What Actually Is A Number?
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- Опубликовано: 4 окт 2024
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Shout to everyone who got it right
You did NOT get it right
@@nonexistent-yt9em why not
@@nonexistent-yt9em they didn't say they did tho
I got it partially right because of fhe thumbnail, I only guessed pi, but not the decimals
I guessed pi (the thumbnail), but not all the decimals
the numerical cinematic universe
Numbers level threat
Reference to LCU 😂
In my earlier classes, my teacher gave a definition. Numbers are symbols devloped by various civilasations to count entities, later evolving into many branches as the needs went on.
Your teacher is the absolute best.
My nephew Julian made up a riddle when he was three: "Q. Who invented numbers? A. The farting butt."
@@GlorifiedTruth your nephew is a genius. I am not even joking. A butt is the best representation of god. Why do you think reality is sh1t? That's right! Because it is sh1t! Meaning it was sh4t. Meaning God is, in fact, a supernatural butt. This is what I believe.
Sometimes we need to distinguish between NUMERALS, which are symbols used to represent numbers, and the NUMBERS themselves.
doesnt say much
2:58 In my ring theory class, I learnt something more general. You can extend any infinite integral domain R to a field (we called it field of quotients of R and represented it by Q(R)). The extension Z to Q is a particular case. And indeed Q is the field of quotients of Z.
I have a question. Why did you specify that the integral domain is infinite? The construction using quotients works perfectly fine and in the finite case.
@@SuperLink2013 a finite integral domain is already a field so it is isomorphic to its field of quotients :)
So I don't see the point of extension.
@@speakingsarcasm9014 oh, fair point, I completely forgot about that. My bad
Ok maths dweebs ;)
Learnt?
7:47 this is actually a common misconception, the definition of i is not the square root of -1, but rather that i² = -1. These two definitions may seem equivalent, but they are actually not due to multivalued square roots and stuff like that. Using the first definition can create paradoxes.
Using i^2=-1 makes the paradox that i has two values
But i and -i are interchangeable, you can just switch them
But it is best to say "i is one of the roots of the equation x^2=-1"
@@eneaganh6319
How is i² = -1 a paradox? Elaborate please
i is one of the roots of -1. The other is -i. Sorted
Can we instead just say ±i = -1^½?
i^3 = -i and i^6 = -1 because i^4 = 1 and i^2 = -1
This has got some ‘we don’t even know what a word is’ energy.
(Wow, thank you for so many likes!)
It’s just a series of letters, seperated by spaces, such as hrrkrkrkrwpfrbrbrbrlablblblblblblwhitoo’ap.
@@asheep7797What about languages without a writing system?
@@Tom-u8qThen invent symbols to represent it. There's a certain conveience to defining mathematics using formal language theory, namely that it makes mathematics "physically concrete" in a sense, and also processed physically.
Jordan Balthazar Peterson has entered the chat
@@stevenfallinge7149you’re sort of missing asheep’s point mate
I think it was bertrands russels definition where he said that, a number is a set of all sets which have the cardinality of that number. for eg: The number 2 is a set of sets that contain 2 elements, the number 7 is a set of all sets that contain 7 elements. so on and so forth. i think thats a really interesting philosophical perspective on math, Hey maybe thats another video idea!
Can this definition apply to number outside of the naturals?
@@Francu8942 no. And it includes "infinite numbers" which we don't usually think as numbers. That actually the definition of cardinal numbers.
Unfortunately, the 'set' of all sets of a given cardinality is actually too big to be a set; it's a proper class. This doesn't mean we can't use it; it just means that we need to keep watch for paradoxes, like Russell's Paradox.
Edit: The set of all sets of zero size is a set; it's simply the set that has the empty set as it sole element.
@@tomkerruish2982 man, don't even start this conversation. Cardinal numbers exist, they were properly defined, OP just invoked something interesting, take it just like that.
Russell's definition was horribly wrong. A set can't have a cardinality of pi or cardinality of a quaternion, unless you get *really* *REALLY* abstract and disconnected from any concrete notion that underpins the idea of "cardinality".
