1 ^ ∞, It's Not What You Think

In this video, we explore the concept of 1 raised to the power of infinity. At first glance, it may seem like the answer should be 1, but as we dive deeper into the topic, we quickly realize that it's not that simple. We discuss the limitations of using limits to try and determine the value of 1^infinity and how the traditional definition of exponentiation doesn't allow for infinity to be used as an exponent. We also explore an alternative approach, using logarithms, to try and determine the value of 1^infinity. We show how logarithms allow us to rewrite exponential expressions as a power of the base, and how in this case, 1^infinity can be rewritten as log(1^infinity) = infinity * log(1) = infinity * 0, which is undefined.

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information. Viewers should always verify the information provided in this video by consulting other reliable sources.

[BriTheMathGuy]

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[Eric-jhmp]

As a physicist, I'm going to forget I ever saw this so I can still do... well a surprising amount of physics that relies on 1^infty = 1 lol.

[a.s.vanhoose]

I rely on the ‘come on man’ theorem on questions like this. The way it works is if there’s a difficult question in math you assert your opinion without backing it up in any way. And if someone challenges it you just say ‘come on man.’

[rednetherbrick]

What we learn in class: 2² = 4
Whats on the homework: 2³ + 3 = ?
The exam:

[Chris_]

The only sensible interpretation is that 1^oo = lim n->oo 1^n and that is 1.

[sanauj]

1:46 The reason why you get different answers is because both the bases are different numbers. They are not equal to 1, just two different numbers that are really really close to 1.

An exact 1 to the power of infinity is equal to one.

[maswinkels]

Sorry but this is wrong. 1 to the power infinity should only be considered indeterminate when we have f(x)^x where x goes to infinity and f(x) goes to 1, and f(x) is not equal to 1. In that case the issue is that we don't know if f(x) goes to 1 "fast enough" to cancel out the effect of the exponent, therefore the result is indeterminate. But if f(x) = 1, then we can prove by complete induction that the result is one.

[merimackara]

"its not what you think"
proceeds to say it is exactly that

[squeezy]

What I don't understand about analysis like this is the whole idea is to get mathematics consistent, correct? So we, like you said, can choose 1∞ to be whatever we want as long as it's mathematically rigorous and consistent. So let's say 1^∞ is 1, is there any other (potentially seemingly unrelated) mathematics that breaks with this definition? Because if there isn't, I don't see any issue with defining it this way.

The example used here where the limit as x -> ∞ of (1 + 1/x)^x = e might be a good counter-example to this point, but the issue I have here is that it doesn't really tend to exactly 1^∞, it's more tending to (1+)^∞, but I guess an argument could be made here that this should diverge to infinity, while it clearly doesn't - this is another issue though.

[draido-dev]

Based on my calculations, this video will be epic

[Emre]

1:35 is completely wrong. Those terms are not 0, they are slightly bigger than 0 and thus you have a slightly bigger number than 1 over infinity that's why it does not work. If you have a plain exact 1 over infinity, it's 1. You can't use normal calculations for special cases like that and then say that the definition is wrong.
Your calculations are wrong.

[degrees]

Also you can approach this like the 0^0 value is approached: if we want a^m a^n = a^(m + n) to hold and we allow ourselves ∞ + ∞ = ∞, then we have a condition 1^∞ 1^∞ = 1^∞, i. e. this should be a multiplicative idempotent. If we want it to be a real or complex number then we already have just two options left: 1 and 0. Now, we can make a couple of arguments to discredit 0 as the potential value, accurately enough to leave 1 as valid. Voilà!

[SomeoneCommenting]

I've seen so many videos about all these things lately, and what amazes me the most is that everybody in their channel talks like if they were the first ones ever trying to find an answer. All these things must be already defined by mathematicians long ago or they wouldn't be able to solve many problems.

[tymberrr]

I remember hearing this once and it will always be stuck in my head: Infinity is not a number, it’s a concept

[Dan-rius]

1:02 No, you cannot approach the question in a multitude of ways; these expressions are not the same as 1 to the power of infinity. This is very misleading.

