicked and realized that the real numbers are insanely larger than imaginable. I tried to explain this to my friend, who is an engineer, and he didn't understand it at all even after 10 or 15 minutes of trying to explain that the rational numbers are basically as good as we're ever going to get. Glad to have a video I can send to people when I need to explain this idea.
8:39 Considering the smallest length of time we can measure is the Planck time (~10^-43 seconds), the most decimals a time measurement could have is 43 digits, i.e. it would be a rational number: some variable 't' (the time it took in seconds from the moment you let go to the moment the ball hit the ground) over 10 tredecillions (1 tredecillion = 10^42).
Can't this number being expressed as "the amount of time, in seconds for this ball to fall from this bridge on this very special moment in time"? (taking into account the ball, the space-time coordinates, the speed of wind, air pressure, heat, etc)
The other problem with infinite precision is that if you have two perfectly precise measurement "observers" detecting precisely when the ball hits the pavement, they will almost certainly disagree on the exact instant the ball makes precise "quantum contact"* with the ground due to the Uncertainty Principle. _*I'm not sure if that's even possible, but let's assume we can divine what "contact" means at the quantum level._
As a rather unimaginative engineer in everyday life, I sometimes like to suspend my disbelief and get drawn into mathematical flights of fantasy such as: • A stopwatch that can measure fractions of a second accurate to 38 decimals ... that works on a 2023 laptop. Which is what-9 GHz with overclocking? (In 1e-38 seconds, even light travels only 4e-30 meters.) • A human whose reactions are that fast. • Rubber bands that are infinitely long and infinitely stretchy.
i consider myself an imaginative person yet i struggle to imagine a liquid nitrogen cooling system in a stock-looking macbook outside with no visible condensation
There's an issue with the conclusion from a physics stand point. The number can be described by comparing the position of every single atom in the universe when the ball is dropped to the position of every single atom in the universe when the ball lands, and there is a finite amount of atoms in the universe. By doing that we isolate only the movement that occurred on the time axis.
You can describe it with physics. Solve s=ut+1/2at^2 for t. That only works in a vacuum but you could extend it with terms for air resistance. Though they'd be very complex and depend on other difficult-to-define numbers like the air temperature, the height of the bridge, the size and surface texture of the ball etc. As an engineer I do find it hard to see the benefit of defining accurately a value of which 34 of the 38 digits are purely fiction and could just as well be 0000 with no loss in usefulness. But I did enjoy the explanation, especially the distinction between different infinities. And, tbh, your presentation of it which was excellent :).
How is this channel not getting way more subscribers? I thought I finished all science vulgarisation RUclips videos and found this one after looking for a specific subject. Why did RUclips never suggested it to me?
The bright side of indescribable numbers is that we don't really have to care that they exist. Any number we can possibly encounter in any context is by definition describable, because at bare minimum a number can be described by specifying the context in which it was encountered.
Hello Jade. I watch all of your videos and love your approach to teaching and applying various mathematical principles. I never comment but today at the end of this video I laughed out loud at the ending when you stared at the laptop sadly realizing your number will never be able to be calculated, before slowly closing it. Moments like this are why I love watching your videos. Keep up the fantastic work making us smile and brightening our minds.
You can measure time exactly using planck time, which is tied into the planck length. Planck time is the time it takes for light to cross one planck length. So in theory, this is as small as you can physically go.
Perhaps one of the biggest problems with trying to find how long it takes for the ball to drop also requires that you have a frame of reference. Are you observing the bank drop and timing from the bridge? From the ground? Are you the ball, timing how long you fall. All three are distorted by their own frames of references and immeasurable beyond some accuracy.
Very captivating! From the title to indescribable numbers. Fantastic! I am a mathematician and I had never thought of indescribable numbers as such, but cool concept.
The amount of time it takes for that instance to occur is subjectively infinite. Or simply, 1. The event is singular. It only gets complicated when you compare it against the rest of reality. Thus a fraction.
It may seem impossible to measure your number, but there's one factor that could force it to be rational. If time is fluid, then your number is almost certainly transcendental, but if time can be broken into absolute units, the same way matter is broken up into electrons, quarks, neutrons, etc, then it must be rational. And if you consider time the 4th dimension, then it can almost certainly be broken up the same way the first 3 dimensions can.
A great video! It explained all the concepts well in a way I could understand and stimulated my curiousity and wonder. Although I was a bit worried at 10:50 that you were going to come even closer and break my screen! 😅
At 16:38 the list shows, as #10, "the first boring number." Which, of course, is interesting, and therefore not boring, and not the first boring number. A well-known paradox, thrown in (I assume) for fun, along with #11, "the answer to the ultimate question of life, the universe, and everything" (42 according to Douglas Adams).
We can subdivide descriptions further. If a number can be calculated to arbitrary precision with a finite length of run of a Turing machine, then it is computable. A Turing machine is an abstraction of computation that makes precise the notion of a finite-sized algorithm, and there are countably many of them. Computable numbers include familiar transcendental numbers like e and pi, but since there is a countable number of them, there are thus infinitely more uncomputable real numbers.
Congratulations, fellow viewer, if this video raised more questions than answered doubts. That's what a good educational video is supposed to do! I'd like to suggest a follow-up video that answers these questions: - So if there is a countable infinity and an uncountable infinity, are there other types of infinities as well? How many? Is there an infinity smaller than countable infinity? Aren't odd numbers, even numbers, and prime numbers smaller than countable infinity? Are there infinities between countable and uncountable? If there are bigger infinities than uncountable, what are they? Complex numbers? Points in 3-D, 4-D, ..., n-D, ..., infinite dimension space? Can we build larger infinities from smaller infinities, like how Dedekind built real numbers from rational numbers?
Actually, there's even better basic question: Does the inability to form pairing with integers implies that the set is bigger? Consider the rationals again. Whether you construct a 1-to-1 mapping with naturals depends on which order you choose to count them. Some of the infinite possible permutations of rationals map to naturals (like the diagonal case) and some don't (like the row-by-row case). It is entirely possible that the set of reals is no bigger than set of naturals and all if its permutations are not 1-to-1 mappings to naturals. As a very rough imperfect analogy, just because a cube can't fit inside a triangle does not mean it's bigger. It may simply be of "incompatible shape" but the same size.
