I first started watching Infinite Series when I was a math undergrad. I've since finished my doctorate (Math: Numerical Analysis) and been working at a research institute for over a year. So glad to see you still making math youtube videos! So funny, this would have been a mind blow to me back then and now I'm watching it because I'm just bored after work on a Tuesday and don't want to think super hard. Thanks!
You have no idea how much we loved Infinite Series! Years later, I still think about it and how great the three of you were. Please keep making more videos!
omg how did I not find this channel sooner?! I used to watch Infinite Series (and some other PBS channels) and have missed your style of teaching! Glad you're still around!
Her and PBS infinite series is an influential part of why i became a math major in the first place. I cant believe theres been 2 years of videos and the algorithm didnt let me know!
Great Numberphile video about that calls them "uncomputable" numbers (saying that the computable transcendentals like pi are still 0% of all numbers): ruclips.net/video/5TkIe60y2GI/видео.html
I think it's a simple fact, owing to the fact that numbers which are roots of polynomials are countable (because rational polynomials are counrable). In that, it is not very dissimilar to or surprising than 100% being irrational. @@odouroushouseant I stopped watching numberphile since they claimed 1+2+...=-1/12, without contextualizing that it's the analytical continuation of Zeta at -1. I thought it is an interview of some research assistants by people who have not much idea about the topics they are asking, and the whole thing is not serious and can even be very dangerous to people who want to learn it.
@@jonathandawson3091 Mathologer did a rebuttal video to the numberphile video. For the last couple years, there have been a few more vids on that series, and one of them that I watched a bit gives some interesting rationale for it. There is a lot of spurious math out there. The example that bothers me the most is the so-called Ross-Littlewood paradox. Matt Parker from standup maths channel jumps on that wagon. So does that young lady, Up and atom. The wikipedia article on it has it wrong. It is bewildering to me why mathematicians have got this wrong for 70 some years, since it was originally described in 1953. What is it about the problem that mathematicians just don't seem to comprehend? The conclusions given in the wiki article are just plain fallacious, along with the general understanding of it in the mathematical community. It is NO paradox at all, and that can be shown with simple mathematics.
I never thought I'd miss videos I, more often than not, don't fully understand (I never studied mathematics above high school concepts). Yet here I am and glad you posted again!
Hahaha, wow that's awesome! I remember that Kelsey's stuff helped me get into the quantum conouting scene my freshman year, but the analysis & set theory stuff she had made was what ultimately wound up leading me down the trail of math lol
Hi Kelsey, it's great to see another video come through by you! I remember I started watching you in high school, around when I had taken calc 1 & 2 from a small community college Your role back with PBS Infinite Series was one of the key inspirations for me to seek out higher education in mathematics, and in a couple weeks I'll be graduating with my MSc in mathematics! Your work was hugely influential in my pursuits, so thank you for continuing on the good work still today :)
This was so well produced, nice job. I can imagine how difficult it must be to balance the scale of rigor vs. simplicity in explaining higher math intution. Really well done - I hope the next part touches on the continuum hypothesis!
Great video! One small suggestion: You say (as most people do) that the decimal expansion of a rational number either terminates or ends with an infinitely repeating sequence of digits. I've found that people understand this more easily if you use only the repeating-sequence definition, while pointing out that the sequence can be a single digit, including 0. So e.g. 1/3 is 0.33333..., and 3/4 is 0.7500000... . That way people don't need to think of the pattern having two separate cases; it's all covered by the repeating sequence pattern.
I was deeply confused when I first saw one of your videos on this channel, because your voice and appearance seemed so familiar but I couldn't remember who you were. Now I know: you were the host of PBS Infinite Series! I loved that channel, I wish it had had a longer runtime. It's nice that you've returned to RUclips as an independent creator; I always appreciate more STEM content in my feed! As an aside, the title of this video reminded me of Gregory Chaitin's paper, "How Real are Real Numbers?". To my mind, the integers are perfect, well-behaved and beloved friends, the rationals are ill-behaved and rowdy hooligans who cause nothing but trouble, and the reals are each and all an eldritch cosmic horror whose very nature induces madness in those foolhardy enough to think about them for too long. If I could draw well, I'd probably depict Z as a friendly neighbour, Q as a shady thug, and R as a simple wooden sign with "Abandon All Hope Ye Who Enter" written on it in big comic sans font.
