Is $1 : 7 = 1 / 8$ or is it $1/7$?

Question

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,

Answer 1

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.

Answer 2

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.

Answer 3

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 $p\in{\Bbb P}^1({\Bbb R})$.

Answer 4

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

Answer 5

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:

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