@@KohuGaly I can answer that for you. When one or both sets are finite, it's pretty straightforward to figure out which is bigger. The difficulty arises only when both are infinite. Particularly because-as you pointed out-depending on how you do the mapping, sometimes set A will seem bigger, and sometimes set B will seem bigger. So mathematicians define it this way: If there is at least one way in which the two sets can be put into a 1-to-1 correspondence with each other, then they have the same cardinality (i.e. size). It doesn't matter if you can find other mappings where one set appears bigger than the other. In fact, you can always find alternative mappings where either of the sets can be made to appear bigger than the other. But, like I said, it doesn't matter. So long as a 1-to-1 mapping exists, the sets are of the same cardinality-other mappings notwithstanding. With Integers and Rationals, you can put them in a 1-to-1 mapping by following the technique discussed in the video. (The video paired Positive Integers to Positive Rationals, but you can easily extend the idea to include all Integers and all Rationals.) That's why they have the same cardinality. Sure, there are other mappings where the Rationals appear bigger, because, after all, the Integers are a proper subset of the Rationals. Strangely enough, you can even find mappings where the Integers appear bigger. But those don't matter, because a 1-to-1 correspondence exists. For the same reason, Odd Numbers and Integers have the same cardinality, though-again-Odd Numbers are a proper subset of the Integers. If no 1-to-1 correspondence exists, we say that the two sets have different cardinalities. Note that you have to _prove_ that a 1-to-1 correspondence is impossible. That's exactly what Cantor did-he _proved_ that Rationals and Reals _cannot_ be put into a 1-to-1 mapping with each other. So they are of different cardinalities. His proof showed that regardless of how you map rational numbers to real numbers, there will always be at least one real number left over that cannot be matched to rational numbers. So the cardinality of real numbers is greater than that of rational numbers. Let me give you my own analogy. Suppose we want to find if 25 is a perfect square. We examine a few potential roots: 3x3=9, 4x4=16, 6x6=36. So we found some numbers, which when squared, _don't_ give 25. Does that mean that 25 is not a perfect square? No, because there exists at least one number, 5, which when squared, does give 25. That's sufficient to say that 25 is a perfect square. The behavior of other numbers when squared doesn't matter. This is analogous to showing that Integers and Rationals _can_ be paired 1-to-1. What about 30, is that a perfect square? Well, 5x5=25, 7x7=49, 10x10=100. So 5, 7, and 10 are not square roots of 30. Does that prove that 30 is not a perfect square? No, that wasn't a satisfactory proof. Maybe an integer that we didn't examine could turn out to be the square root of 30. To prove that 30 is not a perfect square, we have to prove that no integer exists which, when squared, gives 30. (Can you prove that 30 is not a perfect square?) This is analogous to showing that Reals and Rationals can _never_ be paired 1-to-1.
but the number we are talking about is describable and its description is "The EXACT amount of time taken by that specific ball to touch the ground, when released from that particular bridge."
π = C/D For a unit circle we know D. It is 2R. We know π is transcendental, which means it is impossible to measure C to arbitrary precision. This leads to an interesting contradiction. Machinists turn objects on a lathe all the time, and we assume the object they end up with is "round". Assuming their lathes are "perfect" this must mean matter itself is not perfectly smooth. If that is true, then matter itself must be "quantized". We know the smallest unit of length is the Plank distance, and the smallest of time is the Plank time. Both are really small, but known sizes. I'm totally sure if you put a very "round" turned piece of brass under an electron microscope, you will see imperfections that the machinist can't measure with their micrometer. The best I've got is this means that mathematics does not 100% represent physical reality. It is a very good approximation, and a useful tool but physical objects cannot be forced to "conform" to the perfection of mathematics.
Some people time is quantizable, and that Planck time is the shortest unit of time. That means that the time it takes for the ball to fall has an exact value.
Love the ending of the video with indescribable numbers. However, a note of practicality besides infinite abstractions. If measuring the time a ball takes to fall using 38 decimal digits of precision, the tiniest hand position differences would actually matter hugely, of course also the exact time you start measuring vs. the exact time you release the ball, as well as the the exact time the ball hits the ground vs. the exact time you stop measuring time. Minor wind differences along the way would also have a huge impact, as well as minor expansion/contractions of bridge structure, maybe because of temperature differences, or even ground movements. Not controlling any of those variables already would make using 38 decimal digits of precision (in fact even much less) completely pointless. This is the realm of engineering for experiment design, not the realm of the abstract, but details like these matter, in fact hugely. So the starting context dropping the ball simply from your hands under the sun in a possibly windy day, while allegedly trying to measure with 38 decimal digits of precision, is completely off.
@ 1:35 "It makes it more accurate, but it will never be exact" *stops the timer at 12.6524....4375000* with 3 zeros at the end indicating that.. the 0's could indeed continue and the number *could* in fact be completely accurate.
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I hope you enjoyed the wild goose chase through Numberland. If you'd like to learn more about infinity, check out Brilliant's intro to infinity course brilliant.org/upandatom/
I did. I hate having to explain to friends that irrational doesn't mean stupid, but "without a ratio"
Yes, it's a very approachable video on a complex mathematical issue. I am passing it around. I really love mathematics, but it's sometimes difficult to explain this concept to people that are not into math but are curious about many things and will love to know.
"A very normal ball drop led me to infinity"
Literally every young man experiences this at a certain age.
@@donepearce "irrational doesn't mean stupid"
That irrational means illogical or unreasonable. It's not about numbers.
how do I get a two-way ticket to the "Numberland" :) ... how many dimensions are there, does anyone know 🤔
This number was described using language even before it was measured. It's the amount of time it took that ball to hit the ground when dropped from that bridge.
We could create a new set of numbers I would call the "useful numbers" which would be a countably finite set of numbers containing every number any human will ever need for any purpose. In that sense just describing or even thinking about a number would add it to that set, but that set would never be infinite.
Well if you have to measure it exactly you end up with uncountably infinite number of words to describe it considering the gravitational pull of all the objects with mass in universe along with quantum interactions and air resistance it experienced , so impossible....😂
In fact, wouldn't it be describable by relating the time (t) to the mass of the ball (m), acceleration due to gravity (g) and height of the bridge (h)?
… with a certain set of conditions that we could describe here in detail if we had an infinite amount of room but one of the conditions will be the time at which it was dropped so we’ll need to get the release time and then hope it’s an unfathomable miracle that the exact time of day she let go of the ball isn’t a transcendental, so here we go again…
nice try 😊
The physicist in me wants to say you only need 44 decimal places. But the mathematician in me really appreciated how you presented this. I really like your content :)
thanks :) there is a physicist and a mathematician in all of us and they struggle for power
Why 44 decimal places?
Now I wonder the likelyhood of it being an undescribable trancendental if restricted to 44 decimals instead of infinite. Is it now 0 instead of 100 percent?
@@backwashjoe7864 Because 5.39124760 * 10^-44 is the Planck time, the smallest meaningful timestep.
@@backwashjoe7864 A unit of Planck time is 5.39×10−44 seconds. It's the smallest unit of time that makes physical sense, at least according to current theories. There could be smaller segments of time, but we currently can't describe them through physics. That wasn't the point of this video though :)
Excellent abstract math, but for physical events like a ball drop, you have to consider the physics, especially "plank time" which is 10^-43 seconds. Plank time is "the length of time at which no smaller meaningful length can be validly measured". Since the drop time is a finite number of states, all of times can be enumerated.
better to study noncommutative time-frequency nonlocality a la Fields Medal math professor Alain Connes
exactly
@@AryaKohan study Basil J. Hiley
Planck.