While I really like your font choice for Dante's inscription, I think you're grossly exaggerating the messiness of the reals. They're all perfectly intuitive, as are the complex numbers once you get to know them a little. It's the _infinities_ (sometimes called "transfinites") that the real eldritch horrors. If you see anything involving the axiom of choice, … I am reminded of Lovecraft's famous line, "We live on a placid island of [finiteness] in the midst of black seas of infinity, and it was not meant that we should voyage far."
@@ShankarSivarajan And I think you're grossly exaggerating the messiness of the infinities. With the axioms in mind the just become one of the possible type of sets, nothing more.
Measure theory is at the basis of probability theory and all the integrals that appear there. But there are other interpretations of integrals that doesn't touch measure theory, such as in differential geometry.
The nice thing is that also if you consider algebraic numbers, that is, if you add to rationals the irrationals that are rational roots of rational numbers, they are still countable. Trascendent numbers still are "almost all" real numbers.
My favorite numeration of the rationals is the Stern Brocot tree. One of my favorite re-discoveries (I figure most math fans have a few of these) is how to generate the SB tree by multiplying sequences of two 2×2 binary matrices.
You can extend that, and show that the algebraic real numbers are also measure 0, so that is stuff like sqrt(2), cube roots, anything that is a solution to an non zero polynomial with rational coefficients
Maybe you could do a video on how unreasonable and strange the Real numbers are too, particularly in the context of the Computable numbers (which are countable). Pi, e, the square root of 2, and so on are all Computable. Even Chaitin's constants are countable, because they have a one to one correspondence with a countable number of programming languages. Any number anyone has ever described or used in any way is part of a countable set because the symbols to describe it are countable. In what way are the Reals actually real and uncountable when we can't even point to a single one that isn't countable?
I love your video and explanation! But, I don't understand the reason that irrational numbers have measure. Wouldn't each irrational number still just be a single point on the number line? Based on the wording, I presume it has something to do with the sum of an uncountable infinity. I'd love to see an explanation for that!
You're right to call it out; it doesn't make any sense the way it was presented. I don't know the answer, but it must be something like what you're suggesting and was skipped over for simplicity. I would hope that most people are able to notice that this is only half of an explanation, to say one thing is obviously zero and not address why the other thing isn't zero.
I don't know if it helps but ... here goes two rational neighbors take form p/q and (p+1)/q Between these we can always squish unit interval of rationals OR by placing another rational at, say, midpoint where midpoint = [p/q +(p+)q]/2 OR squish irrational unit interval
Yay! New video from KHE! Although given this has a pretty solid point and supporting argument, I'm really curious about the "Part 1" in the title. Where are you going from here with this line of thought?
You left out my favorite math term of all time, "almost everywhere," meaning everywhere except for a finite or countable number of points. It is the 100% of which you speak, but with acknowledgment of the "holes" of measure 0. Thus, "Real numbers are irrational almost everywhere."
The explanation of measure theory makes sense for math, but falls apart for computable numbers. Every computer uses floating point numbers to measure anything that we use, like money, weight, length, area, temperature, bits, bytes, etc. irrational numbers are just too complicated to use, so we just represent them as a rational approximation. In the real world where we live, irrational numbers do not exist.
3:24 *how big* ... you meant *how many* ... there is no biggest rational ... for every rational *M* there exists another even bigger ... for example one such is *M + 1* ... in fact there are countably many such: *M + n* where *n* is a positive integer
A good video! However, the probability of throwing a dart and hitting ANY number whatsoever is ZERO, because any number visuslized in a physical space has a dimension of zero, i.e. do not conflate the idea of a number with the idea of place. Secondly, just because one can not represent an irrational number as a ratio between two integer numbers does not necessarily mean that this fact grants the absolute measure to the infinity that is carved out by the irrational numbers for then one again will commit oneself to the mistake of conflating that which is by definition pinpointably infinite (rationals) with that which is by definition not (irrationals).
here is a funniest thing. 1. between any two rational numbers there's infinitely many irrational numbers. 2. between any two irrational numbers, there's infinitely many rational numbers. 3. there's infinitely many more irrational numbers than rational numbers. how can 2 and 3 be true at the same time? how can there be more irrational numbers than rational numbers if there's infinitely many rational numbers between any two irrationals?
