The 7 Indeterminate Forms that Changed Math Forever

[@dibeos]

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[@lagomoof]

It might be important to clarify that while 0 and 1 are considered numbers, infinity is not. Also, while "1/0 = infinity" and "1/infinity = 0" can be useful shorthands to those aware of the more correct limit definitions, it's misleading to those who aren't, and can be the cause of errors either way. I felt like they were thrown around a bit too carelessly at the beginning. Those shorthands also wrongly imply that "infinity·0 = 1" which was neatly sidestepped at that point of the video.

[@LogosNigrum]

There are systems in which infinity is a number, though, like in the hyperreal numbers.

[@ianfowler9340]

@@LogosNigrum Yes. But for a lot of students of Calculus/Limits who are encountering the concept of infinity for the first time, the concept of hyperreal numbers or the point at infinity is way beyond their pay-grade. IMO, best to leave that for a 3rd year university course in Real Analysis.

[@davidwright8432]

You saved me having to make the same points! Thanks.

[@Alice_Sweicrowe]

​@@ianfowler9340KZbin isn't Uni.

[@AlbertTheGamer-gk7sn]

This is because indeterminate forms are true for ALL values of x. Not just integers, not just rationals, not just reals, not just complex numbers, but the ENTIRE universal set. There are 3 types of functions: 1. One-to-one function: These functions are STRICTLY increasing or decreasing throughout its domain and has no first-order critical points besides the endpoints if they exist. The inverse function is defined throughout its entire domain. 2. Piecewise inverse function: These functions are not one-to-one, but an inverse exists for a PORTION of the function. These functions have at least one first-order critical point. 3. Singular function: The term "singular" in mathematics refers to anything that are not invertible. Because of that, a singular function is a function that doesn't have an inverse. These functions exist as annihilation functions, where all values in its domain get "annihilated" to a single value as its range. Because of that, a function is singular if and only if (iff) its derivative is 0 throughout its domain, similar to how a singular matrix has a determinant of 0. Let's say you have this function f(x)=1. This means that for every value of x inserted, the function returns 1 every time. Therefore, this graph takes the form of a horizontal line at y=1. If we say we want to invert f(x), we need to switch the domains and ranges, and since this function annihilates all values of x and returns 1, meaning the domain of the function is all numbers, but the range of the function is 1, if an inverse exists, it will then map 1 for the domain and, uh-oh, we have a problem here. A function operates like a time-distance relation: You can be in the same position at 2 different Planck Times, but you, at the same Planck Time, cannot exist in 2 different locations. If f(x)=1, f^-1(x) will have f(n!=1) not making sense, as f(x) cannot not equal 1. However, f^-1(1) maps out all real numbers, meaning that it wouldn't be a function, as at the same time the function simultaneously exists everywhere. This means there doesn't exist any continuous piece of this function that makes sense; it only makes sense for that one point, and even for that point, it violates the basic rules for functions. Graphically, to invert a function, you need to reflect it over the line f(x)=x, and reflecting a horizontal line gives you a vertical line, which violates the vertical line test. Finally, using singular functions, we are able to introduce a concept called indeterminate forms. As you can see, for the function f(x)=1, f^-1(1) gives an indeterminate form due to it being a vertical line at x=1. Given a singular function f(x), f^-1(f^(x)) is indeterminate. Even though applying the inverse function after applying the function should result in the identity function f(x), for annihilation functions, it all has lots of paradoxes. An indeterminate form comes from basic algebra, where a declared variable has no value assigned to it, as variables are MEANT to be indeterminate until you assign a certain value of it or a transformation of it. For example, x is indeterminate. With n variables, you need n equations to make the values determinate. Some equations are degenerate and continue to evaluate an indeterminate form, such as x=x, as it is true for all x. Using that, we can prove the 11 indeterminate forms are indeterminate using these formulas: 0x = 0 for all Aleph-Null x, so 0 / 0 is indeterminate. 