This is by far the best video I have seen on the relationship between real, rational, irrational, algebraic and transcendental numbers. So well layed out and tidily expalined!
Agree
The only video I've seen. Early on in the video, my thought, if you can define two points, there are always infinite number of points between them no matter how close they are together.
@@ralphparker not in the physical world, where we have a Planck length
I recommend you check out the Numberphile videos on the same subject. They are so much fun:
- *Transcendental Numbers - Numberphile*
- *All the Numbers - Numberphile*
And the explanation is so beautifully done by Jade turning her head around in a breathtaking way!
Don’t worry! Heisenberg uncertainty principle says that the time it takes for the ball to drop has a little slop to it, so you can almost certainly find a rational number that could plausibly have described the “exact” time to drop for any physical definition of exact :)
I'd also argue that, given that we only have physics to describe a finite temporal resolution(thanks, Planck), you can one hundo find a rational number to describe the exact time, at least within our known physics.
The funny thing is, if you could measure time exactly, you would know nothing about the energy of the ball.
@ check out Alain Connes 2015 talk to physicists on noncommutativity as the origin of time and entropy. Fascinating stuff! See Professor Basil J. Hiley for followup.
"almost certainly" I would go so far as to say certainly. Simply by bisecting the interval, we can achieve any arbitrarily small precision,!
@@PunnamarajVinayakTejas My review of math professor Joseph Mazur's book "The Motion Paradox" - reissued under a different title. Professor Mazur does an expert job of giving the behind-the-scenes wrangling of conceptual philosophy which gave rise to applied science. What is the difference between time and motion exactly? If that question seems too abstract, this book proves the opposite.
Most college graduates assume that Zeno's paradoxes of motion were solved by calculus with its continuous functions. Mazur puts the calculus at the heart of the book, from Descartes and Cavalieri to Galileo, Newton and last but not least Mazur's favorite: Gabrielle-Emilie de Breteuil.
In fact, upon investigation, one finds many top scientists still studying and learning from the anomalies in infinite measurement. Regarding relativity Mazur states the wonder of absolute motion is that it "conspires with our measuring instruments to prevent any possibility of detection."
As Mazur points out "we don't measure with infinitesmial instruments" and so the perceptual illusion of time continuity remains despite the reliance of science on discrete symbols. With attempts at a unification of quantum mechanics and relativity Zeno's paradoxes reemerge with full-force in the "Calabi-Yau manifold." Mazur writes that the original concept of dimension still holds but now means measuring more by abstract reason than by sight.
Although each scientist featured by Mazur appears to have increasingly solved the paradox of motion in the end I think Zeno will be avenged and science will return to right back where it started. There seems to be a deadlocked struggle between discreteness (particle) and continuity (wave) in science and Mazur argues that indeed Nature "makes jumps" despite seeming continuous. But Mazur admits we are left with "splitting operations that can take place only in the mind."
One of my favourite math channel presenters. Infectious enthusiasm and clearly communicated.
Thanks for watching!
plus that fast turn to a back camera.
Jade is incomparable. 😃😍
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
@@ezrasteinberg2016 Comparable only to herself...which is one of those small sets she was describing, yes? :)
10:26 I love how she turns around and slowly but menacingly getting close to the camera
Right? She invaded my personal space bubble through the internet lol
Tell me about it. I watched that section on loop at 0.25x speed
Yeah. 'bummer' got me evry time
This is probably the best explanation of infinite sets I think I’ve ever seen. Thank you. I don’t think I grasped it quite as intuitively until now.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
This was really interesting, but I'm not sure, if I understood correctly. Are the following two statements correct? 1) "The natural numbers are able to assign a unique label to all algebraic numbers." 2) "A single transcendental number contains all natural numbers infinitely many times as intervals of its digits."
That last part where you explained how even language, when pushed to the extreme, is still a countable infinite and so cannot be used to accurately describe real numbers (and beyond) was really great!
Mathematically speaking, you are correct. Physically speaking, I believe that there is a finite smallest length as well as time, so that technically speaking, the amount of time required for the ball to drop is rational.
We don't know if there is a finite smallest length. We just know that there is a smallest (Planck) length that it would even theoretically be possible to measure.
@@John_Fx It would be correct to say that it would be the smallest length *physically possible* which would apply to matter and energy within our universe. Anything smaller than that wouldn't make sense from a Physics perspective, which would include things like dropping a ball. Therefore, the amount of time, as well as the distance that the ball dropped, would be rational.
There is a set of numbers between the agebratic nubers and the transcendent: the computable nunbers. These are numbers, that can be calculated to a given accuracy by a coumputer (or a touring machine).. In this set there are numbers like e and pi. This set is cpuntable infinit too, because the set of comtuter programms is countable infinit.
I thought uncomputable and indescribable numbers were the same?
The transcendental numbers pi and e can be calculated to any given accuracy by a Turing machine,. so they are computable.
@@GarryDumblowski Computable numbers are described by their Turing machine, so they are describable.
The converse is not true. I will describe a number that is not computable. Start by enumerating all Turing machines T_1, T_2, T_3,. ... Set a_i = 0 if T_i halts and set a_i = 1 if T_i does not halt. Now let alpha = sum_i a_i/2^i. The number alpha is describable, because I just described it, but it is not computable.
I recently rewatched Matt Parker's Numberphile video "all the numbers" where he talked about the computable numbers, and the normal numbers(numbers which contain any arbitrary string of numbers)
@@ronald3836how is that number not computeable? If you chose to enumerate the machines by alternating between on that does halt and one that does not then your answer is a geometric series that converges and can be computed to any arbitrary level of precision.
It's 0.0101010... in binary which is 1/3?
Flipping around like a super villain confronting James Bond was very entertaining.
The rule of two -- Darth Bane, Sith lord.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
"No Mr Bond, I expect you to count the transcendental numbers"
Jade, you are such a captivating educator. The point about the number of ways a number could possibly be described being countably infinite was particularly interesting to me. Great video!
Thank you for watching!
Keep it in your pants Moses... she has a boyfriend.
@@thomasp.crenshaw185 he doesnt care
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
@@thomasp.crenshaw185"What is a comment that says more about the speaker than the spoken to?"
this is the first time i am watching a video from this channel, and it blew me away! i did not expect such a crystal clear, nicely paced, logically flowing and informative video... absolutely gripping stuff, thoroughly enjoyed it.. thank you creator!
Infinity is a concept that you can make tons of videos about with completely different angles of approach.
Like, a lot of videos. There should be a number to describe that.
But you van make only a finite amount of videos
@@gabriellasso8808yes, but that just means there's a potentially infinite amount of videos you could still make on the subject 😝
@@sVieira151countably infinite number of videos*
1:30 Computer engineer here, going crazy, looking at what appears to be a common laptop, which has a cpu that runs on the order of few GHz, which means any digits below 9th are physically impossible and just some randomized garbage
I'm just spit-balling here -- don't know if this is actually workable, but . . . .
