1

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.

Joe
  • 22,603
Neev Garg
  • 115
  • 2
    If you didn't have oranges in the first place, then it's meaningless to claim your apples to oranges ratio is $2:0$. – AlvinL Jul 12 '21 at 13:07
  • 2
    This is my first time reading something like this. My English is not good, but what are 0 oranges? So there is no orange in the bowl. How do you find the ratio of a non-existent fruit? – lone student Jul 12 '21 at 13:54
  • 1
    Additional related: Are ratios with zero defined? Also If ratios like 4:0 and 2:0 are defined then how can we determine if they are equivalent? – JMoravitz Jul 12 '21 at 14:02
  • 1
    Also from Wiki : "In most contexts, both numbers are restricted to be positive" I'd assume this is the case for fruits and vegetables as well. – Sarvesh Ravichandran Iyer Jul 12 '21 at 14:05
  • You are actually correct. One useful way to define "ratio" of pairs would be as equivalence classes of nonzero pairs $(x,y)$ such that one is a multiple of the other, namely by defining $(x,y) ≡ (z,w)$ iff there is some nonzero $r$ such that $(r·x,r·y) = (z,w)$, and defining ratios as the classes in $ℝ^2/≡$. In simple terms, for each $x,y∈ℝ$ that are not both zero, we can define $x:y$ to be the set of all $(r·x,r·y)$ where $r∈ℝ_{≠0}$. For example, $(3,6) ∈ 1:2$ and $1:2 = 2:4$. This concept is very useful in mathematics such as in homogeneous coordinates and projective spaces. – user21820 Jul 12 '21 at 19:02
  • 1
    This definition also generalizes naturally and easily to higher dimensions, yielding $1:2:3 = 2:4:6$ for example. Note that the $3$ vertices of a triangle in barycentric coordinates are $1:0:0$ and $0:1:0$ and $0:0:1$. It would be silly to forbid using such ratios just because of a fear of division by zero, since there is no division by zero here! @TeresaLisbon: You may be interested in these. – user21820 Jul 12 '21 at 19:05
  • @AlvinLepik: That's wrong. – user21820 Jul 12 '21 at 19:06
  • @user21820 That is a great context where such notation makes sense. – Sarvesh Ravichandran Iyer Jul 12 '21 at 22:28

2 Answers2

0

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.

JMoravitz
  • 80,908
  • 2
    This also seems acceptable to me. I think that you define ratios to be equivalent by $a:b = c:d \iff ad = bc$, then $2:0$ is allowed (and equivalent to $1:0$). It reminds me of projective space—most lines through the origin are characterized by their slope (a number), except the vertical one. But the vertical line still “counts.” – Matthew Leingang Jul 12 '21 at 15:14
  • Why did you vote to close this question as duplicate of a post which has poor answers? (One has a broken link, and the other assumes a ratio is a division, contrary to your answer.) Moreover, the links you provided in comments are better than the one you closed it as duplicate... Anyway, yours is the only correct answer here. – user21820 Jul 13 '21 at 14:26
  • @user21820 I agree the later posts I linked to were better, however at the time I found them I had already cast a close vote. I would have been unable to change my vote to something else and so opted to just provide the links to the others and leave my vote as is, hoping that others who followed suit with closing as duplicate would have used the better options I linked after the fact. Moderators still have the option to adjust if they so choose, but I think it is fine as is. – JMoravitz Jul 13 '21 at 14:30
  • I see, thanks for explaining. – user21820 Jul 13 '21 at 16:07
-3

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 $2$.

B. Goddard
  • 33,728