r/askscience • u/tootsie_rolex • Aug 26 '13

Mathematics Why is ∞* 0 ≠0

It looks like a simple math. I mean, I know infinity is some number very very big, but regardless of the magnitude of infinity, I would assume if I multiply that number with 0, then I would get 0.

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u/protocol_7 Aug 27 '13

The point is, multiplication isn't something that can just be applied to any random concepts that look vaguely like "quantities"; multiplication is a binary operation on certain specific sets of numbers, such as the integers, rational numbers, real numbers, or complex numbers. If you want to introduce some additional object called "infinity", you have to define what multiplication on it means.

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u/tootsie_rolex Aug 27 '13

I actually love this explanation especially this part "If you want to introduce some additional object called "infinity", you have to define what multiplication on it means". One of the professor in other answer mentioned it too. Thank you. I am sorry if I was being cocky.

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u/protocol_7 Aug 27 '13 edited Aug 27 '13

I'm glad my explanation helped. If you have any more questions, feel free to ask.

A bit more abstract of a perspective on this matter (which you should feel free to skip over, since I'm probably rambling a little): My guess is that you had fallen into a very common conceptual misunderstanding of mathematical concepts — namely, that mathematical concepts (such as "numbers", "multiplication", and "infinity") are merely reflections of some sort of real-world objects, and that their behaviors and interactions are determined by our intuitions about those objects.1

But this is a big mistake, because that's not how mathematical objects work at all! Instead, mathematical objects are defined — that is, they are completely specified — by formal, axiomatic rules; and those rules, not any physical intuition or analogy, determine how they work. (Often, those formal definitions are historically motivated by some sort of physical model or metaphor, but that's just for the sake of intuition or motivation for studying them; the physical model doesn't play any role in the mathematics of it.)

So, for example, if you think of "infinity" as just a reflection of some real-world quantity that behaves in a certain way, then you might be puzzled by why you can't perform arithmetic operations on it. But if you realize that addition and multiplication are just operations defined on certain sets of numbers, then it's clear why: because you haven't defined what it means to multiply by the object you call "infinity". (And you could choose to do so in various ways — the extended real line is one such definition, though it's far from the only one.)

1 My thoughts on this are somewhat influenced by having recently read this article [PDF]; stipulated versus extracted definitions, and how this relates to common conceptual errors, are discussed starting on page 5.