1:00 😉 "The Natural Numbers are 1, 2, 3, ..." . Then immediately after shows they're built from the Peano Axioms and the very first axiom is 0 is a Natural Number. Then immediately goes back to not including 0 in the next slide, and then quickly has a slide saying 0 is in the Whole Numbers but not the Natural Numbers.
Lol, I'm being a little cheeky, but this is an example of how there isn't a universal consensus on whether or not 0 is or isn't a Natural Number. Whether or not it is depends entirely on the context of the person using the set. When you're working from a set theoretic model, for example, the Natural Numbers correspond to the cardinalities of the finite sets, which includes the Empty Set so 0 is a Natural Number. But when you're working in number theory and looking at things like factorizations, etc, 0 is often excluded as a Natural Number for convenience because many theorems that apply for the strictly positive integers 1, 2, 3, etc don't apply to 0.
P.S. Similarly the phrase Whole Numbers is a colloquialism that can refer to various sets.
i think it should be. otherwise the additive identity is meaningless. the additive identity with reference to the successor function and natural numbers would imply 0 is the beginning of the natural numbers yes?
i mean as counting numbers, 0 counts the elements in the null set
not sure tho all this is pretty confusing, any insight?
Strictly speaking, the first axiom shown was not "0 is a natural number", it was "0 is a number"...
@@gerryiles3925 Listen to the video though. "... The Peano Axioms define the Natural Numbers in terms of sets" is what he says while simultaneously flashing the axioms including 0 in that set.
Interestingly, Peano himself defined the natural number axioms twice, once without and once with 0.
@@__christopher__ i swear he's just trolling at this point. still has the math community split nearly 50/50 on this.
If you pronounce it differently, a topical analgesic is a number.
Being a Master in Physics student, it is so good to finally find what a number really is.
a number is a value that specifies relativity between entities, taken as an aspect of all entities concerned considering the sum of those entities as an integral system
Fantastic video, thank you. I'm truly fascinated by nunber theory, and this gave me new things to chew on
I think this can be slightly improved upon by extending this to mention somewhere in it that numbers do not necessarily need to have base 10, so binary, or n-nary numbers, while still equivalent, are a different representation of the natural numbers (and hence real numbers can help constructed from them). Also, you mention that the natural numbers and the even natural numbers have the same size, which is true for the counting measure, but you can extend the idea of a size of a set of numbers by introducing (even just rudimentary) measure theory, for example both sets have the same counting measure, and they also have the same Lebesgue measure, but their counting measure is infinity, and their Lebesgue Measure is 0.
This definitely needs more views man
0:08 I thought I won but I was thinking of the full pi.
what abt non-computable numbers
for example a number that is defined by being the number of steps a turing complete machine would take to halt
They are a subset of the real numbers
@@methatis3013 ohh
I literally guessed pi😳
Me too lol
Maybe because of the thumbnail…
Did you guess pi or pi out to 117 decimal places? Because those are different numbers.
Same. But I thought of pi to the 109 digit 😕
@pvanukoff DARN IT! I guessed pi out to 116 digits. so close...
7:48 i is not defined to be square root of -1 because square root is not defined for negative numbers. Like in video is written i²=-1 is correct way to think of it.
Yeah, yeah. Almost got me this time, Plato.
The very first Peano axiom literally says that 0 is a natural number.
You have forgotten the most important of all, the Dihedron numbers, the true complex numbers!
According to the modern foundation of mathematics aka ZFC, the set of natural numbers is precisely defined as {∅, {∅}, {∅, {∅}, ...}.
The integers and rational numbers then can be defined as equivalence relations of pairs of natural numbers/integers (two integers (a, b) and (c, d) are equivalent if a+d=c+b)
The real numbers are harder to define rigorously, but we know that they are the unique field, which is ordered and complete. There are several ways to construct them from the rational numbers, e.g. equivalence class of cauchy sequences or dedekind cuts.
