The Division Symbol Goes Viral

A few days ago a Twitter user with the handle @Advil posted the following tweet:

As you can see, the tweet was widely “liked” and retweeted.

I was wondering: Is it true?

I don’t know much about the history of the division symbol ÷, although I do know that it is called an obelus. So I cracked open a copy of the famous 1928 text written by the math historian Florian Cajori, A History of Mathematical Notations (Volume 1). OK, I didn’t crack it open as much as open the web page. (All of the information in the blog post comes from pages 239–245 and 268–275 of Cajori’s book.)

From Cajori I learned that ÷ was first used for subtraction, not division—and it was not just used once in a while. It had a long life in that role. The first known such use was by Adam Riese in 1525. It was used for subtraction in a variety of countries by a number of different mathematicians in the 16th, 17th, and 18th centuries. In fact, the last known use of ÷ for subtraction was in 1915!

The first known use of ÷ to represent division was in the 1659 book Teutsche Algebra by the Swiss mathematician Johann Heinrich Rahn. His book was translated into English shortly afterward. However, once the notation started to catch on in England, it became known as Pell’s symbol, in reference to John Pell. This would not be the first instance of a mathematical object having the wrong name attached. But in this case it isn’t clear that it is misnamed; Pell had visited Rahn in Switzerland, so he may have been the originator of the symbol. (Neither mathematician claimed to be the one to introduce the ÷ symbol.)

Apparently ÷ was not the only symbol for division. Leibniz’s notation of : had its proponents too. And according to Cajori, which symbol people used depended on where they lived. He wrote

There are perhaps no symbols which are as 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. There are occasional authors whose practices present exceptions to this general statement of boundaries, but their number is surprisingly small.

As a final neat fact, apparently the Mathematical Association of America tried to rid mathematics of the symbols ÷ and :. In a 1923 report they wrote

Since neither ÷ nor :, as signs of division plays any part in business life, it seems proper to consider only the needs of algebra, and to make more use of the fractional form and (where meaning is clear) of the symbol /, and to drop the symbol ÷ in writing algebraic expressions.

So, what about the original question? Expressing rational numbers as fractions is old. According to Cajori, the use of the horizontal line for fractions dates back at least to al-Hassâr in the 12th century. And Leonardo of Pisa (Fibionacci) used the fractional line in his 1202 Liber Abaci. So in principle, fractions could be the source of the ÷ symbol. But it seems unlikely given the history that Cajori presented. And Cajori does not say that the symbol came from fractions (as far as I could tell), as @Advil asserts.

In fact, I see that the math historian Thony Christie calls @Advil’s claim “folk etymology.



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