@@DavidRabahy that doesn't answer the question. I'm just curious about how can be infinitely many rational numbers between any two irrational numbers but, still, more irrationals than rationals. sadly, it seems the best I'm going to get is the well known "irrationals are uncountable. rationals are countable"
@@WilliamWizer imagine standing at the origin of a Cartesian grid looking outward in any direction. Imagine there are points (mathematical points, i.e. no dimensions) laid out at every integer intersection. If you look in some directions you will see one of those points. But many directions will never come to an integer intersection. The integer intersections represent the rational numbers. The directions that never come to an intersection represent the irrational numbers. Make lists of all of the directions of each type. The rational list can be mapped to the natural numbers. The irrational list will necessarily/provably be incomplete.
Rational comes from ratio not about rationality. A rational number can be obtained by diving two integers. Irrational are impossible to obtain that way
Aren't the rational numbers called such because each rational number is a ratio (an integer divided by a positive integer) and not for being reasonable?
Is "no space" on a line the same as the length 0? Is 0 nothing? Isn't there some difference between the concept of nothing and the concept of zero? Nothing can be divided by 0, but each integer is divided by nothing. In set theory 0 is defined as the empty set. It isn't nothing, it is a set. And 0 is defined as a whole number, but not a natural number. Clever guy who invented the 0. As a figure, not as a positional placeholder to represent other numbers. Brahmagupta (598-668 CE) did it when, ChatGPT generates when I prompt on the topic. I thought it was unknown who did it. I gotta have a closer look at that. I'm also informed that Brahmagupta was the first guy who made the division by zero error. (A human can't make something up without misusing it! Learn by trying, we are all babies after all.)
A very good explanation. But most non-mathematicians will remain confused. You could have mentioned that some irrational numbers, algebraic numbers, are countable. I find it more disturbing that both the rationals are dense in the reals and the reals are dense in the rationals. It is easy to prove but it seems to contradict any reasonable logic.
It's not the rational numbers that are bizarre, it's the real numbers. If you want uncountably infinite numbers, you have to include uncomputable numbers, which are 100% of the real numbers. But we can't unambiguously specify any particular uncomputable number - it's impossible to do so, by definition. So basically all real numbers are these objects which we cannot even reference. For all practical purposes, real numbers which don't fit in any more specific category, like rational or algebraic, simply don't exist. You cannot talk about even one example. If I were in charge of naming things, I'd label "real numbers" as "imaginary numbers" and relabel what was previously called "imaginary numbers" as "perpendicular numbers".
Irrationals and rationals are interleaved, yet former occupy the whole house and latter none of it. This apparent inconsistency is the result of stupid choice of words: rational and irrational which suggests a symmetric dichotomy, that doesnt exist. Every particular "rebel" length that defies the unit system we impose, creates a countably infinite family of related "rationals" that are rational RELATIVE TO the father rebel: for example: humans call the side of an isoceles right triangle "one thing" and the hypotenuse of the triangle rebels against this CHOICE, by sticking its tongue out: "I dont have a describable size: not 3/2, nor 7/5 nor (atoms in andromeda)/(atmos in milky way)", so we name this bitch "root 2", but any related bitch of the form "A/B times root2" where a, b integers, is not irrational from the perspective of the original bitch: root2. only irrational to the chosen unit at the beginning. if root 2 is seen as s rebel, it has its own dense family of rational co-rebels. so does root3, so does root 5, etc. the "set of irrationals" therefore isnt a homogeneous place. there are seperate clans in it. root 2 clan, root 3 clan, etc and any member of clan root 2, is not only irrational WITH respect to 1, but also irrational relative to any member of clan root 3, clan root 5, etc. so the apparerent gargantuan nature of irrational set is our own fucking fault: choosing to call them irrationals insteaad of acknowledgeing their relative sour relations with each other. which is exactly what happens when people are taught to use words in school before having any idea as to what the words mean.
I thought i had it all figured out--the meaning of life, the universe, and everything--but now, thanks to you, I don't think i'll be able to sleep tonight. Loved this video. thanks so much.
Hm. And yet if we were to bet on what number an actually _"randomly"_ and _"blindly"_ physically thrown arrow or dart is going to arrive at or land on a numberline, then I will always bet on that number being a rational number, since that's just more "rational" to do so there - something, something, *Church-Turing Thesis* ,...
