If zero times anything is zero, and infinity times anything is infinity, then what is zero times infinity?

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If zero times anything is zero, and infinity times anything is infinity, then what is zero times infinity?

In: Mathematics

The idea of infinity x “anything” equalling infinity is basically explaining that a number too large to list being multiplied (or even added to) would still be too large to list. But I think this assumes the “anything” is a nonzero number.

I would think that multiplying by zero still equals zero because no matter how many things you multiply by zero, you still have zero.

Infinity is not a number, nor is it undefined. It is simply a concept. It has zero mathematical utility outside of thought experiments. It is impossible to do any mathematical calculations involving “infinity” and “infinity” is usually an answer received when something is wrong in the equation. Many of our current infinities come from an incomplete understanding of physics, they’re not truly infinite.

The idea that **0 * a = 0** for all **a** is a formal axiom for certain types of numbers – that is, it is fundamentally considered to be an intrinsic property of the set of numbers under consideration, not proven but instead assumed. **Infinity** is not part of any of those sets of numbers for which **0 * a = 0 f.a. a** is considered an axiom. Infinity can be defined ambiguously, and so multiplying zero by infinity does not have a single rigorous solution.

“Infinity” is not an element of the standard real number system, so if you want to say something like “infinity times anything is infinity”, you have to choose an alternate system of numbers that actually includes infinite values. There are several of these, which appear in different areas. Using the extended real numbers, which includes points +infinity and -infinity at either end of the real number line, 0 * +infinity and 0 * -infinity are simply undefined. Using the cardinal numbers of set theory, there are many different infinite values, but you still have x * 0 = 0 for any x, whether finite or infinite. As with anything in mathematics, you have to be precise about what you mean by “zero times infinity” before you can talk about it.

It’s not defined.

There’s also the question of “what infinity”, but in this particular case, almost all reasonable ideas of trying to multiply those two together yield “not defined”.

There is one exception that I know of. In measure theory, you usually kinda take zero times (countable) infinity to be 0. The idea there kinda is that, measure theory tries to be a way to discuss all different “measures”, like length, area, volume, etc, and adding together stuff, with zero length for example, gives you end result that’s also zero length. This works up until countable infinity. Beyond that, it no longer works(for example, a line consists of uncountably many points of zero length. That should tell you that uncountably many of zero lengths breaks down this arithmetic)

But “not defined” is the safest answer. The reason being in the title, if infinity times anything is infinity, and zero times anything is zero, and zero ain’t infinity, zero times infinity can’t be defined. But as always in math, you gotta understand the context. Sometimes multiplying zero by infinity makes sense. Admittedly it’s rare, though

An infinity times anything is not always an infinity.

An infinity times any finite number (other than zero) is always an infinity. Finite numbers are the normal numbers we normally deal with, one, ten, pi, square root of two, negatives, one divided by a trillion, they are all finite.

There are also infinitesimal numbers, numbers which are smaller than any finite number (other than zero). Any tiny little number you can think of, it’s larger than an infinitesimal: one-trillionth, one-trillionth of a trillionth, any of that sort, they are all larger than an infinitesimal.

These infinitesimal numbers have little practical use and so are not usually dealt with or talked about. They mostly serve to deal with infinities, and their most important feature is that an infinitesimal times an infinity is not an infinity.

Zero is even less than an infinitesimal, and an infinity times zero is always zero.

It should also be pointed out that an infinity is not a number, it is many number. An infinite number of numbers.