The secrets of long multiplication

In the current discussion about the National Curriculum Review, there is lot of talk of the need to pay attention to long division, and some more cautious suggestions that perhaps we need to start with long multiplication. This give me a pretext to repeat an observation which I discuss, in more pedagogical details, in Sections 9 and 10 of my paper on academia.com.

The following set of formulae continues to circle the Blogosphere:

1 * 1 = 1

11 * 11 = 121

111 * 111 = 12321
1111 * 1111 = 1234321
11111 * 11111 = 123454321
111111 * 111111 = 12345654321
1111111 * 1111111 = 1234567654321
11111111 * 11111111 = 123456787654321
111111111 * 111111111 = 12345678987654321

It was accompanied by usual comments about the intrinsic beauty of mathematics. Indeed, the pattern is beautiful — no doubt in that. But the example nicely illustrates a difference between an amateur and professional approaches to mathematics: professionals are interested not so much in beautiful patterns but in reasons why the patterns cannot be extended without loss of their beauty. In our case, the pattern breaks at the next step:

1111111111 * 1111111111 = 1234567900987654321

The result is no longer symmetric. The reason for that is an interference of a carry, transfer of an unit from one column of digits to another column of more significant digits. The carry arising from the addition of two digits a and b is defined by the rule

c(a,b) =1 if a+b >9 and =0 otherwise.

One can easily check that this is a 2-cocycle from Z/10Z to Z and is responsible for the extension of additive groups

0 -> 10Z -> Z -> Z/10Z -> 0.

This is exactly what cocycles (and cohomology) were invented for: they describe the obstacles for continuation of a certain pattern in behavior algebraic or topological objects. The appearance of cohomology in an elementary arithmetic entertainment piece is inevitable.

And this is why long multiplication is so pivotal concept of elementary mathematics.

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