The complex numbers are just the cartesian product of the reals, with defined multiplication in the way we want
While today the construction you give is universally used as model of the natural numbers in ZFC, it is not true to say that it is defined as that in ZFC. Indeed, there's no axiom in ZFC that says anything about natural numbers (the closest is the axiom of infinity that essentially says that the set you get as natural numbers if you define the natural numbers that way is indeed a set; but it doesn't say that it is the set of natural numbers). The axioms of ZFC are agnostic of numbers, and there are many ways you could define (a model of) the natural numbers in ZFC. The definition you gave is on top of ZFC, not part of it. And ultimately, if you want to show that the construction really gives a model of the natural numbers, you have to prove that the Peano axioms are fulfilled by this construction.
I'd like to add another categlory: Decimal numbers, numbers that can be represented with a finite amount of digits. For example, 1.234, 5.324 and 56.234234 but not 1/3, 4/7, 1/7 nor 1234567890987654322/11.
These are known as terminating decimals.
AFAIK, some people are very unsatisfied with the current definition of real numbers. Not defined rigorously and consistently enough...
number is an expression or relationship between 2 things or an interval
2:12 - speaking as the weirdo who thinks about temperature in Kelvins and cannot stand either degrees Celsius or Fahrenheit, negative numbers are perfectly dispensible with respect to temperature measurements. They should be dispensed with, as far as I'm concerned (even if I understand that our false zeros are more "practical" than the true one). The rest stands.
There is no reason to though. We need a personalized form of measurement that relates to the general public using it.
Celsius and Fahrenheit aren't invented to be forced upon actual researchers for rigorous calculations. Thats just a delusional thought.
Just dont use it brother 😭. Its not that hard to let things that dont have to affect you be. Because there ARE way too many people that need these systems for them to be dispensible.
Fahrenhiets can actually get erased from existence tho I wont really mind.
Kelvin is as arbitrary as Fahrenheit is and as Celsius is
The units are lowercase even when named after people. They are also supposed to be uninflected.
@@antagonisticalex401 ...did you not catch _any_ of the hints that maybe, just maybe, I understand that most people would find it inconvinient to give everyday temperatures in the hundreds of units? Me calling myself "the weirdo" (referring to myself as "the" (only) person like this, and being "weird" for it), and pointing out that °C and °F are more practical than K (while being dissmissive of the value of practicality w.r.t. measurements)? I was being facetious, and wasn't subtle about it. I'm the weirdo who uses Kelvins _for everyday measurements._ This is silly and I know it (hence why I treated "practical" considerations with such scorn), but Kelvins technically are "truer" representations of temperature than °C or °F, so I love playing up that angle. I'm overly dramatic about it, because (a) it's more fun that way and (b) it makes it clear that I don't take myself seriously. Or well, it makes that clear to most people, at least. Even in this reply, I continue to play that character.
On that note, if Fahrenheit can be erased, so can Celsius. They're equally practical (°C gives nice reference points for relevant natural events, while °F gives the full 0-100 range to 'normal' temperatures), but they both get there through the inexcusable sin of setting zero off in space at some arbitrary, senseless point. They both need to go, and all positive references to them burned in a fire.
@@methatis3013 It's less arbitrary. Setting zero equal to zero is the default, where units are concerned. There is no "choice" associated with that; there's a clear, correct answer. Because degrees Celsius/Fahrenheit do not set zero equal to zero, there is an arbitrary choice associated with where the zero point is on each system. With each of kevins, degrees Celcius and degrees Fahrenheit, the size of a unit is arbitrary. With degrees Celcius and degrees Fahrenheit, the zero point is also arbitrary, whereas there is no choice about where the Kelvin scale's zero is.
In any case, my original comment had nothing to do with which systems were more or less arbitrary, and everything to do with the fact that negative temperatures don"t exist. The very concept is nonsensical. There is no tenperature which, when doubled, is colder than it began.
someone decided to make up i because they didn’t want to admit they were wrong, and look what they did
7:52 1/0 is undefined but maybe the answer is a complex number or an imaginary number?
So, you're kinda right, but it's not so easy. I should preface by saying that in the complex numbers, you may approach infinity at any desired angle by choosing an angle from which to have your denominator approach 0. So in a sense 1/0 is "infinity, but in every direction, including both positive infinity and negative infinity, as well as all the rest in between".
