Mathematical notation is not universal, though it is surprisingly international. There is no American way to do mathematical notation, no Muslim way to do mathematical notation, no Capitalist way to do mathematical notation, no Royal way to do mathematical notation.

Yet there are differences in notation and the differences can be interesting and instructive. I’m going to leave aside here small national differences in how numbers are written: what in America would be written as 3,456,789.01 would be written in Germany as 3.456.789,01 and what Americans call ‘decimal point’ is called a ‘decimal comma’ in Germany. There are corresponding differences in how dates are written, and different countries may use different units for currency and measurement.

When I came to the United States I noticed an aspect of mathematical notation I hadn’t seen before, and I liked it. I still do. I’m referring to the convention in American schools to differentiate between the ‘-‘ sign for minus, used for subtraction 5 – 7, and a similar sign written higher, ^{−}7, used for negative numbers. This negative sign, for which I’m not sure there is an official name, can be thought of in different ways.

One way to think of it is like a set of training wheels: something that helps a beginner, but no grown-up would want to be caught dead using one. For beginners learning Hebrew, there is a system of extra markings, called pointing, that you won’t see in typical Hebrew text intended for people fluent in the language, e.g. the Ha’aretz online paper. In Holland, where I grew up, books for children would have explicit syllable break markings, indicated with hyphens. Syllable break markings help a young Dutch reader pronounce the words, for in the Dutch system of writing, the pronunciation of vowel letters depends on whether the vowel occurs in the middle or at the end of a syllable.

Yet you can also defend a sign that marks a number as a negative number as a sign seen as part of the number itself, just as much part of the number as the decimal point. Why *wouldn’t* you think of negative numbers as deserving of a notation separate from the operation of subtraction?

Regardless of how you prefer to think of the negative sign, I’ve observed many students using it, and I’ve been pleasantly surprised at how smoothly they pick up the notation, how fluently they keep the use of the negative sign distinct from the use of the subtraction sign, and how – years later – they drop the use of the negative sign in favor of the ‘-‘ sign without any indication of confusion or hardship. I think there’s a lesson here and a hint that looking at other notational issues might be very rewarding.

Bert, I love this whole look at mathematical notation. It is something I have completely taken for granted and makes a huge difference in the language of math. Love this blog!

Sharon

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