@@BrianDominy Not just the physical non-zero width of the dart is getting in the way of only hitting an irrational number. But again also according to the *Church-Turing Thesis* only rational numbers can be computed and determined but also are actually are computed and determined and with that are only computable and determinable (at least in a finite amount of time computionally). Basically each irrational number is only approximated by a rational number at least computionally: *"Floating Point Numbers - Computerphile"* by Computerphile ( ruclips.net/video/PZRI1IfStY0/видео.htmlsi=hhBDr_XOV4DHDtsy ) *"Approximating Irrational Numbers (Duffin-Schaeffer Conjecture) - Numberphile"* by Numberphile ( ruclips.net/video/1LoSV1sjZFI/видео.htmlsi=t_dh9wv3eKxFPqsL ) Also the function "rand()" isn't as _"random"_ as one might suggest and assume basically for the same reasons: *"Random Numbers with LFSR (Linear Feedback Shift Register) - Computerphile"* by Computerphile ( ruclips.net/video/Ks1pw1X22y4/видео.htmlsi=eY8H1C3Gzi7wnyLw )
Ending the video with "irrational numbers are 100% of the number line" and providing no further context is quite the mic drop. I think I understand why, but it sounds like sequel bait. See you in less than a year hopefully!
0:32 This paradox is illusory --- it's entirely of your own making. You speak of the rationals as a subset of the reals, but it is not necessary to do this. We may instead think of the rationals as a set complete in itself. They are a field.
Cool. When are you gonna talk about algebraic number fields and Galois/Iwasawa theory? This is baby stuff. Your little video is cute and all, but you could've explained it a little more rigorously. Pop sci is a disgrace.
I first started watching Infinite Series when I was a math undergrad. I've since finished my doctorate (Math: Numerical Analysis) and been working at a research institute for over a year. So glad to see you still making math youtube videos! So funny, this would have been a mind blow to me back then and now I'm watching it because I'm just bored after work on a Tuesday and don't want to think super hard. Thanks!
So happy to see you back, KHE! Always loved your PBS videos, please continue enlightening us on RUclips. Amazing video, Welcome Back!
You have no idea how much we loved Infinite Series! Years later, I still think about it and how great the three of you were. Please keep making more videos!
omg how did I not find this channel sooner?! I used to watch Infinite Series (and some other PBS channels) and have missed your style of teaching! Glad you're still around!
Her and PBS infinite series is an influential part of why i became a math major in the first place. I cant believe theres been 2 years of videos and the algorithm didnt let me know!
I think it's pretty cool that 100% of the real numbers aren't just irrational, but more specifically transcendental numbers
Great Numberphile video about that calls them "uncomputable" numbers (saying that the computable transcendentals like pi are still 0% of all numbers): ruclips.net/video/5TkIe60y2GI/видео.html
I think it's a simple fact, owing to the fact that numbers which are roots of polynomials are countable (because rational polynomials are counrable). In that, it is not very dissimilar to or surprising than 100% being irrational.
@@odouroushouseant I stopped watching numberphile since they claimed 1+2+...=-1/12, without contextualizing that it's the analytical continuation of Zeta at -1. I thought it is an interview of some research assistants by people who have not much idea about the topics they are asking, and the whole thing is not serious and can even be very dangerous to people who want to learn it.
In fact, 100% of the real numbers are undefinable, that is, you can't write a sentence that uniquely specifies one of them.
@@odouroushouseant if Im not mistaken, uncomputable and tramscedentals are different classes of numbers
@@jonathandawson3091 Mathologer did a rebuttal video to the numberphile video. For the last couple years, there have been a few more vids on that series, and one of them that I watched a bit gives some interesting rationale for it. There is a lot of spurious math out there. The example that bothers me the most is the so-called Ross-Littlewood paradox. Matt Parker from standup maths channel jumps on that wagon. So does that young lady, Up and atom. The wikipedia article on it has it wrong. It is bewildering to me why mathematicians have got this wrong for 70 some years, since it was originally described in 1953. What is it about the problem that mathematicians just don't seem to comprehend? The conclusions given in the wiki article are just plain fallacious, along with the general understanding of it in the mathematical community. It is NO paradox at all, and that can be shown with simple mathematics.