With the complex numbers, we maintain and even gain useful algebraic properties. But "projectively extending the real line" with a point corresponding to 1/0 ends up losing many of those properties. Suddenly addition, subtraction, and multiplication aren't even closed. Which might prompt us to invent the "wheel algebra" by additionally adding a point corresponding to 0/0, closing those operations back up. But this point itself kinda has no properties we care about. And yet to account for it, we need to completely rewrite the algebraic rules for manipulating multiplication and division by adding on additional multiples of 0 to catch infinite inputs, in order to get that value whenever we expect to.
Actually, the surreal numbers not only contain the real and the hyperreal numbers, they also contain the ordinal numbers. And as that fact already implies, they form a proper class.
And of course just like you can go from real to complex numbers, you can go from surreal to surcomplex numbers. And surquaternions should also be no problem.
Speaking of quaternions, there's also a hierarchy here, although you might question if those should still be called numbers (but then, many people don't call quaternions numbers either). Basically, you can use an uniform process which doubles the number of components in each step. In the first step, you go from the real numbers to the complex numbers, in the second step from the complex numbers to the quaternions, in the third step to the octonions, and so on. However already with the octonions you lose associativity of multiplication.
There are also the nimbers, which form a proper class field of characteristic 2.
A more conventional field of numbers are the algebraic numbers, which form a proper subfield of the complex numbers, and the real algebraic numbers, which themselves form an ordered field that is a proper superset of the rational numbers, but a proper subset of the real numbers. Algebraic numbers are numbers like sqrt(2) which are solutions of rational (or equivalently, integer) polynomials, non-algebraic numbers are called transcendental numbers, and the most famous of those are e and pi.
You skipped the algebraic numbers, constructable numbers, and computable numbers. These are all subsets of Real numbers, but unlike Real numbers, they're actually defined in a way that's actually understandable and they can actually be used. Even the Complex numbers (or rather Complex Algebraic numbers) are easier to define and use than the Real numbers.
P-adic numbers seem counterintuitive... unless you're a programmer. A common function on computer integers is count-trailing-zeroes. This actually gives the 2-adic valuation of the input, or negative logarithm of the 2-adic absolute value. Similarly, the way negative numbers are represented of p-adic numbers should be familiar to any programmer, as its fine by subtracting from zero are carrying all the way to infinity, resulting in the 2-adic -1 being 0b...1111, which is exactly how -1 is represented in 2's complement for a hypothetical infinite-bit integer. There are also strong connections between the representations of fixed point numbers and those of 2-adics.
"Imaginary" is a terrible name for a set of numbers that are exactly as real (if not _more_ so given my first paragraph) than Real numbers. I prefer to call them "spinny numbers", but they ultimately have more to do with rotation than they do with _imagination._ Quaternions are 3D spinny numbers. They don't have a component for each orthogonal _direction,_ but rather a component for each orthogonal axis of rotation _plus one._ 2D only has 1 axis of rotation (that includes the origin): the origin. The remaining component is literally the identity; how much to _not_ rotate, and it has the exact same meaning in the quaternions.
"What's a number" Everything that we decide is a number.
Is a matrix a number? If we decide it is
Though a definition would be nicer
14 minutes before I was 🗿 then after 14 minutes I am 👽
Bro no way i guessed pi. I was expecting you to do something tricky and so i thought that it will be pi and it was.
Nah you had to guess it with 117 digits
What is the set in wich you find 0.999... ?
No octonions?
Sorry, but I don't associate with octonions.
I know short format content on RUclips has its limitations, but you could've touched more on what is lost at each stage of generalization. Negative integers lose corresponce with sizes of finite sets compared to positive integers. Fractions lose unique representation compared to integers. Real numbers lose factorization into integer powers of primes. Complex numbers lose the ability to be ordered. Quaternions lose commutativity of multiplication. And so on.
And p-adics?
@@jam-trousers p-adics branch off of integers, but interestingly fractions now do get unique representation. The loss is more about fractalization of the numbers in terms of distance. There are no longer ranges of numbers everywhere but distinct bins of p-adics with more bins inside them.
How did we get from 1+1=2 to different-sized infinities
I love complex numbers!!!🤩🤩🤩 I mean they are even useful in physics!!!🤩🤩🤩 And also cool!!!😅🤣
They're so spinny and spinny is good. What I can't wrap my head around is why anyone would ever consider calling spinny numbers "imaginary". What could possibly be so "imaginary" about going spinny?
