Im studying calculus this year and one of the lectures included undefined values (*forbidden and unwanted* not my words btw). These included:
1. 0/0
2. oo – oo
3. oo/oo
4. 0*oo
5. 0^0
6. oo^0
7. 1^oo
All of these are extremely weird to me and I don’t really understand them, but the one that strikes me the most is the last one. As a former math competitor and regarded as “gifted” in math, I feel stupid not being able to comprehend this, but most importantly it shatters my belief that math can explain everything and that is has all the answers.
I don’t see infinity as a really big number, I understand it as a concept, and what confuses me the most is seeing infinity treated as number. 1^oo for me doesn’t make any sense. It seems mathematically absurd. Infinity isn’t number, it’s a quantity that can’t be measured. And if it is treated as a number shouldn’t 1^oo = 1?
How come these are “undefined”? Someone please answer, Im losing my mind over this. All explanations are welcome, shallow or deep.
edit: to clarify oo = infinity
In: Mathematics
Not a math expert, but I believe that is because elevating 1 up to infinity would be a neverending process. Infinity is not a number, nor a quantity, it is something that has no end. And as such, you would keep elevating it well beyond the end of time, without reaching a conclusion
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I found [this answer](https://math.stackexchange.com/a/10493) on StackExchange that explains it quite elegantly.
To summarize, the actual fixed expression 1^∞ *will* converge to 1 if taken as lim x-> ∞ (1^(x)). But the thing the math book is warning you about are expressions that contain multiple parts that look like they would *become* 1^∞ when their limits are taken in pieces.
One of the examples given is this expression:
(1 + 1/n)^n
What’s the limit of this if n -> ∞? If you take the part inside the parenthesis on its own, the fraction tends to 1/∞ = 0, so the expression becomes 1. Taking this entire thing to the *n*th power as n->∞ gives us the form 1^∞ . Does this equal one? No. In fact, the limit of this expression as n->∞ is the constant *e*, of all things! Weird, right?
The thing is, if you have two functions, f(x) and g(x), composed together like f(x)^g(x) , where x->∞ leads to f(x) -> 1 and g(x) -> ∞, the limit of the composition could be *any number of things*. It all depends on what f(x) and g(x) actually are. They all *look* like they go to 1^∞, but they all behave differently. That is why 1^∞ is indeterminate form.
To think about it another way, if you have an expression that can be split into two pieces, f(x) and g(x), those two pieces can approach their limits at different “speeds”. For most limits, this effect doesn’t really matter, one of those pieces will likely be in a position where its “speed” is so much faster than the other that the effect of the other function gets completely nullified. For these, you’re allowed to take “naive” limits (sub in ∞ for all the variables and do “algebra” with it, treating divisions by ∞ as zero) of expressions. But if we get constructs where those speeds actually matter, the final limit could end up being basically anything depending on what those speeds actually are relative to one another. So naive approaches to evaluating limits that completely disregard this kind of nuance will almost always fail to arrive at the correct answer. Any expression that reduces to one of the “indeterminate forms” after a naive limit is taken is one of these expressions. This list of forms is warning you that, “hey, the speeds actually matter here, we need to use an approach that takes this into account”. Methods like L’hopital’s rule will allow you to do this for some expressions. That’s what all those funny calculus rules are for, they’re special methods that take these fighting “speeds” into account when calculating the limit. Any expression that becomes 1^∞ after one of these naive limits happens to fall into this territory, needing a special rule to calculate correctly.
∞ is alt-5 or opt-5 on a mac.
It seems like you have a perfectly good explanation in your hands already: ∞ isn’t a number, and operations like addition, multiplication, and exponentiation require two numbers as inputs, and give back a third (not necessarily different) number as a result.
Treating ∞ as a number is sloppy practice: you should always be trying to write lim (x » ∞) of 1^x instead of 1^∞. (» is not the right glyph, but I can’t find right-arrow on my keyboard)
Meanwhile, lim (x » ∞) of 1^x is perfectly well defined: it’s 1. If you have a sequence of real numbers that tends toward infinity, for any difference ∂ you care to name (edit: ∂ > 0), there is a number *n* such that, after *n* elements of the sequence, the difference between 1^a (where *a* is from your sequence) and 1 is less than ∂. Specifically, that distance is zero, because 1 raised to any real number gives 1.