[scarletevans]

2:55 an interesting reference here could be mentioning that X^Y is set of all functions from Y to X, then say about Von Neumann's definition of natural numbers.
Empty set is 0, while 1 = {0}, thus if we think about ∞ as the set Y with infinitely many elements, then 1^∞ can be interpret as {0}^Y, i.e. set of all functions from Y into {0}. However, it contains only one constant function that is always equal to 0 😄
The result is still not a number, but the same could be said about "1^∞", so it's an interesting interpretation of it 🙂

[angelmendez-rivera]

Okay, so before I start giving mathematical explanations, I need to get this out of the way: mathematics educators and communicators seriously need to stop talking about "indeterminate forms" as a legitimate mathematical concept, because I think they are doing a great disservice to people who are trying to get a deeper understanding of the mathematics of calculus and arithmetic. *There is no such a thing as an 'indeterminate form.'* No, in mathematics, there are two, and only two classes of expressions: those which are well-defined, and those which are undefined, or ill-defined (I take "undefined" and "ill-defined" to mean the same thing, as there ultimately is no coherent way for the two to be meaningfully distinguished). An expression like 2^3 is well-defined. An expression like 1^∞ is not. The latter is in the exact same category as 1/0, ln(0), 0/0, 0•∞, and so on. Please, let us start calling these things for what they are: undefined expressions, because that is all that they are.

An expression is just a string of symbols. We can take symbols to either be fundamental in signficance (i.e., the symbol by itself means something), or we can take a few symbols that, when combined into a string, that string has fundamental significance (e.g., the string "sin, " which is comprised of the symbols 's, ' 'i, ' 'n, ' symbols which have no significance of their own, but when combined into the string "sin, " they form a string with some actual mathematical significance). When you combine multiple fundamental, 'atomic' strings into an expression, the significance of this composite expression depends, in some way, on the significance of the fundamental symbols, and how that significance emerges from the combination is dictated by rules of notation. The expression 1^∞ is an expression with three atoms: the atom "1, " the atom "^, " and the atom "∞." We need to discuss how these individual symbols are defined, before we can say anything about 1^∞. What is the symbol 1 defined as? Well, we are secretly working with some fundamental type of mathematical structure obeying certain axioms formulated in some type of logic. This structure is called the ordered field of real numbers, and these are ultimately just some abstract objects that do nothing except satisfy certain properties we call 'axioms, ' properties that are meant to be understood rules for how these objects interact, and these objects we take as being primitive or fundamental in this context: they are not defined in terms of anything else, they kind of just "are" for the sake of being alone. These real numbers satisfy one axiom, that there exists some y, such that for all x, x•y = y•x = x. This axiom is one of the defining properties of •, a binary operation, and when we use the other axioms, we can prove formally that this y object satisfying the above property is unique. Since it is unique, we give it a name. The symbol "1" is that name. So, 1 is defined by the property that it uniquely satisfies: that 1•x = x•1 = 1, for all x. That is one symbol out of the way. Now, what is ^ defined as? It is a function, specifically a function on the Cartesian product between R+ and R, with R as the codomain. R stands for the ordered field of real numbers, the fundamental structure we are interested in studying, and R+ refers to those real numbers which satisfy 0 < x, they are called positive real numbers. ^ is defined by the property that x^1 = x, that if x < y, then (1 + 1)^x < (1 + 1)^y, and that x^(y + z) = (x^y)•(x^z). Notice how, in turn, this requires understanding the definitions of < and +, which are all defined by axioms themselves. Everyone here intuitively understands how these objects work, so I will not bother with every detail, but the point is, it matters to know where the foundations of the discussion lie.

The one symbol here that is troublesome here is the symbol ∞. This symbol is technically well-defined: it is the maximum of the Dedekind-McNeille completion of the real numbers, which is a mathematical structure that extends the real numbers in a very specific way. However, the problem is that the way the symbol is defined, it does not for any arithmetic. There is no mathematically consistent way to extend to the domains of + and • as functions to include ∞ in the domain, and continue having well-defined expressions. We have tried, but because of the axioms these structures are defined by, it can be proven that there is no way to make it work. Things like ∞ – ∞ and 0•∞ will always cause problems. So, arithemetic expressions with ∞ are *undefined.* They are *undefined, * because the way mathematical definitions work makes them a very specific kind of thing. Mathematical definitions have to make reference to mathematical properties: to well-defined mathematical concepts, and we have to ensure that the definition does not secretly hide a contradiction within it (which makes it incoherent), and we have to make sure that whatever properties are being referenced are actually uniquely satisfied if the definition is meant to denote a particular object, and not an entire class of objects. Since the axioms make it impossible for ∞ to be in the domain of any suitable extensions of +, •, and ^, the objects that satisfy the properties that the expression 1^∞ would make a reference to simply do not exist. In other words, the expression does not refer to anything at all! It has no actual significance, and this what we mean when we say the expressions are undefined. Definitions in mathematics are not an arbitrary endeavor, like everyone likes to say. There is some degree of arbitrariety, yes, but only after you get past many walls of restrictions that must be satisfied.