It should be obvious why spinny numbers are so useful in physics. There are more things in physics that go spinny than that don't. Planets go spinny; subatomic particles go spinny. Electric potential works in a more abstract space, but within said space it also goes spinny.
π = 2 in Riemann Paradox and Sphere Geometry System Incorporated...
Tau = 2π = 2^π = 2^2 = 2 × 2 = 4
Oh shoot, I guessed phi out to 1618 decimal places. I'm wondering why you picked the arbitrary 117, probably to be a 'random number' sounds like a number people wouldn't pick right?
I guessed 6.34, quite close to the double of pi.. coincidence
Great vid but you could have simplified the p-adic definition, you sort of skipped past it. P-adics are amazing, so different from other number systems.
Never heard of hyperreals or surreals before, thank you for that
I feel like we are starting to bring in quality in a more.. systematic way. Idk. Like how one Infinity was technically bigger, but actually the same size
Meanwhile the other multicomolex numbers, perplex numbers, the reimann sphere and literally every other algebraic field: bruh
0:09 Well, I guessed correctly!
I told my son he could have something he wanted (don't remember what) if he guessed the number between 1 and 10 that I had written down. It was 3 + the square root of 2.
You've got 0 for nothing and 1 for something. Once you add more than that, you've opened Pandora's box, in my opinion. Nothing "intuitive" about the counting numbers once you reach super-astronomical numbers that can't reasonably be used to quantify anything.
Maybe numbers are artifacts of human thinking, the way sawdust is left behind by a craftsman.
Engineering must then be the art of finding out how much sawdust you can add to a rice krispy before people notice
@rarebeeph1783 Yeah, sounds legit
Initially a value. But something which represents something else, no?
His voice just impregnated me like wtf just happened
uhmmmm pi = circumfrence / diameter, checmkate mathemeticians
We need new names for the reals and imaginaries, especially since the discovery of the quaternions and octonions. All numbers have the same degree of reality, the meaning of which is a subject for philosophy. It would be nice to keep the symbol i for sqrt(-1), but it would be equally nice to have a new interpretation for it.
Iirc, Gauss wanted to call complex numbers "lateral" numbers
Complex numbers are 2D spinny numbers. Quaternions are 3D spinny numbers. (Not 4D. 4D spinny numbers are actually 8D, but also aren't octonions because spinny is inherently associative, while octonions _aren't_ associative.)
If Complex numbers are spinny numbers since they rotate things, then the logical next step would be to call Real numbers "stretchy", because they scale things.
Numbers, not to be confused with numerals which are symbols used to represent numbers.
Was disappointed that dual numbers didn't put in an appearance as they're kind of like a simplified version of the hyperreals. Just one extra element, epsilon, which is greater than 0 but when squared, gives 0. Many of the calculus tricks of the hyperreals work well with the duals instead.
Dual epsilon is a cousin to the imaginary unit i, but there is another: j. This isn't the quaternion j, nor the engineer's j; these both square to -1 whether they're the same j or not. I'm talking about the split-complex j, which when squared gives 1, but is not itself equal to 1. Split-complex numbers are weird, man.
They're not weird. They're just _hyperbolic._ Dual numbers are flat by comparison, and complex numbers are elliptic. The three perfectly correspond with the three distinct curvatures for geometry.
Also you can add another element that squares to 1 but is neither the split-complex j nor the Real number 1, and on its own it doesn't do much, but depending on how you define its interaction with the prior j, you can't end up conjuring a Complex i just out of their combination.
There are no numbers beyond real numbers, unless you want change how you define them. Filling the entire number line means no spaces left to fill. Add to it mean whatever you were dealing with was only some part of the numbers line
There is no what if i go further because you haven't gone far enough to cover the real numbers and yet or even the natural numbers
The trouble is that there is no standard mathematical definition of "number" we'd have to change in the first place. There are things that mathematicians call numbers (i.e. the things discussed in this video), and by that they are called numbers, they are "numbers"--that's how language works.