0 * infinity and infinity / infinity are variants of 0 / 0, as 0^-1 = infinity. infinity + x = infinity for all Aleph-Null x, so infinity - infinity is indeterminate. The infinitieth root of x is 1 for all Aleph-Null x, so 1^infinity is indeterminate. 1^x = 1 for all Aleph-Null x, so log_1(1) is indeterminate. 0^x is either 0, 1, or infinity, so log_0(0), log)_0(infinity), log_infinity(0), and log_infinity(infinity) are all indeterminate. Using these equations, we can prove which forms are NOT indeterminate, to see how many indeterminate forms are there. Calculi operate on indeterminate forms, as a derivative is equal to 0 / 0, and integrals are 0*infinity, so we can see how many calculi are there. For example, the product/geometric calculus consists of the product derivative, which is limit as h goes to 1 of log_h(xh/x), which is log_1(1), and the product integral, which is 1^infinity. Infinity is a fixed-point of X = X+1, so when dealing arithmetic with infinity, we use a variable x to denote the fixed-point, which is useful on checking if forms are indeterminate. Case 1: 0*infinity, x=0*infinity=0+0+0+0+0+...+0+0+0+0=0+(0+0+0+0+...+0+0+0+0), so x = x+0, and x = x. Indeterminate. Case 2: 1^infinity, x = 1^infinity = 1*1*1*1*1*...1*1*1*1*1=1*(1*1*1*1*...1*1*1*1*1), so x = 1*x, and x = x. Indeterminate. Case 3: 1^^infinity, x = 1^^infinity=1^1^1^1^1...1^1^1^1^1=1^(1^1^1^1^1...1^1^1^1^1), so x = 1^x. This only is true if x=1, so 1^^infinity is NOT indeterminate. However, if x=infinity, it equals an indeterminate form, so it is considered indeterminate in the complex world. Case 4: 0^infinity, x = 0^infinity = 0*0*0*0*0*...0*0*0*0*0=0*(0*0*0*0*...0*0*0*0*0), so x = 0*x, and x = 0. Not indeterminate in the real world. Case 5: 0^^infinity, x = 0^^infinity = 0^0^0^0^0...0^0^0^0^0=0^(0^0^0^0^0...0^0^0^0^0), so x = 0^x, but NO SOLUTIONS EXIST. This is because 0^^x = 1 if x is even, and 0 if x is odd. Same thing goes with 0^^^x, 0^^^^x, and 0{n}x. In Boolean algebra, if a function has no solutions, it evaluates to false. If it is an indeterminate form, it evaluates to true. If it is only true for a value or a finite set of values, it returns that value or set of values. Therefore, 0/0=0*infinity=infinity/infinity=infinity-infinity=infinity^0=1^infinity=log_1(1)=log_0(0)=log_infinity(0)=log_0(infinity)=log_infinity(infinity)=True Using this, we can derive a Boolean algebra from equations: Union of solutions: ∪ Intersection of solutions: ∩ Non-solutions: ∥ Exclusive solution: x ⊘ y = x ∩ ∥y "Xor" solution: x ⊗ y = (x ⊘ y) ∪ (y ⊘ x) Here are some rules derived from this: Identity laws: x ∩ I = x, x ∪ ∞ = x where I ∈ indeterminate forms and ∞ = n/0 where n ≠ 0 Domination laws: x ∩ ∞ = ∞, x ∪ I = I Idempotent laws: x ∩ x = x, x ∪ x = x Commutative laws: x ∩ y = y ∩ x, x ∪ y = y ∪ x Associative laws: x ∩ (y ∩ z) = (x ∩ y) ∩ z, x ∪ (y ∪ z) = (x ∪ y) ∪ z Distributive laws: x ∩ (y ∪ z) = (x ∩ y) ∪ (x ∩ z), x ∪ (y ∩ z) = (x ∪ y) ∩ (x ∪ z) Negation laws: x ∩ ∥x = ∞, x ∪ ∥x = I Double negation law: ∥∥x = x DeMorgan's Laws: ∥(x ∩ y) = ∥x ∪ ∥y, ∥(x ∪ y) = ∥x ∩ ∥y Using this, we can tell: For any Boolean algebra, we must have a correctness, an incorrectness, an addition, a multiplication, a negator, and an excludor. Therefore: Logic: truth ⊤, falsity ⊥, and ∧, or ∨, not ~, implies -->, biconditional , xor ⊕ Set theory: Universality U, Nullity ∅, Intersection ⋂, Union ⋃, Compliment C, Exclusive \, xor ⊕ Arithmetic Boolean algebra: Enabled 1, Disabled 0, Times * ∏, Plus + ∑, Prime ', xor ⊕ Equations: Indeterminate I, Infinity ∞, Intersection of solutions ∩, Union of solutions ∪, Non-solution ∥, Exclusive ⊘, xor ⊗

[@justyceleague698]

Write a book if you haven't already this is great

[@attackofthejiggli]

Damn man went way too hard for a KZbin comment

[@methatis3013]

