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In a certain (non-mathematical) Stack Exchange, when I wrote $n : m = n / m$ where $n$ and $m$ are positive integers, one of the moderators said "No! $n : m$ is usually the notation for "$n$ parts in $(n + m)$ parts vs. $m$ parts in $(n + m)$ parts, thus it means $n / (m + n)$." And many participants agreed with that and kept on saying I was wrong.

My question is...

Is there any background, educational say, that makes them that insistent? First I thought I would show them the definition of colon ideals to convince them of their false faith, but on second thought I concluded that would have made things worse.

* added * On having a look at the answer by dREaM, I think I have to add the context.

Someone asked to provide clarification (=translation) of certain passage from a fiction that runs like "the (average) physical capacity/ablility of a human kind is one-seventh of that of a vampire." As the original appeder asked if one-seventh = $1 : 7$, I said "yes, one-seventh $= 1 : 7 = 1 / 7$." Then came the frenzy.

Thus I had no idea why the moderator brought in $8 = 1 + 7$ (is there any point in "adding" the capacity/ability of the human and that of the vampire in the discussion!?)

* added (again) * Thank you very much for sharing me your time. I marked @Hans Lundmark's answer as the best one because he pointed me to the historical evidence. And I thank others as well, especially those who pointed out that the dear moderator might have mistaken mere ratio with odds and probability,

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  • $\begingroup$ It depends on the situation. For example if you divide something in two parts, you have divided it in a ratio of 1:1 but each part is 1/2 of the whole. But sometimes we can use them interchangably, $\endgroup$
    – Kartik
    Jul 14 '15 at 10:32
  • $\begingroup$ Anyone used to quoting probability as odds views it the way the moderator did. "An even match gives each player 50:50 odds"...in that case the probability attached to one player is clearly $\frac{50}{50+50} = \frac{1}{2}$. $\endgroup$
    – lulu
    Jul 14 '15 at 10:33
  • $\begingroup$ The properties of both a:b and a/b are the same it just depends on the context. $\endgroup$
    – Kartik
    Jul 14 '15 at 10:34
  • $\begingroup$ In my corner of the world, $:$ is the (horizontal) division symbol, so $n : m = \frac{n}{m}$. Other parts of the world have different conventions, some use $\div$ as the division symbol (which nowadays is also recognised here since it appears on calculators, but when I was young, it elicited a strong "What the heck is this?" reaction). $\endgroup$ Jul 14 '15 at 11:07
  • $\begingroup$ But it is also possible to talk about three subsets and say their proportions are $50 : 30 : 20$. If that is not "equal" to a number, why should proportion $50 : 30$ be equal to a number (without thinking what is intended)? $\endgroup$
    – GEdgar
    Jul 14 '15 at 12:20
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You have good historical reasons for interpreting $n:m$ as $n/m$.

From http://jeff560.tripod.com/operation.html:

The colon (:) was used in 1633 in a text entitled Johnson Arithmetik; In two Bookes (2nd ed.: London, 1633). However Johnson only used the symbol to indicate fractions (for example three-fourths was written 3:4); he did not use the symbol for division "dissociated from the idea of a fraction" (Cajori vol. 1, page 276).

Gottfried Wilhelm Leibniz (1646-1716) used : for both ratio and division in 1684 in the Acta eruditorum (Cajori vol. 1, page 295).

In Cajori's book (A History of Mathematical Notations, available on Google Books) there are some further interesting statements. For example:

There are perhaps no symbols which are as completely observant of political boundaries as are ÷ and : as symbols for division. The former belongs to Great Britain, the British dominions, and the United States. The latter belongs to Continental Europe and the Latin-American countries.

But perhaps it should be added that just because something was once used, it's not necessarily a good idea to keep using it. Although the colon notation for division lived on well into the 20th century at least, nowadays it's probably fair to say that it's been replaced by the slash, and that colon in most people's mind is strongly associated to geometric proportions.

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  • $\begingroup$ Thanks! It is a convincing evidence, indeed. $\endgroup$
    – eltonjohn
    Jul 14 '15 at 13:27
  • $\begingroup$ <colon in most people's mind is strongly associated to geometric proportions> Yes, but how about colon ideals? :-) I know mathematicians, let alone algebraists, are minorities.... $\endgroup$
    – eltonjohn
    Jul 14 '15 at 13:57
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If there are eight people and $1$ of them is tall and the rest short then the ratio of tall people to short people is $1:7$. However the ratio of tall people to all the people is $1:8$.