I think we are all comfortable with pretending that her set-up (minus the limitations of the laptop) is capable of perfect precision, right?
What if she has a measuring system with multiple known time delay lines. So for one event, we could get hundreds of measurements of the time. Those delays could be calibrated to fall within certain known fractions of a CPU cycle. By comparing the measurements at the different "beats" we could get precision to many more decimal places (I think).
So, I'm certain that is the setup that she used :-)
@@fewwiggle Dude was clicking stop by hand. I don't think the speed of the processor is the biggest source of inaccuracy in that measurement :D.
Calm yo' ass down, dude! You need to get high. Seriously. Try it.
Given that Planck units seem to be discreet, it's entirely possible that ever digit after the 44th might indeed be a 0
That was sort of like what I was thinking, Quantum Mechanics tells us that there are fundamental limits to how precise any physical property can be defined, thus there will ALWAYS be a rational value that uniquely specifies a given possible value from all other possible values. Mathematics might have infintes and infintesimals, but the Physical Universe doesn't
@@RichardDamonThat proves you can’t have a perfect circle. Right?
@@RichardDamon " thus there will ALWAYS be a rational value that uniquely specifies a given possible value from all other possible values."
Nope, not even close.
All it means is that there is no 1 true value as it is always a range of possibilities. But for timescales that can still be shorter than a planck-time, it is just fundamentally impossible to measure.
@@ABaumstumpf No, it isn't a "Measurement" phenomenon, it is that time is actually indeterminate at that scale, so time doesn't exist finer than that. You could say that the idea of a "Precise Time" doesn't exist. Just as no integer exists between 1 and 2, no time exists between one time quanta and the next.
@@RichardDamon "it is that time is actually indeterminate at that scale, so time doesn't exist finer than that."
Nice claim, but again - that is not what is says.
Also "No, it isn't a "Measurement" phenomenon"
I never claimed that - so why the strawmen?
"Just as no integer exists between 1 and 2, no time exists between one time quanta and the next."
Which is 100% wrong.
I never get tired of listening to Jade explain bijections and the cardinality of sets. And I never will.
Injective is dual to surjective synthesizes bijective or isomorphism (duality).
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Feels like finally i understood countably infinite and uncountably infinite sets. Thankyou for making such videos.
I love that final shot, with a bit of jazz at the end. Who hasn't closed a laptop with that feeling.
YEY! Finally a video that's not one of those terrible shorts.
We like normal videos. ☺
EDIT: Let me clarify that by "terrible" I mean the format not the content. 🙃
If I wouldn't like Jade's videos I wouldn't watch and subscribe to the chanel. I just don't like the shorts format and I always skip those.
😂
@@upandatomYour shorts are good, too, just less exciting to open the app and find! Definitely *not* terrible.
@@CDCI3Yeah terrible is a strong word. They're nice. This is just better
There is an argument to be made that the existence of the planck measurements (time, length, etc.) proves that irrational numbers do not exist in the real world. The planck measurements are the smallest meaningful measurements in our current framework of physics, and since everything is made up from integer multiples of these, you can't actually have an irrational distance in the real world.
Edit: This would only be true under the condition that the planck measurements are actual limits of time and space, and not simply limits of the ability to measure them. This is still an unsolved problem however.
energy seems infinitely divisible because the plank energy is very high
I thought Planck units were just what can be, in theory, meaningfully measured. As in, it doesn't mean there's nothing smaller, it's just that we would never be able to measure it.
Do numbers in general exist?
@@tomshieff You're right, it is currently unknown if the planck measurements are actually real or just limits of measurement. They can be calculated using the uncertainty principle and our current understanding of gravity (relativity), so until someone finds an accurate model of quantum gravity, no one knows if the planck measurements are actually real.
The limits that define the Planck units are fundamental, but it's definitely not known that everything is integer multiples of them.
This is similar to the idea behind loop quantum gravity, which experiments have found no evidence of (and good, albeit not 100% definitive, evidence against).
Fascinating video! The end bit about counting descriptions got me thinking about cellular automata and language modeling, which I've never cut through before. It's really great to see, think about, experience, how branches of knowledge coalesce. Thanks, Jade!
Synthetic a priori knowledge -- Immanuel Kant.
Knowledge is dual according to Immanuel Kant.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Assuming zero delay in starting and stopping a timer. You could use a numberless analogue stopwatch to measure the drop of your ball. The arc travelled by the stopwatch pointer is precisely how long it took your ball to drop.
For the measurement to remain precise, it can not be described using numbers because numbers introduce divisions.
I've always been dissatisfied with the explanations of transcendental numbers, but this is the best so far. Thank you Jade. Fun to think about the time it takes for the ball to fall being an actual value but never being able to write it down.
Hat's off on the needle in the haystack analogy! I really liked that. Great episode overall!
Great video! It's like we're tiny little babies in the universe, understanding nothing around us.
Penso que não é bem assim, entendemos muitas coisas, porém somos limitados e essa limitação não se dá basicamente em nossa capacidade mental, ela se define mais pelo nosso tamanho em proporção ao universo ao redor, não entendo que haja algo que a mente humana esteja impossibilitada de entender para sempre, mas certamente há muito que não podemos alcançar, medir, ver etc, ou seja, coisas que são necessárias para que possamos chegar no entendimento. Não é uma questão de inteligência, mas de nossas limitações em relação ao contexto onde estamos inseridos. Pior que, uma das coisas que entendemos, é que tinha que ser assim, pelo menos nesse universo nessa vida, não haveria como ser diferente.
we have invented maths, but cant understand everything in maths.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Nice to see you again. Your content is so well presented and comprehensible. So happy to be a “patron”.
Glad to see this comment. It reminded me to become a Patreon
are you not really a patron?
I really liked how you aligned the things we must know (like the definition of countable infinity and transcendental numbers etc.) in an ordered track.
These definitions feel like a schoolyard argument.
"Ofc i can count your set! I have infinite numbers to count with!"
"Nu-uh! My set is an uncountable infinity! I win!"
You described that infinitesimal just fine tho.. it's the gravitational distance in time from the top of that bridge to the earth plus or minus human and computational error!
The problem is you haven't described it precisely enough to make it reproducible.
The ratio of the diameter of a circle to its circumference is reproducible because it's based on theoretically perfect geometric objects.
If you were to reproduce her experiment, the probability of measuring the exact same number is essentially zero because there's no way for you to perfectly replicate the experiment to infinite precision. All of the subtle forces of gravity from the Earth, Sun, Moon, etc... would all be different and would result in a different measurement.
So for you to describe her number, your description would also have to specify the exact initial conditions of her experiment including for example the starting height, time it was dropped, etc.., but those numbers when measured to infinite precision are ALSO likely to be indescribable.
So there actually isn't any way to perfectly describe her experiment to reproduce her exact number.
The way you described it doesn't specify her specific number, it specifies an infinite set of numbers.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
_Max Planck has entered the chat_
😂
Once your accuracy reaches a number around a Planck time you achieved the max accuracy possible.
