What is the proper definition of a "Factor"?

Question

The definition I found on most websites was "A natural number $x$ is a factor of a natural number $y$ if $\frac{y}x$ leaves no remainder."

This definition seemed correct until I searched "What are the factors of $\sqrt2$?" The majority of the people seemed to say that the factors of $\sqrt2$ are $1$ and $\sqrt2$. This does not make sense, as $\sqrt2$ is not a whole number, and though we technically can't find its proper value, it's still clear that $\sqrt2$ is a decimal number and therefore it should have no factors.

Also, if $\sqrt2$ is a factor of $\sqrt2$, then $\sqrt2$ should also be an factor of $2$.

So once again, what is the proper definition of a factor in mathematics, and do numbers other than the natural numbers have factors?

Answer 1

A factor¹ or divisor of $x$ is a $y$ such that for some $z$ we have $yz=x$

Now, notice that I did not specify what $x,y,z$ should be. This is where I suspect your confusion comes from.

Typically, when talking about integers, if we don't specify otherwise in the context, $x,y,z$ are all assumed to be integers as well. Sometimes, when $x$ is a natural number, we only actually mean natural (non-negative) $y$. In particular, without specifying any particular context, the question of whether something is a factor of $\sqrt 2$ (or any other non-integer) is mostly nonsensical.

In general, the definition makes sense whenever there is some kind of multiplication that makes sense, for example in any ring, as mentioned in the comments (or semiring, as in the case of natural numbers, or...).

In high-school adjacent contexts, one example other than the integers or the naturals is the field of real numbers, but in this case, the concept is not very useful: in the reals (like in any field), every nonzero number is a divisor of every other number.

Another, slightly more ambitious, but arguably still high-school adjacent, example is the ring of Gaussian integers, where for example $1+i$ is a divisor of $2$.

You might also encounter other rings of quadratic integers, for example the ring $\mathbf Z[\sqrt 2]$, where indeed $\sqrt 2$ is a (prime) factor of $2$ and of itself.

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1: There is a small caveat here in that in some contexts, there might be a distinction between a left factor and a right factor, but I think that is beyond the scope here.