I never thought I'd miss videos I, more often than not, don't fully understand (I never studied mathematics above high school concepts). Yet here I am and glad you posted again!
Glad to see you back! Your videos really helped me learn Module Lattice Post Quantum crypto for a project I was working on. Thank you!
Hahaha, wow that's awesome! I remember that Kelsey's stuff helped me get into the quantum conouting scene my freshman year, but the analysis & set theory stuff she had made was what ultimately wound up leading me down the trail of math lol
Instantly subscribed! Great Video! It's so awesome to see you're on RUclips again! I loved your PBS Infinite Series videos!
This was basically the intro lecture for real analysis in university.
Love it and love your videos!
Hi Kelsey, it's great to see another video come through by you! I remember I started watching you in high school, around when I had taken calc 1 & 2 from a small community college
Your role back with PBS Infinite Series was one of the key inspirations for me to seek out higher education in mathematics, and in a couple weeks I'll be graduating with my MSc in mathematics! Your work was hugely influential in my pursuits, so thank you for continuing on the good work still today :)
Omg I instantly recognized you I love you have been doing videos on your own since the pbs series ended. Great stuff :)
So glad to have found your new home on YT, Kelsey. Really enjoyed this, but I see I've got some catching up to do!
This was so well produced, nice job. I can imagine how difficult it must be to balance the scale of rigor vs. simplicity in explaining higher math intution. Really well done - I hope the next part touches on the continuum hypothesis!
Great video! One small suggestion: You say (as most people do) that the decimal expansion of a rational number either terminates or ends with an infinitely repeating sequence of digits. I've found that people understand this more easily if you use only the repeating-sequence definition, while pointing out that the sequence can be a single digit, including 0. So e.g. 1/3 is 0.33333..., and 3/4 is 0.7500000... . That way people don't need to think of the pattern having two separate cases; it's all covered by the repeating sequence pattern.
indeed. But the underlying problem is that real numbers aren't the same kind of thing as a rational at all, right?
I was deeply confused when I first saw one of your videos on this channel, because your voice and appearance seemed so familiar but I couldn't remember who you were. Now I know: you were the host of PBS Infinite Series! I loved that channel, I wish it had had a longer runtime. It's nice that you've returned to RUclips as an independent creator; I always appreciate more STEM content in my feed! As an aside, the title of this video reminded me of Gregory Chaitin's paper, "How Real are Real Numbers?".
To my mind, the integers are perfect, well-behaved and beloved friends, the rationals are ill-behaved and rowdy hooligans who cause nothing but trouble, and the reals are each and all an eldritch cosmic horror whose very nature induces madness in those foolhardy enough to think about them for too long.
If I could draw well, I'd probably depict Z as a friendly neighbour, Q as a shady thug, and R as a simple wooden sign with "Abandon All Hope Ye Who Enter" written on it in big comic sans font.
While I really like your font choice for Dante's inscription, I think you're grossly exaggerating the messiness of the reals. They're all perfectly intuitive, as are the complex numbers once you get to know them a little. It's the _infinities_ (sometimes called "transfinites") that the real eldritch horrors. If you see anything involving the axiom of choice, … I am reminded of Lovecraft's famous line, "We live on a placid island of [finiteness] in the midst of black seas of infinity, and it was not meant that we should voyage far."
Integers are perfect and well behaved? _laughs in number theory and diophantine equations_
@@ShankarSivarajan And I think you're grossly exaggerating the messiness of the infinities. With the axioms in mind the just become one of the possible type of sets, nothing more.
Oh! I recognized you instantly from that PBS Infinite show. Nice to see you again.
Another Chalk Talk!? More Kelsey? What a pleasant surprise!
Big fan of these kinds of videos - please keep em coming!
She's back!
Very nice video! There are so many cool and quick concepts like this in math, I've been waiting for a channel like this.
Yay! So great to see you back happy holidays 🎁!
OH MY GOD YOU'RE BACK
I'VE WAITED SEVEN YEARS
IMMEDIATE SUBSCRIPTION
rational numbers are ratio-nal, as far as I know.
Yup. They aren't reasoning or reasonable, they are ratios.
Came here to say that.