Which means it actually is rational. Though I guess then the question becomes, what exactly counts as the start and the end. At the level of Planck times, the concept of "the moment it touches the floor" probably isn't so clear cut.
@1vader I think that the instant that either the acceleration is first zero, or the velocity is first zero after release, would be two good candidates for defining "touching the floor". It depends on whether you want to define "touching the floor" to be defined as when it exerts enough force to have altered the object's velocity, or to define it as when the objectsvdownward velocity is cancelled. But then again, air resistance would affect those two points, so maybe it needs to be specified as occuring in a vacuum? 🤷🏻♂️
@@AMcAFaves That's still thinking on a way too macro level. Not all of the atoms of the ball will come to a stop at the same time. And at the scale of plank times, we could differentiate stuff on an even smaller scale where the particles might not even have something like a defined location. Actually, because of Heisenberg's uncertainty principle, I guess it wouldn't even be possible to know.
@@1vader Good point. I suppose the closest we could come would only be some sort of statistical model of the atoms in the object.
@@1vaderBrilliant.... Our perception of touching holding seeing are actually averaged out of infinitessimely smaller events we can't perceive
The funniest, the most pleasant, kind and warm person that makes a very arid topic/domain which is math to be likeable by people that usually are away of math (NOT mt case...). Anyway: it is my youtube favorite channel, it does not matter what type, above everything. Pur and simple I adore the way she is and how much she enjoys what she does and of course I like her. When I want to have a nice mood, I watch one of her math videos which always give deep knowledge expressed simply, funny, enjoyable, so that many can get the point...
as an educator, I learned that its not the teaching method (which yours is great, BTW), its the emotions and enthusiasm of the educator that really inspires students. And in this department - you are 100%
This is by far your funniest video so far. I love your brand of humor; this is my flavor of dork.
The ending had me laughing out loud!
😂
If you choose the right unit of measure, the ball-drop-number can be algebraic or rational or a natural number or even just 1. It might be hard to choose the right unit, though.
Sure, you could declare that the ball takes exactly 1 Bleep to reach the ground. But would that be usefull ?
As soon as you want to translate your Bleep unit into another unit like the second, you'll be back to square one.
Unless you decide your Bleep unit is the new standard, and any time measurement must be expressed in Bleeps. However if your goal is to avoid dealing with strange numbers, then you must create a new unit fit for any new measurement you make, and forget about comparing results between experiments. Maybe that's not such a good idea after all. :p
@@theslay66 - Did you fail to understand my last sentence? I tried to make it succinct so the reader would have a little "aha" moment. Was it effective for you?
@@rubiks6 Your last sentence doesn't solve the problem in any way.
Rational is dual to irrational -- numbers.
Rational, analytic (noumenal) is dual to empirical, synthetic (phenomenal) -- Immanuel Kant.
Subgroups are dual to subfields -- the Galois Correspondence.
Homology (convergence, syntropy) is dual to co-homology (divergence, entropy).
The integers are self dual as they are their own conjugates -- the conjugate root theorem.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron.
"Always two there are" -- Yoda.
Concept are dual to percepts" -- the mind duality of Immanuel Kant.
Mathematicians create new concepts from their perceptions or observations all the time -- conceptualization is a syntropic process -- teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
@@hyperduality2838 - Okeydoke.
What a beautifully deliveredl lesson. I feel there were several times where it could have become very confusing. You kept everything crystal clear. Thank you very much.
😊
Your palpable despondence at the end of the video was brilliant!
You are such a lovely science communicator! Good work. High school teachers have now new support material for their classes.
2:55 Reminds me of doing drawing octagons in Paint. Trying to make the sides equally long, having the straight sides 14 pixels long would make the skewed sides about 10 pixels (14 / √2). As I can only draw whole pixels, at that scale it's as precise as it gets. Making a pattern of nested octagons, it doesn't scale well while drawing larger octagons around the first one and trying to keep everything nice and centered. So this vid tells me/confirms that, even if I start with an octagon with straight sides of 14,000,000,000 pixels or even larger, I can only approach the length of the skewed sides and scaling up or down from that will never work out perfectly.
How did you type the square root symbol? Was it on mobile?
This was an awesome video! Im glad i found your channel. You related all of these concepts together and answered some questions ive had for a while now. Dont stop making videos!
Thank you for watching and supporting :)
Just a pleasure as always to watch all your exuberant content on anything that captures your attention, Jade! Your enthusiasm and passion for learning are just but more than contagious! It's been my immense pleasure and honor to have been your audience for almost more than three years now, and it is with utmost admiration that I admit that I am very proud to sponsor your content to all my friends and family members, and bug them constantly with all my ramblings about science and its wonders in our Nature :)
I hope you keep spreading your contagious energies here forever, and I am thrilled to be a member of our little community here!
Best,
Sai
Dear Jade, great video. I love how passionate you are about describing mathematical and physical nuances.
As for the time measurement description (or any other measurement), there is a handy tool - measurement uncertainty (and the way to express it and measurement results). It would be great to watch you explaining probability issues connected to them.
Great explanation on infinities and types of real numbers. My only slight nitpick, based on the title I thought this was going to have something to do with the fundamental properties of time… like I was like “yeah, can we measure time? I feel like all we can really do is find very regular sources of a periodic signal and call that time… let’s learn more on this interesting topic”… but the concept of this video really applies to any exact physical measurement, not just time (which as you correctly alluded to, the question of what constitutes a measurement and how to minimize error is itself a endlessly complex subject).
Anyways, still really enjoyed the great explanations, and loved your personality. Thanks for sharing!
Your way to present everything in very ascending order is really ahhh!
I was all thinking, put in an accelerometer and measure it electronically... but then I got an amazing math lesson instead! Great video! :)
At the beginning, I thought you were going to talk about using continued fractions to find the fraction of a decimal. Great video though, as always!
Me too!
At one moment I thought the Stern-Brocot Tree would make an appearance… a clean way to show every positive rational number (well, by clean I mean there are no duplicates)… and you can do a binary search on it with your irrational number and each further depth on your search leads you to a better approximation, already in reduced form.
i learned more in this video than in my 3 years of high school.
my math teachers never talked about real numbers this deep (maybe because its not required in exams)
I understand you were trying to explain this in the most simple, easy to follow way possible, and yet! What a mistery.
The Infinite Precision Dilemma is the perfect name for this…glad you found a good way to succinctly describe it!
Thanks for helping!
I see a video from jade. Immediately gotta click on it. You're my favorite educational content creator. Please never stop 😘
thank you so much!
@@upandatom Simply wonderful - having you been taking drama courses, too?
We are all Jade. 👏
It feels interesting to rediscover this channel after having seen her explain knot theory related to the painting on Tom Scott's channel over 4 years ago.
This was a great video and it seems like there is a whole lot of material/videos I should catch up on here.
