Tuesday, September 14, 2010

a very useful formula

I am puzzled that my children's secondary school doesn't seem to teach the most elementary form of the binomial theorem, i.e.

(a+b)2 = a2 + 2ab + b2
(a-b)2 = a2 - 2ab + b2
(a+b)(a-b) = a2 - b2

In German selective secondary schools (Gymnasium) you get these 3 equations drilled into your head until you know them in your sleep, backward and forward (well at least this was still the case when I went to school), and rightly so, as they are incredibly useful for everything from quadratic equations through to mental arithmetics (allowing you to calculate things like 53 * 47 in a flash). They are known as "binomische Formeln" and have jokingly been attributed to a mathematician called Binomi, though of course the name refers to the fact that they are about algebraic expressions based on two terms.

Looking them up on Wiki, I found that the German entry Binomische Formel, which explains their usefulness in great detail, is linked to the English entry Binomial theorem, which is about the more general version [(a+b)n], applying to powers of all sorts. The latter would of course be too difficult for most pupils, so I am wondering whether it's only the German system that has come up with the the idea of rebranding the simplest case to make it accessible and useful. (A quick check of the wiki entries in Spanish, French, and Dutch reveals they also focus on the general formula.)

Any clues to this mystery appreciated.

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