I would say it makes more sense to relate $1:7$ with $\frac{1}{8}$ because it implies $\frac{1}{8}$ of the total satisfy the property.

On the other hand one could say $1:7=\frac{1}{7}$ because it means the number of tall persons is a seventh of the rest, although this is less natural in my opinion.

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    $\begingroup$ Why should we try to add context to a purely arithmetic expression ? All of ratio and proportion that I have read says 1:7 = 1/7 ! $\endgroup$ Jul 14 '15 at 12:04
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    $\begingroup$ @trueblueanil Don't mean to be silly but if you talk about purely arithmetic expression then don't use exclamation mark in the end of sentences because $\frac 17 \neq \frac 1 {5040}$ $\endgroup$ Jul 14 '15 at 12:39
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    $\begingroup$ @Renato:I agree, I should not have added context !! (that's not the double factorial). $\endgroup$ Jul 14 '15 at 12:52
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    $\begingroup$ @dREaM: You write "..more sense to relate 1:7 to 1/8.." but go on to "..1:7 = 1/7.." [ 1:7 equals 1/7 ] Doesn't that tell the story ? $\endgroup$ Jul 14 '15 at 12:59
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    $\begingroup$ @eltonjohn even worse $\frac 17 !$ can be read as the Gamma function at $x= \frac 87$ $\endgroup$ Jul 14 '15 at 13:09
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Arithmetically, 1:7 definitely means 1/7

May be, the moderator was comparing it with equating odds of 1:7 with a probability of 1/7 (as, alas, numerous people do, taking the two terms to be synonymous), which 0f course is incorrect and the probability is 1/8

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  • $\begingroup$ <the moderator was comparing it with equating odds of 1:7 with a probability of 1/7> Aha. Well, it's quite probable. $\endgroup$
    – eltonjohn
    Jul 14 '15 at 14:19
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A reasonable definition would be

$$a_0 : \cdots : a_{n-1} = \frac{1}{a_0+\cdots+a_{n-1}}(a_0,\ldots,a_{n-1})$$

For example, under this definition, we have:

$$1:7 = (1/8,7/8)$$

Exercise. Show that $(a:b) = (a':b')$ iff $a/a' = b/b'.$

By the way, this can be used to take affine combinations. Given a $k$-tuple of real numbers $a$ and a $k$-tuple of vectors $x$, define:

$$a \bullet x = \sum_{j<k} a_j x_j$$

For example, $(1,7) \bullet (x,y)$ is the linear combination $x+7y$, and $(1:7) \bullet (x,y)$ is the affine combination $\frac{1}{8}x + \frac{7}{8}y$.

Edit. I just noticed these operations show up naturally in probability theory. Suppose Amy, Betty and Carl are playing a game that consists of independent minigames, played sequentially. The first player to win a minigame wins the whole thing. Let $p_A,p_B$ and $p_C$ denote the respective probabilities of winning a minigame. These needn't add to $1$; the rule is that if no one wins the minigame, then the process repeats, until a victor has emerged. Let $P_A,P_B$ and $P_C$ denote the respective probabilities of winning the whole game. Then:

$$(P_A,P_B,P_C) = (p_A : p_B : p_C)$$

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  • $\begingroup$ I do love that homogeneous coordinate system! $\endgroup$
    – eltonjohn
    Jul 14 '15 at 12:14
  • $\begingroup$ @eltonjohn, not 100% sure what you mean, but glad you like it :) $\endgroup$ Jul 14 '15 at 12:17
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If an amount of money is shared among A and B in the ratio $3:5$ then A gets ${3\over 8}$ of the total, but ${3\over5}$ as much as B.

In my view an expression of the form $a:b$ is NAN (not a number) but a way of talking to be parsed in real time. Mathematically the pair $(a,b)$ can be considered as homogeneous coordinates of a point $p\in{\Bbb P}^1({\Bbb R})$.

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  • $\begingroup$ Je crois que c'est la question de comparaison, pas celle de partage. Or, je ne remarque pas quelque chose? À propos, j'aime la parabole d'espace projectif! $\endgroup$
    – eltonjohn
    Jul 14 '15 at 13:36

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