There's no reason to prefer cardinality over ordinality for measuring size. Indeed, I would say ordinality is far more accurate. When you do that you see the rationals are roughly omega^2, whereas the integers are roughly omega. As for Cantor's "proof" of the uncountability of the reals, that turns out to be wrong. I published a proof that his "proof" was wrong on RUclips a while back, but it had an error in it. Shortly thereafter I fixed my proof, but I have not republished yet due to lack of time. It's fairly simple to show that Cantor's idea of "cardinality" is flawed and only ordinallity exists as a measure of size (or order). The basic idea is to establish there must be an ordinal number of digits for real numbers. We can then show that no matter what ordinal number of digits you choose, when you try to make a list it will always be longer than wide. This shows the diagonal does not cross all elements of the list, and subsequently, Cantor's proof falls.
An uncomputable number is Chaitin’s constant, or more precisely, family of constants. It is related to the probability that some Turing machine will halt, something that there exists no Turing machine for determining. Another one is constructed from a base-2 representation of a number between 0 and 1, where each trailing digit gets its value from whether or not a Turing machine associated with it will halt; each digit has its own Turing machine.
But... we can describe that number, can't we? That number is equal to the number generated on a computer by a man attempting to measure the time it takes for a ball dropped by Jade from a bridge of X height plus the height from the bridge to Jade's hands, with the ball falling at the rate of acceleration 32.17 ft/s^2, and your man with a reaction time of Y to press the start and stop button. I don't know what number that is exactly but isn't whatever that number is equal to the description above?
How precisely can you define that velocity you mentioned however? You will run in the same problem eventually, since you are just shifting on what part of the equation of the event needs to be exactly defined.
You will also run into the whole issue of uncertainty at a certain level of precision where it becomes involved in quantum dynamics and that is just opening another can of worms outside of just the mathematical principles of it.
But wouldn't that be describing an approximation of the number? Like, if you try to reproduce the number based in this description, you would get very similar results, sure, but you can't guarantee the same number to pop up? I'm not a mathematician tho, so idk
I have a similar question. The problem only arises because of the units you select. You could just say, the time it took is 1 Ballbridge, which is the amount of time it takes for this ball to fall from this bridge under these conditions. Then the problematic number becomes how many seconds there are in a Ballbridge, which you can just describe as "the number of seconds in a Ballbridge." What is the number you can't do this for? If you talk about it, you've described it.
Each of the other numbers you mention (height) has the same problem of being hard (impossible) to know exactly. Theoretically. In a physical world made up of quantum elements, "infinitely exact" isn't a meaningful concept.
In theory, technically, the number you get by the result of a given experimental process can be described as "the result of (description of the experimental process)", that that description does not actually tell us anything meaningful - the whole point of describing a number that is the answer to some question is to be able to understand that answer in some greater, more meaningful context.
There is also the problem that a given description may not actually describe a specific number, or may require as part of its description other 'merely describable' numbers - for example, your suggestion involves things like the acceleration due to gravity at Earth's surface (itself only an approximation, and a number which varies from moment to moment, with latitude and longitude, and with height above the surface), and also involves descriptions of individuals who are not constant with time, performing actions with initial conditions that are not specifiable with exactness except to describe them 'as what they are' in the same ultimately unhelpful way.
Which is all to say that, in a very technical sense, the number is technically describable as the result of a very specific description of events which happened, described unambiguously (though only descriptively) in their time and location, the actual value of that number becomes utterly inaccessible - Describable numbers need not be computable.
We then reach an interesting philosophical possibility - we are assuming that all mathematically possible numbers are physically possible. We know that, for example, pi (square units) is the area of a circle that is one unit in radius... but *can we physically make a circle*? Does it physically matter if we cannot exactly express pi if we cannot *make* pi in the physical world? After all, there is some evidence that space is, in a sense, 'pixelated', quantized with some minimum possible meaningful distance, meaning any attempt at any physical circle will only be 'approximately' a circle. Likewise, even with something as simple as sqrt(2), can we say with certainty that a triangle with unit legs can physically exist to the point that its hypotenuse is an irrational value, given that the length of that hypotenuse must be some whole number multiple of that minimum possible length (so must therefore be rational).
Perhaps it is not so surprising that we cannot meaningfully describe most of the numbers that can mathematically exist, since, even as big as the observable universe is, it is finite, and thus can only contain a finite number of combinations of things... If the entire universe, and all 10^80 particles it contained were a mechanism for storing binary data, it could only represent 2^(10^80) different unique states - a very very large number, but still finite.
This was brilliant. Accessible yet profound. Like an episode of James Burke's Connections. I'm convinced that Jade is among the world's best STEM communicators.
But . . . did you get your ball back?
Haha no we didn’t 🥲
Ironically, My daughter who is getting a math minor with her bio degree and I were just talking about the sizes of infinity the other day related to the infinite hotel experiment. Because I am that nerdy. When I saw your video I had to check it out. I have to say you brilliantly explained this concept so well. I remember in engineering school in Calculus, that we just got 'infinity' but never considered different types of infinity. I never thought about it until years later. Thank you!
Thank you for watching! Maybe you can show it to your daughter :)
@@upandatom I forwarded the link to my daughter. As a math junkie I know she will love it.
Well, for calculus, you don't need the different types of infinity. All you need to know is if a quantity is bounded or grows unbounded. That's true for science, engineering, and many other fields of math as well. Even within math, the different infinities arise only when you get deep into Axiomatic Set Theory. Which-however fascinating it is in its own right-isn't useful in science and engineering.
After all, human knowledge is so vast, it's impossible to learn all of it in one lifetime, let alone a mere 4 years of college. If you want to graduate in a reasonable amount of time-so that you can leave academia, come out into the real world, solve real-world problems, and earn a living-it shouldn't come as a surprise that you were taught very specific subjects that were relevant to your major, while vastly more subjects were left untaught. Luckily, we all have the option to learn more if we are so inclined.
@@nHansWell said. As I approach retirement I still learn new things every single day. Keep my brain active. I'm too nerdy to do otherwise :)
I know you probably know this already (I just like talking about math), but there is a way to write 1/3 in (repeating) decimal notation: 1/3 = 0.(3) = \lim_{x \to \infty} \sum_{i=0}^{x} {3 \over {10^i}}. (And, as a quick random mention, it has some funky properties like 0.(9) = 1)
There is a Planck constant that limits the accuracy of distance measurments, when you measure the time - you measure the distance too, as each object have some size. So the exatct time will be limited by Planck constant. The minimum amount of time that it is possible to measure - is when a photon passes a one Planck distance. And you can mesure time relative to the number of Planck distances that photon passes at the same period. In this units time measurment will always be an integer.
A lot of RUclipsrs have been doing videos on infinity and this one is on my list of favorites!!! Great job!
Thank you! Yes it’s a fascinating concept :)
@@upandatom it really is!
I am quite glad that this lady has gotten to fulfill her lifelong ambition of dropping a soccer ball off a very tall bridge! She did quite a good job of at least trying to explain something that cannot be made sense of, at least at this time. The number of unexplainable in our universe is, quite literally, infinite.