However, when it comes to infinities. There very much is a "what if I go further," and in fact in set theory it's almost impossible not to. When we refer to "the natural numbers", we're invoking a set which has more elements than any finite set. When we refer to "the real numbers"--a set explicitly constructed in terms of the rationals and therefore of the naturals--we're invoking a set which has more elements than the naturals. These are examples of sets with cardinalities that have "gone further".
In some poorly behaved games such as "infinite chess" (arbitrarily expandable chess, if you prefer), there are positions which, for any natural number of your choosing, one may delay checkmate longer than that many turns. We call the first of these positions "mate-in-omega". A position in which your best move is to move to a mate-in-omega position, then, is "mate-in-(omega+1)". And so on. These are examples of order types which have "gone further". The hyperreals to my understanding are in some sense just extending the countable ordinals (as such) with a sense of subtraction and division, and the surreals to my understanding do the same for all ordinals.
@rarebeeph1783 if you can go further you haven't reached infinite yet.
If infinite can extend past infinite than the supposed empty set contains it, it meats the criteria x≠x.
@@SunShine-xc6dh You have reached infinite as soon as you exceed the finite. That does not mean you can't still exceed it by a greater degree.
Here's some food for thought: one of the axioms of Zermelo-Frankel (ZF) set theory, is that there exists one infinite set. If you accept that one exists, together with the other rules of set theory, you necessarily come to the conclusions I've been telling you. If you do not accept that it exists without some similarly powerful alternative, as in Peano arithmetic, you lose the ability to talk about real numbers or calculus.
Technically, alphabets are number that can represent communication. Everything is number.
my first guess was pi but then I thought that was too obvious and changed my mind to e
seven hundred and twenty?
how do you get quaternals? sqrt(-I) is just (-sqrt(2)+sqrt(2)I)
For quaternions we add new imaginary values and extend the definition i^2=-1 into i^2=j^2=k^2=ijk=-1. We know i,j,k aren't equal, because if they were, ijk would be -i, not -1. Thus, we get a basis for the quaternions
@@dacomputernerd4096 so how is i^2=j^2 if i=/= j? (=/= is the unequal symbol)
@@kales901 They are three different values. The implication is that since i^2=ijk, i=jk. However, similar logic means that since k^2=ijk, k=ij. And since k^2=ijk, k^3 = kijk, so k^2 = -1 = j^2 = kij, so j=ki. So there's not any real or complex number which acts this way and is also a root of -1, and so these don't behave as such.
The problem is you're asking the question from a complex number view; it's like asking how can i^2 be -1 when all squares are positive. It's just how quaternions are defined.
@@dacomputernerd4096
if i² = j², why can't we just root both sides and get i = ±j ?
@@xXJ4FARGAMERXx nope. That would lead to a contradiction (i=j or -j -> ij or -ij = j^2 -> k or -k = -1 -> k^2 = (-1)^2 -> -1 = 1) So i can't equal j or -j, and i^2 = j^2 must not imply that i=j or -j. In the quaternion world, that rule breaks down.
I dont actually know if square root is defined for quaternions. Note that on all my examples I only use addition, subtraction, multiplication, and division, taking care to multiply or divide on the same side. And squaring single values, but that's just multiplication
Looking into it, it does exist, but there are infinite roots of -1 so two things squaring to -1 only actually tells us that their magnitudes are 1 and they have a real part of 0
10:20 FYI the word is Infinitesimal, not "infant-esimal". 😄
Yes, technically, but literally no one in history has pronounced it like that. Or maybe I'm just "rationalizing" my own mispronunciation.
@@GlorifiedTruth I pronounce it infinitesimal. 🤷♂️
Nice!
8:23 I'd say quaternions, and even higher-dimensional 2^n-ions are more accurate & precise. And, better, because they can store more information. Yes, complex numbers are useful, but it's when you try to say/subtly imply that there's no practicality for higher-dimensional complex numbers, then it gets bad.
Higher-dimensional complex numbers (including ones defined in terms of arbitrarily large exponents to 2, such as the power 2^128) can be very useful in some contexts, including physics, neural networks & (more recreationally) thinking about higher dimensions of reality.
