Seriously, It’s Just Division

Don’t get caught up on the concept of “fractions”.

A photo of a portion of a mathematics book defining “fraction”.
Photo by the author of Milne’s 1893 “Elements of Arithmetic”

There is one topic students of mathematics consistently struggle with, to the point that it has become legendary: Fractions.

I teach Algebra II. Fractions don’t exist.

I’m not saying, of course, that 1/2 and 5/31 aren’t things that might occur. I mean that I encourage students to stop obsessing on “fractions” as an isolated concept.

The History of Division Notation

In “Elements of Arithmetic” (1893), William J. Milne writes, “(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.”

Let’s examine the origin of the ÷ sign itself. My primary source will be Florian Cajori’s “A History of Mathematical Notations” (2012 Dover printing; originally published 1928/1929).

The earliest consistent way of depicting division with notation rather than words was through fractions. This is attributed (§235) to the 12th Century author al-Hassar. (Chronology note: This is several centuries after al-Khwarizmi, who gave us the al-Jabr. Not only did al-Khwarizmi not have algebraic notation available to him, he didn’t even have the full complement of mathematical operators!)

In the 16th Century, we find ) being used (§236), as in 4)8 representing 8/4 = 2. The fullest notation was a)b(c, where ac = b. We still use this notation today in elementary school, but we also use a vinculum (a line written over a numeric expression to group it) and act as if the ) and the vinculum are a single character.

Cajori traces the ÷ symbol to 1659 and Johann Heinrich Rahn (§237). It was largely ignored in Rahn’s Switzerland but embraced in England, where it got falsely attributed to John Pell, who had met and discussed mathematics with Rahn.

Rahn’s precise motivation for the symbol is not given, but Leibniz (§238) makes his clear. Beginning in 1684, he uses : rather than ÷, but writes that it is a typographic convenience. The preferred method is generally to write division in fractional form, but “very often however it is desirable to avoid this and to continue on the same line… so that a:b means a divided by b.”

Modern Notation

This history placed four symbols for division in competition. These are all mathematically equivalent ways of writing one half, in apparent historical order: 1/2, 2)1, 1÷2, 1:2.

As I’ve mentioned, 2)1 has largely been abandoned except at elementary school levels, where it isn’t used to write a numerical value but rather to set up a problem to be solved by the student.

1:2 is still used throughout Europe; I have a German-language 9/10 Klasse Algebra I book from the publisher Mentor Lern-Hilfe (1999), for instance, which uses it, although the fractional notation is highly preferred.

Cajori comments that, as of his writing, the obelus (÷) and the colon (:) represent a political divide that is especially profound in mathematics: “The former belongs to Great Britain, the British dominions, and the United States. The latter belongs to Continental Europe and the Latin-American countries” (§240).

Looking at the Wikipedia entry for “Division” in various languages, we find a preference for the obelus on the English, Spanish, French, Korean, Vietnamese, and Chinese pages, and a preference for the colon on the German, Italian, and Russian pages.

However, what is universal throughout is a preference for the vinculum (fraction bar) notation. Both 2÷5 and 2:5 are generally seen as inferior to ⅖.

This is for several reasons. Pragmatically, the vinculum notation is clearer to interpret when it’s done properly, particularly for complicated cases. Compare \(\frac{3x^2}{4x-1} = (3x^2) \div (4x – 1)\).

Secondly, because of the modern computer keyboard layout, it is far easier to type 3/4 than 3÷4 (although 3 : 4 is also readily available).

Most importantly, from a mathematician’s perspective, the “fraction bar” notation simply represents division. That is, \(\frac{3}{4} = 3/4 = 3 \div 4\).

There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.

I suspect (without explicit confirmation) that both the obelus and the colon were chosen specifically because they suggest the fractional notation: The two dots represent the numerator and the denominator. In the case of the obelus, we also write the fraction bar itself. Cajori even reports at least one occurrence of ·/· (compare to 3/4).

Despite this, so many students get the message that division and fractions are unrelated or only vaguely related concepts. In those places where the colon is used specifically for proportions, this might be seen as yet a third idea.

Conceptually, fractions may be a thing unto themselves (or not… I’m unconvinced either way). In notation, though, there is no meaningful mathematical difference between fractional notation or using the slash or obelus. It always simply means division, mathematically speaking.

Indeed, as I explained above, fractional notation was the earliest way of writing division, existing for about four centuries before any of the modern competitors.

When students get caught up on the notion that the fractional notation only represents fractions, a concept that they’ve come to fear or loathe, it can get in the way of forward educational progress.

It’s. Just. Division.

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