∞⁰ = ∞, 1, or undefined. Which is it?

A couple days ago I wrote an article about the Ramanujan Summation, which to cut a long story short is a mathematical series that looks something like this:

If you want to read the article, click here. I prove this fact in the article along with two other equally interesting equations. This is actually where I stumbled into the idea for this very article. After publishing the Ramanujan Summation, I got a comment about my use of the commutativity of an infinitely countable set. Commutativity is the idea that if you have 1+2+3, reorder the terms does not change the outcome. So 1+2+3=1+3+2, you can but the terms in any order and the answer will still always be 6. I use this property to prove the above equation in my other article, but forceOfHabit brought up an interesting point, does this hold for an infinite set of numbers?

“Its intuitively obvious there are twice as many positive integers as even positive integers. But if we take the sequence of positive integers and multiply them all by 2 we get the sequence of even positive integers. But multiplying every member of the sequence by 2 doesn’t change the number of members. So there are exactly the same number of positive integers as even positive integers. So which is it? Twice as many or the same number?” — forceOfHabit

And honestly, I didn’t know the answer to this. But it had peaked my interest, so I decided to research it a little more. I went down a Wikipedia wormhole through different branches of the mathematics, learning some interesting facts along the way, and ended up at cardinality. Cardinality deals with sets and is how you would describe the number of elements in a set. For example, the set {1,2,3} has 3 elements or a cardinality of 3.

Using cardinality, we can start to get a grip on the questions above. I researched a little further and found an interesting part of cardinality called Cardinal Arithmetic which are arithmetic operations that can be performed on cardinal numbers that generalize the ordinary operations for natural numbers. To put it in lamens terms, they are a special set of operations that work specifically for cardinal numbers, each with their own definition. For example, if you have two sets A and B with cardinalities 3 and 4 respectively, then we denote this as |A| = 3 and |B| = 4. Then |A| + |B| = |A ∪ B|. Of course, this is the same as just adding numerical values of |A| and |B|, the fact that it is defined this way shows how there are arithmetic operations that can be created for specific sets (providing the operation meets certain criteria).

Using cardinal arithmetic, it has been proven not only that the number of points in a real number line is equal to the number of points in any segment of that line. It sounds very counter-intuitive, but then again, so is the question above, which is why I like to think they are similar. Obviously, this is in no way a formal or even a valid proof, but I would posit that if you consider them in the same sense, then the answer to forceOfHabit’s question is option b; the same number of integers.

But on the other hand, I may be completely wrong, and that is the perplexity of infinity. There is so much that is not known about it because it is just a concept. There is no way to measure infinity because by definition it is unmeasurable and that in and of itself is a difficult concept to wrap your head around. I think my 1st year mathematic’s professor summed up infinity pretty well: “I hate infinity. It’s not a number, but we treat it like one, but we shouldn’t. It’s a concept, not a mathematical value, so if any of you use it as such, you may as well drop the course!”


Now for my favourite number in the entire world. You ask someone what their favourite number is (after running out of small-talk about the weather of course), and they will probably say something relating to a birthday or a lucky number they believe in. But ask me, and I will tell you 0. It’s not a lucky number, nor a birthday or anniversary, but it is by far the most interesting to me.

For starters, it has a value, but no value. If you add it to another number, it stays the same. Subtract it, stays the same. But when you multiply it, you get 0, no matter what you multiply it by.

1 x 0? 0.

123456789876543212345678987654321 x 0? 0.

And when you divide it, you get 0 regardless of what the denominator it (bar 1 number, stay tuned for that). 0 / 1234 is still zero

But when you diving by zero, you get some really wacky stuff. I’m talking dodging bullets in the matrix level crazy. Anyone who has taken an algebra class knows we cannot divide by zero, because it is undefined. We classify it as undefined because if you are trying to divide 6 by zero, it is analogous to asking the question “What number times 0 equals six?” We know that no number exists to satisfy that, so division by zero does not follow the normal rules of division. Hence, we disregard it. But, if we forget that rule for a second, division by zero can become a very neat tool to ‘prove’ completely ridiculous things. For example:

Let a = b. Then
a² = ab
a² + a² = a² + ab
2a² — 2ab = a² + ab — 2ab
2(a² — ab) = 1(a² — ab) #Magical step occurs here
2 = 1

There we go, I just proved that 2 = 1 and broke mathematics! The reason this works is because of the magical step, dividing both sides by a² — ab, but if you look at the original statement, a = b, so a² = ab, in other words a² — ab = 0. This is division by zero, which is undefined for this exact reason. It is also why mathematicians avoid it like the plague.

Fortunately it is actually the third option. I could go through how when it is in the form of a limit, it is an indeterminate form, but I think a well-known friend from Apple describes it best:

“Imagine that you have 0 cookies and you split them evenly among 0 friends. How many cookies does each person get? See, it doesn’t make sense. And Cookie Monster is sad that there are no cookies. And you are sad that you have no friends.” — Siri (really, try asking Siri “what’s 0 divided by 0?”)

A more complicated question involving zero, what is 0⁰? Well by definition, if you have a to the power of b, then the result would by a multiplied by itself b number of times. So it must be zero right? Because any number multiplied by zero is zero. But we also know that a⁰ = 1 (for all a ≠ 0), so maybe it should be 1? Or should it be undefined like division by 0?
This has been long debated in mathematics, and there are arguments for both sides as to what the real answer should be. There is an interesting website here that gives arguments for both sides, but the main ones are as follows: On the 0⁰ should be undefined side, we have:

  1. We know a⁰ = 1 (for all a ≠ 0), but a⁰ = 1 (for all a > 0). This contradiction means that 0⁰ should be undefined

On the 0⁰ = 1 side, we have:

  1. For the binomial theorem to hold for x = 0, we need 0⁰ = 1
  2. 0⁰ represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1 (this is also the same reason why anything else raised to the power of 0 is 1).

So whats the answer? Well we still don’t have a concrete answer. Most people would agree that is is indeterminate (since x^y as a function of two variables is not continuous at the origin). But both sides have valid arguments, and until someone can come up with a concrete proof claiming one or the other, it’s really impossible to claim if either one is true.


Now you may be wondering what happens if you combine the two. What is ∞ x 0? How about ∞⁰? Well the problem comes back to infinity, in that it is just a concept. There is no way to measure it, you can’t have a infinite number of gummy bears or an infinite amount of ice cream (though I’m sure we all wish we could).

Most of the time, the answer is undefined. These are all examples of questions that do not have an answer, because we cant give a meaningful value to a concept like infinity. Of course there is the odd exception, like 0^∞, which has a sort-of-value of 0. If you take the limit of 0^n as n tends to infinity, it is zero. But those are rare cases, and even then 0^∞ is still technically not equal to 0, it just gets very very close to it.

So you see, infinity is a very interesting thing because it is so tangible and so abstract at the same time. You see it all the time in mathematical textbooks and equations, but we still don’t have a concrete definition or value for what it is.

Zero is just awesome because it does it’s own thing. Sometimes it likes to play by the rules, sometimes it does it’s own thing, and occasionally it lock’s itself in a room and refuses to collaborate with anyone.

Both have their own redeeming qualities, which are very useful in the field of mathematics. They also have their own quirks, which can be useful and sometimes, and a pain in the butt at others. But while that’s just one of the facts of life, it is the perplexity of infinity and zero.

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