They really aren't practical. If you need more imaginary units, in 90% of cases it's just better to use tuples instead of octonions etc
@@methatis3013 Complex numbers eliminate the need for tuples altogether. (a,b) on the RxR plane can be represented as a single number, which is a+bi, on the RxI plane. (a,b,c,d) can be represented as a+bi+cj+dk on the RxI1xI2xI3 plane, and so on & so forth. It's just more elegant & less tedious to use complex numbers. And, octonions are, infact, a more compact way of representing an 8-tuple.
@@methatis3013 I find tuples quite impractical anyhow. n-tuples are annoying to deal with. There is no other way to express an n-tuple of numbers compactly.
I'm curious. What makes you think tuples or ordered pairs are more practical than complex numbers?
0:58 is 0 a natural number? I think it should be. In fact you later include it in the non-standard "W" whole number set, but why not the "counting numbers" without 0 and N with 0?
It's usually down to the convention. In the context of number theory, 0 is usually not counted as a natural number
@@methatis3013 So "natural number" and "positive integer" have no distinction. That's why I'm not fond of it, to be honest. My preference - and I know my preference matters a hill of moldy beans - is that the counting numbers, or whole numbers, denote 1, 2, 3, .... We don't count starting with 0, so "counting numbers" seems more intuitive, and then natural numbers could be distinct, describing quantities we could possibly have, including an empty set. I know it's all names, and I'm being silly about this.
@@Qermaq well people usually say "positive integers" or "non-negative integers" to avoid any ambiguity which can arise when saying "natural numbers". Again, it is a convention after all and there are as many standards as there are different opinions, which is A LOT when it comes to math. Even in my uni, 2 different professors use 2 different notations for the same thing in many instances. Don't think about it too much, it's just people being people
Calling it a limitless field of study is quite ironic
I did guess lol! pi is the first number that comes to my mind
Vsauce?
7:38 You could just say "the Cartesian product of R and R*i". It's not that hard.
AKA R[i]
Can we please start calling the Imaginary numbers Lateral numbers.
I personally call them spinny numbers, because they spin.
The reason they called imaginary numbers because of historic roots.
Also, they have no direct real and tangible representation to tangible objects. Like you can have 3 oranges. -12 degrees farenheit, pi litres of water, we can imagine those, we have intuition. But i? We cannot imagine i oranges, 2i degrees farenheit, 4i litres of water, what does it even look like? As if it only exists in mathematics and drawings with no real life representation.
Its the same reason why physicists are hesitant to add i until schrodinger. He forced his hand to add i in his calculations because thats the only way to make sense of it all. Proving that i is not as imaginary as we once thought.
Why are you late ?
I guessed 2763
Pi is actually a letter but ok
damn
10:03 Talking down to your viewers and trying to explain every concept you think is even slightly foreign to them is jarring, extremely so after a while.
I'd say a number is any element of an algebraic structure
That would make functions numbers as well!
@@RUclips_username_not_found correct!
Define "algebraic structure", and, what type? Are you specifically talking about, rings, lattices, modules, groups, or fields?
@@Gordy-io8sb I'm unsure about the definition of an algebraic structure but it's some sort of object containing sets and operations. All of your examples would be aplicable. I think this way of defining number would be the best because it's general in the way that all normally recognized numbers such as the ones in the video would be included, and it isn't as arbitrary as other definitions i've seen that for example would exclude functions.
@@soup1649 So functions, sets, matrices are numbers as well ? Ok, Fine.. be it. It is not like I have something to say on the matter.
12?
I hardly know her!
I'm sorry for saying this BUT:
1. Asking What is a number? And instead of defining one just describe different types of them makes a bad impresion. If you are really that smart you should know the difference. This looks like you don't understand what you're describing.
2. Your definition of p-adic numbers wasn't the best nor was the one of infinite ordinal nubers.
If there are readers interested in these topics I recommend searching these topics on youtube the frist two to three results are the best videos.
3. (added after editing) There is in fact a definitoin of numbers. It use empty sets. This video explains the definitoin well ruclips.net/video/dKtsjQtigag/видео.html
I don't want to be a hater of this channel I just can't stand when math isn't taught as it should be. With no talking about HOW when faced with question WHY.
WHAT IS i^^i?