Ratio in Latin has some meanings. One of them is 'relational'. So rational comes from the relation of numbers, and is therefore rational.
amazing to see you again since Infinite series!!❤❤❤❤
Measure theory is at the basis of probability theory and all the integrals that appear there. But there are other interpretations of integrals that doesn't touch measure theory, such as in differential geometry.
i remember her from PBS series. she is very good teacher. so happy to see her back on youtube. this should be the next 3B1B.
The nice thing is that also if you consider algebraic numbers, that is, if you add to rationals the irrationals that are rational roots of rational numbers, they are still countable. Trascendent numbers still are "almost all" real numbers.
My favorite numeration of the rationals is the Stern Brocot tree. One of my favorite re-discoveries (I figure most math fans have a few of these) is how to generate the SB tree by multiplying sequences of two 2×2 binary matrices.
She's back! ❤
Beautiful stuff, please continue making videos :)
It's great news to hear from U again 😊
You managed to fit a lot into six minutes there, I'm impressed.
She is back 🎉🎉🎉
Look who's back! Missed her so much!
You can extend that, and show that the algebraic real numbers are also measure 0, so that is stuff like sqrt(2), cube roots, anything that is a solution to an non zero polynomial with rational coefficients
Great to see you back on youtube.
Kelsey! You're back!
Great u r back!!!
Great content. Years ago there was another great video from your team on numbers and their extension for the property of closure.
Thanks, it was listed in this video's description
A Hierarchy of Infinites | PBS Infinites Series
With luck and more power to you. Hoping for more great episodes.
It's also interesting that real numbers are defined by limits of convergent rational sequences
wow, thank yo so much for your work!
An early Christmas present 🎉
Welcome back
The queen is back
Very nice video! I need to remember this the next time I teach cardinality
Welcome back!
A fun graph, I forget who it is named after:
___
| 1, when x is rational
y = | 0, when x is irrational
|__
Dirichlet function!
Maybe you could do a video on how unreasonable and strange the Real numbers are too, particularly in the context of the Computable numbers (which are countable). Pi, e, the square root of 2, and so on are all Computable. Even Chaitin's constants are countable, because they have a one to one correspondence with a countable number of programming languages. Any number anyone has ever described or used in any way is part of a countable set because the symbols to describe it are countable. In what way are the Reals actually real and uncountable when we can't even point to a single one that isn't countable?
I love your video and explanation! But, I don't understand the reason that irrational numbers have measure. Wouldn't each irrational number still just be a single point on the number line? Based on the wording, I presume it has something to do with the sum of an uncountable infinity. I'd love to see an explanation for that!
You're right to call it out; it doesn't make any sense the way it was presented. I don't know the answer, but it must be something like what you're suggesting and was skipped over for simplicity. I would hope that most people are able to notice that this is only half of an explanation, to say one thing is obviously zero and not address why the other thing isn't zero.
I don't know if it helps but ... here goes two rational neighbors take form p/q and (p+1)/q
Between these we can always squish unit interval of rationals OR by placing another rational at, say, midpoint where midpoint = [p/q +(p+)q]/2 OR squish irrational unit interval
After this lecture, you understand everything she said and nothing at the same time....
What a fantasic video!
Yay! New video from KHE! Although given this has a pretty solid point and supporting argument, I'm really curious about the "Part 1" in the title. Where are you going from here with this line of thought?
Early Christmas! I hope you'll find the time, energy, and funding to upload more.
wow, never seen the difference between countable and uncountable like that before
I hope there will be more video!
Yeah
Kelsey!
You left out my favorite math term of all time, "almost everywhere," meaning everywhere except for a finite or countable number of points. It is the 100% of which you speak, but with acknowledgment of the "holes" of measure 0. Thus, "Real numbers are irrational almost everywhere."
Best day ever
Hey I remember you from PBS math! I absolutely loved your show. But then you disappeared.
Is this your new channel? I will subscribe then.
Very cool.
The explanation of measure theory makes sense for math, but falls apart for computable numbers. Every computer uses floating point numbers to measure anything that we use, like money, weight, length, area, temperature, bits, bytes, etc. irrational numbers are just too complicated to use, so we just represent them as a rational approximation. In the real world where we live, irrational numbers do not exist.
What are those signs (tattoos?) on your fingers???
