Can a ratio $a:b$ have $b=0$?

Question

Suppose that there are $2$ apples and $0$ oranges in a fruit bowl. Can we say that the ratio of apples to oranges is $2:0$, or is this not allowed because it involves division by zero? It seems that division and ratios should be separate concepts.

Answer 1

A ratio with zero has the same problems as a fraction with zero in the denominator. If you have $2:1$, you can multiply both numbers by a constant and still have the same ratio: $2:1 = 4:2 = 7:3.5$.

But do you think the ratios $2:0$, $4:0$, $1000:0$ should be equal? If I have $2$ apples for every $0$ oranges, then I also have $1000$ apples for every $0$ oranges. Yet where are my $1000$ apples? I seem to have only $2$.

Answer 2

I am of the opinion that when treating it as ratios (which should be kept distinct in your mind from fractions) having a zero on either side (so long as not both sides) is perfectly acceptable.

Ratios to me mean the following: Given a ratio, $a:b$ (which is equivalent to other equivalent ratios such as $3a:3b$ and so on) the ratio describes what proportion of the combined collection of objects will be of one type versus another. $\frac{a}{a+b}$ of the objects will be of the first type. This works for all $a,b$... even if one was zero (unless both were zero, in which case it is still undefined).

While it is true that for when $a$ and $b$ are both nonzero, you can use the ratio to say "if I have $a$ objects of the first type then I necessarily have exactly $b$ objects of the second type" that does not work when $a$ is zero... but that is perfectly fine because that wasn't how ratios are defined in the first place. It is merely a convenient result you can use about them in the restrictive case that neither $a$ nor $b$ are zero.