I appreciate the changeing camera angles, you turning swiftly & most importantly, slowly walking towards the the camera instead zooming with it.
Time is a measurement of motion. Just like inches are a measurement of length. Measuring time with a clock is like measuring inches with a ruler. How precise do you want to be? Currently, clocks measure length down to the frequency/ wavelength of the microwave beam. What happens when you boost the frequency up to the X-ray/Gamma ray frequencies? You shorten the wavelength and the distance the photon has to travel to complete a cycle. Precision goes from 10 to 10th to 10 to 20th/25th.
1:32 adding more digits increases precision. You made it more accurate when you eliminated the errors from reaction time and processing time etc
Certainly you can eliminate the mortal reaction factor. They do it all the time with sports competition, etc. For example, this thing could be based on a computer and cameras.
Isn't time considered (according to our current knowledge) quantized with the plank-time (the time in which light travels one plank-length) being the shortest amount of time being able to pass?
Hey this is great as a maths tutorial but the premise may not be true here due to some underlying physics.
The smallest distance that has any physical meaning is the planck length, and the smallest grain of time that has any meaning is the length of time causality (speed of light c) takes to travel the minimum meaningful distance.
This means we can likely break time down into finite, or quantised, grains. Using this minimum length of time as a unit, we can then count how long the ball takes to reach the ground.
This of course depends on your interpretation of physics and whether you think plancks length is truly the smallest distance and granular, but in this case it would be genuinely possible :)
Wonderful video, really well edited and presented. Very clear and fun way of explaining mathematical traits of numbers.
for measuring length or time it's sufficient to use a natural number, as they are always multiplies of the Planck length and Planck time. I thought you are going to take it to a different problem: the gravitation of the start point is slightly less than the end point down there, so the clocks run at a different speed. Of course, the own clock of the ball runs at yet another speed. And it is not clear what means to measure the time difference between two distant events at the points at which clocks cannot be synchronized
I had this exact realization during a measure theory course recently. We were discussing something about how you can approximate any measurable function within epsilon with a continuous function and something in my brain clicked and realized that the real numbers are insanely larger than imaginable. I tried to explain this to my friend, who is an engineer, and he didn't understand it at all even after 10 or 15 minutes of trying to explain that the rational numbers are basically as good as we're ever going to get. Glad to have a video I can send to people when I need to explain this idea.
8:39 Considering the smallest length of time we can measure is the Planck time (~10^-43 seconds), the most decimals a time measurement could have is 43 digits, i.e. it would be a rational number: some variable 't' (the time it took in seconds from the moment you let go to the moment the ball hit the ground) over 10 tredecillions (1 tredecillion = 10^42).
Can't this number being expressed as "the amount of time, in seconds for this ball to fall from this bridge on this very special moment in time"? (taking into account the ball, the space-time coordinates, the speed of wind, air pressure, heat, etc)
The other problem with infinite precision is that if you have two perfectly precise measurement "observers" detecting precisely when the ball hits the pavement, they will almost certainly disagree on the exact instant the ball makes precise "quantum contact"* with the ground due to the Uncertainty Principle.
_*I'm not sure if that's even possible, but let's assume we can divine what "contact" means at the quantum level._
As a rather unimaginative engineer in everyday life, I sometimes like to suspend my disbelief and get drawn into mathematical flights of fantasy such as:
• A stopwatch that can measure fractions of a second accurate to 38 decimals ... that works on a 2023 laptop. Which is what-9 GHz with overclocking? (In 1e-38 seconds, even light travels only 4e-30 meters.)
• A human whose reactions are that fast.
• Rubber bands that are infinitely long and infinitely stretchy.
i consider myself an imaginative person yet i struggle to imagine a liquid nitrogen cooling system in a stock-looking macbook outside with no visible condensation
There's an issue with the conclusion from a physics stand point. The number can be described by comparing the position of every single atom in the universe when the ball is dropped to the position of every single atom in the universe when the ball lands, and there is a finite amount of atoms in the universe. By doing that we isolate only the movement that occurred on the time axis.
You can describe it with physics. Solve s=ut+1/2at^2 for t. That only works in a vacuum but you could extend it with terms for air resistance. Though they'd be very complex and depend on other difficult-to-define numbers like the air temperature, the height of the bridge, the size and surface texture of the ball etc.
As an engineer I do find it hard to see the benefit of defining accurately a value of which 34 of the 38 digits are purely fiction and could just as well be 0000 with no loss in usefulness. But I did enjoy the explanation, especially the distinction between different infinities. And, tbh, your presentation of it which was excellent :).
Thank you for the highly pleasurable mental stimulation.
How is this channel not getting way more subscribers? I thought I finished all science vulgarisation RUclips videos and found this one after looking for a specific subject. Why did RUclips never suggested it to me?
The bright side of indescribable numbers is that we don't really have to care that they exist. Any number we can possibly encounter in any context is by definition describable, because at bare minimum a number can be described by specifying the context in which it was encountered.
Legend says the ball is still falling.
Hello Jade. I watch all of your videos and love your approach to teaching and applying various mathematical principles. I never comment but today at the end of this video I laughed out loud at the ending when you stared at the laptop sadly realizing your number will never be able to be calculated, before slowly closing it. Moments like this are why I love watching your videos. Keep up the fantastic work making us smile and brightening our minds.
thank you so much :)
Yeah, that got me too (to laugh out gently). ;-)
So nice.
Thanks!( to Jade) and to you, putting into words what I thought to say as well.
You can measure time exactly using planck time, which is tied into the planck length. Planck time is the time it takes for light to cross one planck length. So in theory, this is as small as you can physically go.
Perhaps one of the biggest problems with trying to find how long it takes for the ball to drop also requires that you have a frame of reference. Are you observing the bank drop and timing from the bridge? From the ground? Are you the ball, timing how long you fall. All three are distorted by their own frames of references and immeasurable beyond some accuracy.
Very captivating! From the title to indescribable numbers. Fantastic! I am a mathematician and I had never thought of indescribable numbers as such, but cool concept.
very well done. best explanation of the various kinds numbers I've seen so far.
I enjoyed that a lot! A fun and approachable manner of describing what is a fundamental (but surprisingly complex) part of mathematics! Well done!
The amount of time it takes for that instance to occur is subjectively infinite. Or simply, 1. The event is singular. It only gets complicated when you compare it against the rest of reality. Thus a fraction.
It may seem impossible to measure your number, but there's one factor that could force it to be rational. If time is fluid, then your number is almost certainly transcendental, but if time can be broken into absolute units, the same way matter is broken up into electrons, quarks, neutrons, etc, then it must be rational. And if you consider time the 4th dimension, then it can almost certainly be broken up the same way the first 3 dimensions can.
A great video! It explained all the concepts well in a way I could understand and stimulated my curiousity and wonder.