3:24 *how big* ... you meant *how many* ... there is no biggest rational ... for every rational *M* there exists another even bigger ... for example one such is *M + 1* ... in fact there are countably many such: *M + n* where *n* is a positive integer
I was looking at my number line and found a rational number. What do I win?
You're baaaaaack !
A good video! However, the probability of throwing a dart and hitting ANY number whatsoever is ZERO, because any number visuslized in a physical space has a dimension of zero, i.e. do not conflate the idea of a number with the idea of place.
Secondly, just because one can not represent an irrational number as a ratio between two integer numbers does not necessarily mean that this fact grants the absolute measure to the infinity that is carved out by the irrational numbers for then one again will commit oneself to the mistake of conflating that which is by definition pinpointably infinite (rationals) with that which is by definition not (irrationals).
I'm thinking it's more a reflection on mathematicians, rather than rational numbers.
I would like to understand the proof for this, i saw it before but i didnt get it
here is a funniest thing.
1. between any two rational numbers there's infinitely many irrational numbers.
2. between any two irrational numbers, there's infinitely many rational numbers.
3. there's infinitely many more irrational numbers than rational numbers.
how can 2 and 3 be true at the same time?
how can there be more irrational numbers than rational numbers if there's infinitely many rational numbers between any two irrationals?
The irrationals remain uncountable no matter how small an interval is examined.
@@DavidRabahy that doesn't answer the question.
I'm just curious about how can be infinitely many rational numbers between any two irrational numbers but, still, more irrationals than rationals.
sadly, it seems the best I'm going to get is the well known "irrationals are uncountable. rationals are countable"
@@WilliamWizer imagine standing at the origin of a Cartesian grid looking outward in any direction. Imagine there are points (mathematical points, i.e. no dimensions) laid out at every integer intersection. If you look in some directions you will see one of those points. But many directions will never come to an integer intersection. The integer intersections represent the rational numbers. The directions that never come to an intersection represent the irrational numbers. Make lists of all of the directions of each type. The rational list can be mapped to the natural numbers. The irrational list will necessarily/provably be incomplete.
Are you the PBS Infinite Series host?
WTF. The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.
Rational comes from ratio not about rationality. A rational number can be obtained by diving two integers. Irrational are impossible to obtain that way
Aren't the rational numbers called such because each rational number is a ratio (an integer divided by a positive integer) and not for being reasonable?
Doesn't these problems come from the systematization of measure on top of real numbers?
Patsing infinities can be useful but not in trying to explain quantities
ratio-nal numbers. Heavy on the ratio.
the real numbers is the set of rational numbers and irrational numbers.
Rational number is more aking to the sense of army Ration than someone acting Rationally.
Is "no space" on a line the same as the length 0? Is 0 nothing? Isn't there some difference between the concept of nothing and the concept of zero? Nothing can be divided by 0, but each integer is divided by nothing. In set theory 0 is defined as the empty set. It isn't nothing, it is a set. And 0 is defined as a whole number, but not a natural number.
Clever guy who invented the 0. As a figure, not as a positional placeholder to represent other numbers. Brahmagupta (598-668 CE) did it when, ChatGPT generates when I prompt on the topic. I thought it was unknown who did it. I gotta have a closer look at that. I'm also informed that Brahmagupta was the first guy who made the division by zero error. (A human can't make something up without misusing it! Learn by trying, we are all babies after all.)
"everywhere but nowhere" means a dense set of measure 0.
A very good explanation. But most non-mathematicians will remain confused. You could have mentioned that some irrational numbers, algebraic numbers, are countable. I find it more disturbing that both the rationals are dense in the reals and the reals are dense in the rationals. It is easy to prove but it seems to contradict any reasonable logic.
Did you know that name of rational numbers comes from "ratio"? So, maybe they are ratio-nal after all.
It's not the rational numbers that are bizarre, it's the real numbers. If you want uncountably infinite numbers, you have to include uncomputable numbers, which are 100% of the real numbers. But we can't unambiguously specify any particular uncomputable number - it's impossible to do so, by definition. So basically all real numbers are these objects which we cannot even reference. For all practical purposes, real numbers which don't fit in any more specific category, like rational or algebraic, simply don't exist. You cannot talk about even one example.
