4

According to answer in this post ratios with zero on either side of them are defined: Are ratios with zero defined?

But it raises another problem. How can we know if ratios 4:0 and 2:0 are equivalent?

Normally we claim that ratios a:b and c:d are equivalent if fractions a/b and c/d are equal. It works when a=0 and c=0, but breaks down when b=0 and d=0 due to division by zero.

So what do we do? Intuitively it seems that all a:0 and c:0 must be equivalent, but we lack formal criterion to back up this intuition.

KarmaPeasant
  • 838
  • 3
    If by the ratio $a:b$ we literally mean $a$ ÷ $b$, then $4:0$ is not defined. However, in a situation like the linked problem, there's no reason why we choose to list the numbers in that order—we could have written $0:4$ as easily as $4:0$. That's why such ratios are defined if either number doesn't equal $0$. – Greg Martin Oct 19 '20 at 05:11
  • Division by $0$ is not defined , hence the ratio $a:0$ is not defined either. We could take the reverse ratio $0:a$ which would be the same for all $a$. In this sense the ratios $a:0$ are all equal. – Peter Oct 19 '20 at 05:17
  • 1
    @GregMartin But 4:0 does make sense, unlike 4/0. For an example, I can have collected 4 fruits, out of which 4 are apples and 0 are pears. Maybe there are alternatives to modeling ratios as fractions? – KarmaPeasant Oct 19 '20 at 05:22
  • 1
    @Peter But 4:0 does make sense, unlike 4/0. For an example, I can have collected 4 fruits, out of which 4 are apples and 0 are pears. Maybe there are alternatives to modeling ratios as fractions? – KarmaPeasant Oct 19 '20 at 05:23
  • And if we, in response of problem with division by zero, just switch our ratio to 0:4, then we indirectly acknoweledge that 4:0 makes sense as a ratio. – KarmaPeasant Oct 19 '20 at 05:25

2 Answers2

4

Really the mathematical object that models ratios, in this sense, most closely is real projective space. The $n$-dimensional real projective space is defined to be the set of $(n+1)$-tuples of real numbers, other than $(0,\dots,0)$, under the equivalence relation that $(a_0,\dots,a_n) \sim (b_0,\dots,b_n)$ if and only if there exists a (nonzero) real number $\lambda$ such that $a_j=\lambda b_j$ for all $0\le j\le n$. This is the space of lines through the origin in $\Bbb R^{n+1}$.

For example, $1$-dimensional real projective space, called the real projective line, is the set of all (equivalence classes of) ordered pairs $(x,y)\ne(0,0)$ under the equivalence relation $(x,y)\sim(\lambda x,\lambda y)$. There is one equivalence class for each real number $m$ (the slope of the line), namely all points of the form $(\lambda,m\lambda)$; there is an additional equivalence class (the "line of infinite slope") of all points of the form $(0,\lambda)$. (In both cases $\lambda\ne0$.) This is how we want ratios to behave: the ratio $10:2$ is the same as the ratio $5:1$.

Greg Martin
  • 78,820
4

It depends on what a ratio is.

By that, I mean it depends on what the mathematical definition of the word ratio is in your case.

Let me explain. Remember, we are talking mathematics here. And mathematics deals with statements about mathematical objects, and mathematical objects have strict definitions.

For example, we can talk about a fraction $\frac{a}{b}$, because the expression "$\frac{a}{b}$" has a definition that we all agree on. And we all know that the definition does not cover the case when $b=0$, which means that, by definition, the fraction $\frac{a}0$ does not exist.

Your question is about when two ratios are equivalent and when they are not. Before you ask this question mathematically, you need to determine two things:

  1. What a ratio is.
  2. What "this ratio and that ratio are equivalent" means.

Now, point 1 is easy. A ratio is an expression of the type $a:b$, where $a$ and $b$ are two real numbers.

How about point two? For point two, we must determine a rigorous definition of when $a:b$ and $c:d$ are equivalent. Formally, this means defining an equivalence relation on the set of all possible ratios.