Now, what is the funny business with indeterminate forms? Why do people insist to interpret 1^∞ as being a "shorthand expression" for lim f(x)^g(x) (x —> p), specifically in the cases that lim f(x) (x —> p) = 1 and lim g(x) (x —> p) = ∞ (whatever this expression even is supposed to mean)? The problem here is that these two notations actually make reference to two completely unrelated mathematical topics that are different. We need to stop making an association and pretending these symbols are synonymous. Do we use the symbols floor(0) and lim floor (x —> 0) to mean the same thing? No, we do not, and the questions "what is floor(0)?" and "what is lim floor(–x^2) (x —> 0)?" have completely different answers. floor(0) = 0, while lim floor(–x^2) (x —> 0) = –1. Incidentally, lim floor(x^2) (x —> 0) = 0, which is not equal to –1. Does that mean floor(0) is "an indeterminate form"? Hmm??? I ask this in a completely serious fashion, just to highlight how, um, inaccurate (to put it politely) the notion of "indeterminate forms" is. No, floor(0) is not an indeterminate form. No one thinks of it as one. We all know that floor(0) = 0, and if I ask you "what is lim floo(x) (x —> 0)?, " then you would, correctly say, that the expression is undefined/ill-defined. Alternatively, you would say that the limit being referenced by the expression does not actually exist: there is no mathematical object that we can call "the limit of floor(x), x —> 0, " because the definition is not satisfied by any objects at all. So, despite this being the case, why are people still insisting on this indeterminate form concept? I do not understand it. The question "what is 1^∞?" is a completely different question from "if lim f (x —> p) = 1 and lim g (x — p) = ∞, then what is lim f(x)^g(x) (x —> p) equal to?, " and the two questions have different answers. For the former question, the expression given is undefined. *Why* is it undefined? To answer that, you have to get into a discussion of Dedekind-McNeille completions as mathematical structures, and why their properties do not allow for + and • as operations. The latter question, though, has a simple answer: it depends on the properties of f and g as functions. Do you see how these are different questions? One question is about arithemetic, abstract algebra, and about mathematical structures. The other question is about functions of real numbers and their asymptotic behaviors. Completely unrelated topics. So, why does the education system insist in treating them as the same question and as the same topic? I have no idea why, but we need to stop doing it. And, yes, students do need to understand that knowing lim f (x —> p) = 1 and lim g (x —> p) = ∞ is *not* sufficient information to determine what lim f^g (x —> p) is. But using the language of "indeterminate forms" to teach this is unhelpful, and very misleading. Firstly, it gives people the impression that being "indeterminate" is somehow fundamentally different from "undefined." Secondly, it reinforces the already too common misconception that to evaluate limits, you just "plug in" and see what happens (not how limits work at all). Thirdly, it leads to people developing a bunch of mystical, crackpot nonsense like "ah, so this number/expression is equal to all numbers at once, which proves we are all one [insert more mysticism]." Fourthly, it is just an unnecessarily obfsucating, convoluted way of telling the student "we do not have sufficent information to answer the question." Fifthly, whenever we bring up limits to answer questions like the value of 1^∞, we do a disservice by actually failing to answer the question, and instead, answering a different, unrelated question.

[AM_-wghj]

The limit of q ^ n when n goes to infinity depends on the value of q
q = -1 undefined since it's an oscillating function like sin and cos
-1 < q < 1 it's 0
q > 1 it's infinity
q = 1 it's actually equal to 1 but only because it's the variable n which is part of Z

Assuming the value is exactly 1 and not (1+) like in that example, it's undefined if the variable is a real number and 1 if the variable is from Z

[venkataseshasai]

X^infinite= 1 if X is absolute 1.But if X is trending to 1 then, X^infinite is not equal to 1.

[wscamel]

2:32 - that is really interesting how we have gone from something obvious to something impossible