If I were in charge of naming things, I'd label "real numbers" as "imaginary numbers" and relabel what was previously called "imaginary numbers" as "perpendicular numbers".
perhaps a lot of the time when you say 'zero' it might be better to say 'infitesimly small'
Irrationals and rationals are interleaved, yet former occupy the whole house and latter none of it.
This apparent inconsistency is the result of stupid choice of words: rational and irrational which suggests a symmetric dichotomy, that doesnt exist.
Every particular "rebel" length that defies the unit system we impose, creates a countably infinite family of related "rationals" that are rational RELATIVE TO the father rebel: for example: humans call the side of an isoceles right triangle "one thing" and the hypotenuse of the triangle rebels against this CHOICE, by sticking its tongue out: "I dont have a describable size: not 3/2, nor 7/5 nor (atoms in andromeda)/(atmos in milky way)", so we name this bitch "root 2", but any related bitch of the form "A/B times root2" where a, b integers, is not irrational from the perspective of the original bitch: root2. only irrational to the chosen unit at the beginning. if root 2 is seen as s rebel, it has its own dense family of rational co-rebels.
so does root3, so does root 5, etc.
the "set of irrationals" therefore isnt a homogeneous place. there are seperate clans in it. root 2 clan, root 3 clan, etc and any member of clan root 2, is not only irrational WITH respect to 1, but also irrational relative to any member of clan root 3, clan root 5, etc.
so the apparerent gargantuan nature of irrational set is our own fucking fault: choosing to call them irrationals insteaad of acknowledgeing their relative sour relations with each other. which is exactly what happens when people are taught to use words in school before having any idea as to what the words mean.
So the rationals are dense, countable and have Lebesgue measure zero? Wow, so deep.. what is this primary school?
Thought the channel was dead haha
I thought i had it all figured out--the meaning of life, the universe, and everything--but now, thanks to you, I don't think i'll be able to sleep tonight. Loved this video. thanks so much.
Hm. And yet if we were to bet on what number an actually _"randomly"_ and _"blindly"_ physically thrown arrow or dart is going to arrive at or land on a numberline, then I will always bet on that number being a rational number, since that's just more "rational" to do so there - something, something, *Church-Turing Thesis* ,...
An actual dart has a non-zero width, so it has a 100% chance of hitting a rational number!
@@BrianDominy Not just the physical non-zero width of the dart is getting in the way of only hitting an irrational number. But again also according to the *Church-Turing Thesis* only rational numbers can be computed and determined but also are actually are computed and determined and with that are only computable and determinable (at least in a finite amount of time computionally). Basically each irrational number is only approximated by a rational number at least computionally:
*"Floating Point Numbers - Computerphile"* by Computerphile ( ruclips.net/video/PZRI1IfStY0/видео.htmlsi=hhBDr_XOV4DHDtsy )
*"Approximating Irrational Numbers (Duffin-Schaeffer Conjecture) - Numberphile"* by Numberphile ( ruclips.net/video/1LoSV1sjZFI/видео.htmlsi=t_dh9wv3eKxFPqsL )
Also the function "rand()" isn't as _"random"_ as one might suggest and assume basically for the same reasons:
*"Random Numbers with LFSR (Linear Feedback Shift Register) - Computerphile"* by Computerphile ( ruclips.net/video/Ks1pw1X22y4/видео.htmlsi=eY8H1C3Gzi7wnyLw )
Ending the video with "irrational numbers are 100% of the number line" and providing no further context is quite the mic drop.
I think I understand why, but it sounds like sequel bait. See you in less than a year hopefully!
0:32 This paradox is illusory --- it's entirely of your own making. You speak of the rationals as a subset of the reals, but it is not necessary to do this. We may instead think of the rationals as a set complete in itself. They are a field.
That's fine, but rational numbers ARE a subset of real numbers, as integers are a subset of both, and all are a subset of the complex numbers.
Cool. When are you gonna talk about algebraic number fields and Galois/Iwasawa theory? This is baby stuff.
Your little video is cute and all, but you could've explained it a little more rigorously. Pop sci is a disgrace.
Too bad mathematicians are not linguists.
"Rational" numbers are based on a "ratio", not on "reason".
"Mathesis" means "learning", girl!
i dont care
Measure Theory; so ... Axiom of Choice? Axiom of Determinacy?
I'm going to need that p2 please