​@@justyceleague698a lot of this is nonsense and some of it is straight up wrong. A one-to-one function does not need to be strictly monotone. There are uncountably many bijections that aren't monotone even over a single interval. Furthermore, given the definition of a "piecewise invertible" function, any function satisfies that definition, since you can restrict them all to a single point. OP's comment has A LOT of errors in it, it's pretty much pure misinformation

[@attackofthejiggli]

@@methatis3013 Damn man went too hard correcting a KZbin comment

[@simonwillover4175]

1:40 and that's why call x/0 undefined for real xs, complex xs, and even matrix xs. Becuase, (x/0)'s value is not real, it's not complex, and it's not a matrix. It would belong in a different number system if you defined it, and in that number system, there would exist some y = x/0, where y*0 != 0, but instead y*0 = x. This makes no sense in any common number system, and the few mathematicians who do play around with the number systems that allow this have not found a use for it yet. It's not a fact that you can't divide by 0. It is a fact that defining the value you get from dividing by 0 is useless. Hopefully, in the future, someone will find a reason to divide by 0, but no one has found it yet. For now we will be stuck with undefined and inifinity being the only answers (fun fact: modern computers define 1/0 as infinity).

[@hirotempest2647]

Idk what I had one day, but I made some weird shit that gave me as a result that: "infinity equals 1" I need to find that paper, but, yeah, I was probably drunk or some shi

[@dibeos]

@@hirotempest2647 hahahahah that’s the best comment we got so far 😂 please find that sheet of paper. I’m curious

[@ValidatingUsername]

You should do a video on base 0, base 1, and 0*ln(0). 2:15 The additive identity kicks in too.

[@markwrede8878]

Nature works upon slopes, best characterized by certain primes. These are quickly exhausted under multiplication, however, slopes more flat than multiplication can produce may be characterized by slopes of immense Pythagorean triangles where the hypotenuses are but one or two integers larger than the longest leg, i.e. b+1 or b+2. While b+3 is congruent, no integers come to satisfy it.

[@ianfowler9340]

You have to very careful when you say that the limit "= +inf". This is simply an agreed upon shorthand notation used as a stand in for " increases without bound" and that should be made clear before we start using an = sign in this context. When we write lim[f(x)] = +inf, the equal sign, in this special case, does not have the same meaning as lim[f(x)] = L. In such a case the limit does not exists, but the symbol +inf indicates the behaviour of the function as x -->a. Expressions like: "it's unclear what the limit of this infinity is" and "the x becomes infinity" and "it's unclear how fast each infinity is reached" are not correct. Infinity is not a destination. It cannot be reached. A real variable cannot become infinity. A limit cannot equal infinity - it can tend toward infinity. Again, "=" +/ inf is just a short hand stand-in notation. I get what you are trying to explain in the video and I applaud your efforts. We just all have to be careful with our exact wording and be clear about what symbols we use actually mean.

[@livedandletdie]

WE do not write Lim of x = +inf, we write lim ƒ(x) → +inf → is approaches. And infinity is a perfectly reasonable argument for limits, and integrals and derivatives... the amount of times l'hospitâl has made me make sense of 0/0 inf/0 0/inf or 1/inf or 1/0 or 0/1 or inf/inf for that matter, is in the hundreds. They're not well defined, but we can still use them to find well defined values.

[@ianfowler9340]

@@livedandletdie I think we are talking about 2 slightly different things. We do indeed write "lim[f(x)] = +inf" all the time in place of f(x) ---> +inf. I'm not suggesting we stop using either notation. And I never contested the value of LH's rule in that it helps us make sense of indeterminant forms - by the hundreds. The notation :x --> +inf also ferfectly fine if we understand that it means "as x increases without bound". Newbies often get the worng idea when they see x ---> +inf as they sometimes interpret +inf as a "place" we are approaching. But I still stand by statements like "replace x with infinity" or "then x becomes infinity" or "it's unclear how fast each infinity is reached" are simply not correct as no real number ever "becomes" infinity.

[@ONRIPRESENCE]

This video further motivates my interest in linguistical mathematics and descriptive linguistics.

[@dibeos]

@@ONRIPRESENCE really? What do you say say? Can you expand on it?

[@ONRIPRESENCE]

I'm basically just using descriptive linguistics to map out the various ways of labeling which technical terms in the sciences are abstract and how to use abstractions to better understand their differences with things that are specific. Similar to how the term "indeterminate" is used in mathematics as an abstract concept of sorts, rather than something with a specific value.

