Zero vs Infinity relation type

Question

I'm not sure it should be asked here or in philosophy. Bertrand Russell in his book "Introduction to Mathematical Philosophy" in chapter 7 when discussing rational numbers on page 66 says:

"It will be observed that zero and infinity, alone among ratios, are not one-one. Zero is one-many, and infinity is many-one."

I cannot get the point of above sentence and the difference meant by Russell in relation type of zero and infinity.

Clarifying is appreciated.

Answer 1

See page 64; he says :

We shall define the fraction $m/n$ as being that relation which holds between two inductive numbers $x, y$ when $xn = ym$. This definition enables us to prove that $m/n$ is a one-one relation, provided neither $m$ nor $n$ is zero.

This symply means that, for each fraction $m/n$, for every "input" $x$ it is univocally defined which $y$ satisfy the equation :

$xn = ym$.

Thus the relation defined by the fraction $m/n$ is one-to-one.

Then we have :

From the above definition it is clear that the fraction $m/1$ is that relation between two integers $x$ and $y$ which consists in the fact that $x = my$.

Then he consider :

It will be seen that $0/n$ is always the same relation, whatever inductive number $n$ may be; it is, in short, the relation of $0$ to any other inductive cardinal.

This simply means that :

for all $n \ne 0$, we have that the relation : $xn=y0$ [see previous : $xn = ym$] is satisfied only for $x=0$, whatever $y$ is.

Thus, the relation $0/n$ (for each $n \ne 0$) is a one-to-many relation which holds between $0$ and any other number.