WE NEED THIS
We have got this already bruhh...
@@namratashrivastava389 when?
As Matt Parker puts it, "it's about a fifth"
@@namratashrivastava389 when?
@@allozovsky bruh
Numbers, irrational numbers, transcendental numbers, complex numbers, variables as placeholder numbers, p-ardic numbers, prime numbers, et cetera blah blah
at the end i think numbers are... a unit.
"a unit"? Wtf is that supposed to mean?
@@samueldeandrade8535 if you have 5 fingers, fingers are a unit. if you run at the speed of 5km/h, km/h is a unit.
@@symmetricfivefold well ... all measurement starts with an unit. That's absolutely right. Then you may add and combine other units. Numbers were like that. But they are not "a unit". But, ok, in some sense, you are right.
GRRR cardinality isnt "size" its "the number of *types* of ways to organize everything in a set"
sorry this i sound like 🤓
That’s not cardinality.
I'm not sure what you mean. Two sets have different cardinalities iff one can't be injected into the other, implying that one has more elements than that other.
And in fact, there are uncountably many countable ordinals (types of ways to well-order a countable set). So by your definition, wouldn't any countably infinite set be uncountable?
e
7:58 you know someone doesn't know anything about Mathematics when he or she says "complex numbers originated to give solutions to equations like x²+1=0".
No need to ragebait people into giving you more attention dawg. And if you're serious on the unnecessary hate then thats honestly even more pathetic lmao.
@@antagonisticalex401 ragebait? What rage are you talking about? It seems that you are the one trying to get attention ... Also, "hate"? What's hateful about my comment? In my opinion, this youtuber doesn't know Math. There is nothing hateful about this.
I don't know how complex numbers originated. But it seems to me that his layman explanation is quite right... R is a field in which some polynomials don't have any root. We wanted a new field so that every polynomial has a root in it. Extension from R to C was algebraically motivated.
@@speakingsarcasm9014 nope. Complex numbers originated because of cubic equations that we knew had to have solutions, real solutions. But the formula discovered would make us deal sometimes with the square root of a negative number. It has nothing to do with "wanting a new field so that every polynomial has a root in it". Just to give some more info, a cubic equation like
x³+3px-2q = 0
has
³√u + ³√v
as one solution, in which
u = q + √(q²+p³)
v = q - √(q²+p³)
You can easily check that. But look what happens when we try to apply it to
x³-6x+4 = 0
Notice that x=2 is a solution,
2³-6×2+4 = 8-12+4 = 0 ✓
The equation has
q = -2 and p = -2
so we would have to calculate q²+p³,
q²+p³
= (-2)²+(-2)³
= 4-8
= -4
Using that in the formula,
³√(-2+√(-4)) + ³√(-2-√(-4))
should be a solution. What happened is mathematicians realised that if we treat √(-4) as
2i, with i² = -1,
then the terms inside the cubic roots can be calculate in the form
a+bi
Indeed,
1±i = ³√(-2±2i)
in such way that
³√(-2+√(-4)) + ³√(-2-√(-4))
= (1+i)+(1-i)
= 2
the solution we knew!!!
So, if we allow numbers of the form
a+bi, with i²=-1,
we could find solutions even if the term inside the square root was negative. This is NOT the same as wanting to give solutions to equations. The equations already had solutions, but the formula would only work to find them if we dared to calculate with complex numbers. And THAT's the origin of complex numbers.
You're welcome.
If you notice something wrong, what can easily happen in comments like this, tell me and I will correct it.
@@speakingsarcasm9014 just to complement my reply, the fact that C is algebraically closed over R, which means any polynomial equation with coefficients in R have a root in C, was something we found AFTER the definition of the complex numbers, not at all the reason why they were defined. It just so happens that C have this such property.
I'm the 666th viewer XDDD
77 views in 5 minutes you fell off
Bro asked what is a number and proceeds to talk about anything but question in the title
Screw this garbage
What do you mean, the entire runtime is discussing different types of number? In the absence of a single coherent definition (which I'm not sure even exists), a series of examples is the next best thing.
@@rarebeeph1783 which is exactly my problem.
Video is not called "types of numbers", it just goes around spitting fun facts