[@glitchyfruit2503]

lim n->♾️, 1^n = 1 lim x->1, lim n->♾️, x^n = indeterminate{0,+♾️} Love math

[@karlbjorn1831]

9:09 would that be 0 + inf ? I don’t see how that would form 0*inf

[@dibeos]

@@karlbjorn1831 you are correct, it would be 0 + inf. We meant that if you analyze the reciprocal of the original function, which was in this case x/(1+x^2)=1/(1+x^2) * x, when x goes to infinity, we get 0*inf. it’s just another way of seeing the indeterminate form. The point here is that it’s really easy to manipulate the expressions in order to transform one indeterminate form into another, which also emphasizes their absurdity

[@syphon5899]

Amazing animation

[@dibeos]

@@syphon5899 thanks! 😎

[@clympsarchery]

Theres another one! limit as x aproaches infinity of (-1)^x

[@dibeos]

@@clympsarchery yeah, this is an undefined limit. But it is not considered an indeterminate form, because it just oscillates. It is not a matter of “which function tends faster to zero or infinity”

[@Alice_Sweicrowe]

Since Graham's number is so large that it can't be physically written we could potentially use that in place of Infinity. Or, to be cheeky, we could use 10^88, which is the approximate number of atoms in the universe. Some would argue that that would be limitless as well. We'll have to wait until somebody can get there to find out.

[@TimTuinman-e2d]

pretty sure 8 divides 8 evaluates to true

[@dibeos]

@@TimTuinman-e2d yes 😎👌🏻

[@Alice_Sweicrowe]

You know, a sign for infinitesimals would solve a couple issues here.

[@dibeos]

@@Alice_Sweicrowe I agree with you. A sign for infinitesimals (distinct from the of infinity) would be useful

[@Alice_Sweicrowe]

@@dibeos They already used the perfect symbol for it as well; •

[@methatis3013]

It mostly wouldn't. Infinitesimals are all the same size, i.e. ε/ε = 1. However, if we have 2 functions f and g such that f->0 and g-> 0 as they approach the same point, the limit of the resulting ratio f/g can be any real number, ±♾️ or it can simply be undefined. It depends on specifically how f and g are given

[@Alice_Sweicrowe]

@@methatis3013 Yeah, a short hand for both. Linguistics helps. Maths is such a confusing natlang/conlang. 🥲🙃🫠

[@bhbr-xb6po]

3:05 ooooofff

[@SobTim-eu3xu]

New video, new like)

[@ImpossibleEvan]

Small correction n/0=±∞ not ∞ (where n is non-zero value)

[@dibeos]

@@ImpossibleEvan yesssss thanks!!!! 😎

[@ValkyRiver]

Oh, that’s unsigned infinity (from the projective number line)

[@shadowbirdz8685]

Mathematics seven deadly sins

[@dibeos]

@@shadowbirdz8685 oh yeah 😎🤟🏻

[@petohunor6737]

Infinite arithmetic is nonsense.

[@dibeos]

@@petohunor6737 really? Why?

[@ethanbartiromo2888]

There’s so much stuff in this video that seems like you guys are trying to explain something that you don’t TRULY understand, because you’re just giving examples and using quite bad notation. I believe you guys believe these things are true because of these examples, but don’t truly KNOW that these things are true. I know this is for non math people, but would you rather have people tell someone else something they heard, but have no clue why, or would you rather have people be able to back it up? If you heard someone say something that a politician said, but had nothing to back it up, then would you believe them just because they said that they know it’s true because so and so said it?

[@danielreyesalarcon1891]

This is just individuals doing their best to show what they know. They may not Be as eloquent as others but they do a fair job and therefore you should be easy on them.

[@victoramezcua4713]

For God's sake, DON'T write 1 + 1 = 2 + 1 = 3 + 1 = 4 + 1 = ... as stupidly is taught in elementary school

[@dibeos]

@@victoramezcua4713 my bad, I didn’t know that by writing it somebody would actually be tempted to believe that 1+1=2+1

[@victoramezcua4713]

It's not that I or other discerning person who knows some math would believe that 1 + 1 = 2 +1 =... it's true. Teaching math to Non- math majors in college made me realize that this incorrect way of writing pseudo equalities in elementary school persists in a pernicious way, all the way to college for people who for the most part dislike math. So why perpetuate this, considering that your video is trying to illustrate math principles?