Although I was a bit worried at 10:50 that you were going to come even closer and break my screen! 😅
😂
By the way, time is discrete. The smallest time is about 10^-44s (Planck's time). You can not divide further below that, not in this universe anyway.
At 16:38 the list shows, as #10, "the first boring number." Which, of course, is interesting, and therefore not boring, and not the first boring number. A well-known paradox, thrown in (I assume) for fun, along with #11, "the answer to the ultimate question of life, the universe, and everything" (42 according to Douglas Adams).
We can subdivide descriptions further. If a number can be calculated to arbitrary precision with a finite length of run of a Turing machine, then it is computable. A Turing machine is an abstraction of computation that makes precise the notion of a finite-sized algorithm, and there are countably many of them. Computable numbers include familiar transcendental numbers like e and pi, but since there is a countable number of them, there are thus infinitely more uncomputable real numbers.
Congratulations, fellow viewer, if this video raised more questions than answered doubts. That's what a good educational video is supposed to do! I'd like to suggest a follow-up video that answers these questions:
- So if there is a countable infinity and an uncountable infinity, are there other types of infinities as well? How many? Is there an infinity smaller than countable infinity? Aren't odd numbers, even numbers, and prime numbers smaller than countable infinity? Are there infinities between countable and uncountable? If there are bigger infinities than uncountable, what are they? Complex numbers? Points in 3-D, 4-D, ..., n-D, ..., infinite dimension space? Can we build larger infinities from smaller infinities, like how Dedekind built real numbers from rational numbers?
Actually, there's even better basic question: Does the inability to form pairing with integers implies that the set is bigger?
Consider the rationals again. Whether you construct a 1-to-1 mapping with naturals depends on which order you choose to count them. Some of the infinite possible permutations of rationals map to naturals (like the diagonal case) and some don't (like the row-by-row case). It is entirely possible that the set of reals is no bigger than set of naturals and all if its permutations are not 1-to-1 mappings to naturals.
As a very rough imperfect analogy, just because a cube can't fit inside a triangle does not mean it's bigger. It may simply be of "incompatible shape" but the same size.
@@KohuGaly I can answer that for you. When one or both sets are finite, it's pretty straightforward to figure out which is bigger. The difficulty arises only when both are infinite. Particularly because-as you pointed out-depending on how you do the mapping, sometimes set A will seem bigger, and sometimes set B will seem bigger.
So mathematicians define it this way: If there is at least one way in which the two sets can be put into a 1-to-1 correspondence with each other, then they have the same cardinality (i.e. size). It doesn't matter if you can find other mappings where one set appears bigger than the other. In fact, you can always find alternative mappings where either of the sets can be made to appear bigger than the other. But, like I said, it doesn't matter. So long as a 1-to-1 mapping exists, the sets are of the same cardinality-other mappings notwithstanding.
With Integers and Rationals, you can put them in a 1-to-1 mapping by following the technique discussed in the video. (The video paired Positive Integers to Positive Rationals, but you can easily extend the idea to include all Integers and all Rationals.) That's why they have the same cardinality. Sure, there are other mappings where the Rationals appear bigger, because, after all, the Integers are a proper subset of the Rationals. Strangely enough, you can even find mappings where the Integers appear bigger. But those don't matter, because a 1-to-1 correspondence exists.
For the same reason, Odd Numbers and Integers have the same cardinality, though-again-Odd Numbers are a proper subset of the Integers.
If no 1-to-1 correspondence exists, we say that the two sets have different cardinalities. Note that you have to _prove_ that a 1-to-1 correspondence is impossible. That's exactly what Cantor did-he _proved_ that Rationals and Reals _cannot_ be put into a 1-to-1 mapping with each other. So they are of different cardinalities. His proof showed that regardless of how you map rational numbers to real numbers, there will always be at least one real number left over that cannot be matched to rational numbers. So the cardinality of real numbers is greater than that of rational numbers.
Let me give you my own analogy. Suppose we want to find if 25 is a perfect square. We examine a few potential roots: 3x3=9, 4x4=16, 6x6=36. So we found some numbers, which when squared, _don't_ give 25. Does that mean that 25 is not a perfect square? No, because there exists at least one number, 5, which when squared, does give 25. That's sufficient to say that 25 is a perfect square. The behavior of other numbers when squared doesn't matter. This is analogous to showing that Integers and Rationals _can_ be paired 1-to-1.
What about 30, is that a perfect square? Well, 5x5=25, 7x7=49, 10x10=100. So 5, 7, and 10 are not square roots of 30. Does that prove that 30 is not a perfect square? No, that wasn't a satisfactory proof. Maybe an integer that we didn't examine could turn out to be the square root of 30. To prove that 30 is not a perfect square, we have to prove that no integer exists which, when squared, gives 30. (Can you prove that 30 is not a perfect square?) This is analogous to showing that Reals and Rationals can _never_ be paired 1-to-1.
but the number we are talking about is describable and its description is "The EXACT amount of time taken by that specific ball to touch the ground, when released from that particular bridge."
One down. Uncountably many to go! Better get cracking!
π = C/D For a unit circle we know D. It is 2R. We know π is transcendental, which means it is impossible to measure C to arbitrary precision. This leads to an interesting contradiction. Machinists turn objects on a lathe all the time, and we assume the object they end up with is "round". Assuming their lathes are "perfect" this must mean matter itself is not perfectly smooth. If that is true, then matter itself must be "quantized". We know the smallest unit of length is the Plank distance, and the smallest of time is the Plank time. Both are really small, but known sizes. I'm totally sure if you put a very "round" turned piece of brass under an electron microscope, you will see imperfections that the machinist can't measure with their micrometer. The best I've got is this means that mathematics does not 100% represent physical reality. It is a very good approximation, and a useful tool but physical objects cannot be forced to "conform" to the perfection of mathematics.
Matter isn't quantized. Angular momentum and charges are.
Some people time is quantizable, and that Planck time is the shortest unit of time. That means that the time it takes for the ball to fall has an exact value.
Love the ending of the video with indescribable numbers. However, a note of practicality besides infinite abstractions. If measuring the time a ball takes to fall using 38 decimal digits of precision, the tiniest hand position differences would actually matter hugely, of course also the exact time you start measuring vs. the exact time you release the ball, as well as the the exact time the ball hits the ground vs. the exact time you stop measuring time. Minor wind differences along the way would also have a huge impact, as well as minor expansion/contractions of bridge structure, maybe because of temperature differences, or even ground movements. Not controlling any of those variables already would make using 38 decimal digits of precision (in fact even much less) completely pointless. This is the realm of engineering for experiment design, not the realm of the abstract, but details like these matter, in fact hugely. So the starting context dropping the ball simply from your hands under the sun in a possibly windy day, while allegedly trying to measure with 38 decimal digits of precision, is completely off.
@ 1:35 "It makes it more accurate, but it will never be exact" *stops the timer at 12.6524....4375000* with 3 zeros at the end indicating that.. the 0's could indeed continue and the number *could* in fact be completely accurate.