The typical definition is that $a:b$ is equivalent to $c:d$ if $\frac{a}{b}=\frac{c}{d}$.

This definition works well when none of the numbers is zero, however, as you corretly pointed out, it fails when $b=d=0$. In that case, the definition, as usually written out, technically says that the two ratios are not equivalent.

What's weirder, the definition claims that $0:a$ is equivalend to $0:c$, but $a:0$ is not equivalent to $c:0$.

The conclusion you should draw from the above is that the typically stated definition of ratio equivalence is, in a sense, "not good". It works fine for nonzero cases, but for zero cases, it returns strange results. Note that the definition is not, mathematically speaking, incorrect (mathematical definitions cannot be incorrect), but it is not useful. It does not model the concept of ratio that we want it to model.

So, a better definition of when two ratios are equivalent is needed. The best (also pointed out by @GregMartin in his answer is to say that

  1. $0:0$ is not a ratio
  2. $a:b$ is equivalent to $c:d$ if there exists $\lambda\in\mathbb R$ such that $c=\lambda a$ and $d=\lambda b$.

You can easily see that using this definition, $0:4$ is equivalent to $0:2$.

5xum
  • 123,496
  • 6
  • 128
  • 204
  • 1
    One detail: wouldn't it be more correct to say, that we don't know if a:0 and c:0 are equivalent when we rely on fraction to find equivalent ratios? Both ratios would be undefined. If we compare undefined result with another undefined result, then we can't be neither sure that they are equivalent nor sure that they are not. – KarmaPeasant Oct 19 '20 at 06:07
  • 1
    @user161005 Depends on how you read the "fraction-y" definition. If you read it as "$a:b$ is equivalent to $c:d$ if and only if $\frac{a}{b}=\frac{c}{d}$ is a true statement", then by definition, $a:0$ and $c:0$ are not equivalent, because the statement $\frac{a}{0}=\frac{d}{0}$ is not a true statement. Note that it's not really "false", it's just undefined. Alternatively, you could say the definition is actually not a definition, because it does not define equivalence for zero-containing ratios. Either way, the conclusion is the definition must be improved. – 5xum Oct 19 '20 at 06:13
  • 1
    But if the statement is undefined, doesn't it mean that we don't know if it's a true statement or not? So it would be a misnomer to call it not true statement. – KarmaPeasant Oct 19 '20 at 06:26
  • @user161005 Technically, a statement is either true or not. So if it is a statement, it is certainly a not true statement. – 5xum Oct 19 '20 at 06:29
  • I agree that statements are either true or false, but how can you be certain that it's not a true statement? – KarmaPeasant Oct 19 '20 at 06:32
  • @user161005 Something undefined cannot equal something else, can it? – 5xum Oct 19 '20 at 06:43
  • So, if I have 1/0, then by your logic it can't be equal to 2, as it's undefined. So we would need to conclude that 1/0 is either bigger or smaller than 2. The same goes for every other number instead of 2, when we compare it to 1/0. But it's absurd, how can a number to be NOT equal to ALL numbers with which it can ever be compared? It would be also mean that 1/0 is greater or smaller than 1/0, which doesn't make sense either. Or maybe we could say that comparison of something undefined with something defined is in itself undefined. But it would mean that we don't know if it's equal or not. – KarmaPeasant Oct 19 '20 at 06:56
  • @user161005 "So, if I have 1/0, then by your logic it can't be equal to 2, as it's undefined. So we would need to conclude that 1/0 is either bigger or smaller than 2"

    "elephant" is not equal to "$2$". Does that mean that "elephant" is either bigger or smaller than "$2$"?

    $\mathbb R$ is not equal to $2$. Does that mean that $\mathbb R$ is either bigger or smaller than $2$?

     – 5xum Oct 19 '20 at 07:19
  • @user161005 But in any case, we are veering away from my main point, which remains: fractions are not a good way of defining equivalence of ratios, and there exists a better way, which is (as the other answer by Greg Martin shows) closely tied to projective spaces. – 5xum Oct 19 '20 at 07:27