[@dibeos]

@@victoramezcua4713 you are right! We will not do it anymore. Thanks for helping us to improve! 😆

[@crigsbe]

Please, just learn to reason like a mathematician. You only present a plausibel explanation for n x 0 = 0. This is not proof at all !!! Assume n to be any chosen number. Now n x 0 = n x (0 + 0) = n x 0 + n x 0 therefore nx0 = nx0 + nx0 Subtract of both side nx0 and you get 0 = nx0. Reasoning: The first = mark is valid, because 0 is the neutral element of addition. That is: 0 + y = 0 for any y. Here we have y = 0 and it follows: 0 + 0 = 0. The second = mark is valid, because the multiplication is distributive over the additon, that is: n x (0 + 0) = nx0 + nx0. The following = mark is valid, because each number has an inverse element corresponding to additon, that is: The number plus it's inverse add up to the neutral element of additon. Therefore nx0 + (--nx0 ) = 0. It follows: The left side is 0 and the right side is nx0 of the = mark. Summary: n x 0 = 0 for any chosen number n. QED

[@idotestbottler9706]

Why do you randomly change gender

[@dibeos]

@@idotestbottler9706 hahahah why? Do we look alike?

[@TheHebrewMathematician]

The video was great! But in the last section when you talked about 0⁰ what you said isn't true. By definition we know that 0⁰=1, you may argue that there are limits which go towards an expression of the form 0⁰ and they're value isn't 1, but it doesn't mean that 0⁰ is undefined.

[@fullfungo]

0° is NOT defined. It is very rare to see a textbook define it to be 1, because of the exact problems shown in this video. Defining 0°=1 is less accepted then, say, 0!=1. It is just a preference of the author, which you seem to share, even though it is not universal.

[@TheHebrewMathematician]

@@fullfungo I don't know where you're from but *generally* the US doesn't use "common" notation because the education there is absulot trash in math and cs. Now for the definition, we know that by definition for sets A,B follows that |A| ^ |B| = |B → A| So if A,B are both the empty set then 0⁰ = |∅ → ∅| = |{∅}| = 1 And the same argument also works for n! = |{f : [n] → [n] | f is a bijection}|

[@fullfungo]

@@TheHebrewMathematician “we know that by definition […]” Where did you get this definition from? I haven’t personally seen it in any math text books I had. I would like to see the context where it is defined.

[@Dent42]

@@fullfungo Defined recursively, any number to the power of 0 will equal 1, because 1 is the empty product

[@NothingOnceTwice]

​@@fullfungoI know that its definition is not agreed upon and can vary but it is definitely NOT rare to see it in textbooks and is used more often than leaving it undefined

[@ravenecho2410]

Inf - inf = (-inf, 0, inf) + c It just does 1/0 = +/- inf It just does 0 / 0 = c It just does C/inf = 0+ +/-0*inf = +/-1 sign(c) = sign(0)*inf it just does

[@justyceleague698]

Nope because that would lead to the contradiction of all numbers being equal to eachother and that's just not possible

[@Shadramelk]

I have a bone to nitpick with your video setup - the depth of field is too small, and the earrings keep swinging in and out of focus as you speak. It feels incredibly annoying in an otherwise interesting video

[@googleyoutubechannel8554]

I've never been convinced that 'infinity' offers any benefit to any type of algebra, system, or model, over just having a concept of 'incomparably large in relation'. I have a feeling if math weren't developed via handwritten quill and blackboard scratching, half of these symbolic concepts wouldn't have been given the primacy they are in math. An understanding of information and processing would have staved off this culture of symbolism and all the arguing that went into number theory, 'proofs' and arguing over who's incomplete symbolic system is better.

[@dibeos]

@googleyoutubechannel8554 the infinite and the infinitesimal are definitely more than symbols

[@arthuuuuuuu]

Maybe actually studying math would convince you.

[@marigold2257]

I did not think someone could have an opinion this incomprehensibly stupid but you did. Literally take one calculus class or one set theory class or literally any class that goes beyond basic algebra and you will see the utility of infinity, actually now that I think about it you specifically might not since your mind seems so closed to the idea anyway but anyone willing to accept new ideas would probably get it.

[@zackbuildit88]

You don't know what math is then. Just, like, that's it. I hope this is rage bait because math, *real* math, *IS* proofs. Real math *IS* logic and extrapolation. Real, actual math, that mathematicians have value in, *IS* the abstract. If you don't include that in math you *are not doing math at all.*

[@nzuckman]

Projective geometry my dude

[@perpetgholl5742]

the infinite is not a value. that's why everything szid in this video, is BS.

[@mosescheung5794]

correct, that's what everything in this video comes about: infinity is not merely a value but it's a limit outside the most used number system.