To be more precise, we’re actually talking about the limit of 1^n as n approaches infinity.
Perhaps the easiest way to understand this would be to consider the concept of a left-hand and right-hand limit. That is, we choose something that bounds our limit arbitrarily close on either side.
If you’ve got the limit of 2^n as n approaches infinity, it’s pretty easy. The limit would be bounded by (2-d)^n and (2+d)^n, where d is some arbitrarily small number – and both of those evaluate to the same limit as 2^n itself for very small values of d.
But what happens when we use (1-d)^n and (1+d)^n ? In the first case, our limit is zero (no matter how small d may be). In the second case, our limit is infinite (again, no matter how small d may be). So all we can really say is that the limit of 1^n as n goes to infinity is somewhere between zero and infinite.
Sidetrack: if you say “it shatters my belief that math can explain everything and that is has all the answers” then you should take a look at Godel’s incompleteness theorem. No formal system can both be complete and consistent. Math isn’t as all-encompassing as you might think.
Algebraically, these all contain an instance of 0/0. Which is bad.
We can say that 1/0, 5/0, -8/0 etc are all equal to infinity and everything is totally fine and consistent and works out (though, you do need +infinity = -infinity for this to be nice). Dividing by zero, in most cases, actually causes no issue and using limits is a way to make sense of this in a precise way without having to actually consider infinity as a point.
The issues pop up when you divide zero by zero. All of the “proofs” of inconsistent things involving division by zero end up happening because they try to divide zero by zero somewhere, not simply because they are dividing b zero.
All of these expressions use 0/0 in them, so the are indeterminate, since there isn’t really any nice, consistent way to assign a value to 0/0 – either trivial, hyper complicated and unhelpful, or leads to contradiction.
1. 0/0
2. oo – oo = (1/0) – (1/0) = (0-0)/0 = 0/0
3. oo/oo = (1/0) / (1/0) = 0/0
4. 0*oo = 0*(1/0) = 0/0
5. 0^0 = A^(oo*0) = A^(0/0) – for any 0<A<1
6. oo^0 = B^(oo*0) = B^(0/0) – for any B>1
7. 1^oo = A^(0*oo) = A^(0/0) – for any A!=0
For any of these to have a value, even if it is infinity, you need to make sense of 0/0, which cannot be done.
You can take one to an infinite power, you just can’t call it infinity.
The problem with “infinity” is that it doesn’t fall on the number line. It’s defined as the extreme limit that the number line approaches, but never actually reaches, and because of that, it’s not a number, in and of itself. That’s why you can’t use it in operations like exponentiation. To do that, you’d need a different way of thinking about numbers: something that can handle actual numbers which are infinitely large.
*This system exists*. They’re called surreal numbers, and they include values which are infinitely large or infinitessimally small. The most famous is probably ℵ₀ (pronounced “aleph null”) which could be called the answer to the question “How many natural numbers are there?”. Note the phrasing: this is about how many there are, not what the biggest one is. The reasons that this is important kind of fall out of ELI5 territory, but the end result is that this is a number, and you can plug it into exponentiation if you have something that understands it. According to Wolfram Alpha, 1 to the ℵ₀ power is 1. There are other surreal numbers too, some of them even larger than ℵ₀, and as long as everyone understands that you’re working in surreal numbers, the math works out.
One fun fact: surreal numbers are large enough that you can’t really change their values by adding or subtracting any finite value from them. This has led to the joke “ℵ₀ bottles of beer on the wall / ℵ₀ bottles of beer / Take one down, pass it around / ℵ₀ bottles of beer on the wall”.
Wow.. Im sorry, But can someone put this in a bit more clear perspective? explain like im 3 or something?
There are 2 reasons that these formulas are forbidden and unwanted:
1. Infinity and zero have many different definitions. This is especially important with Taylor series, where you can have an infinite sum of numbers go either to a finite or infinite number depending on the series.
2. When a solution goes to infinity or zero, it’s often a trivial case to a real world problem that has multiple possible answers. It’s like saying you’re going to solve both sides of an equation my just multiplying them both by zero. Yes it works under the rules of math, but it’s useless.
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