[Originally published on Blogger, Monday, August 28, 2006]
In Molier’s Le bourgeois gentilhomme, Monsieur Jourdain was surprised to learn that he had been speaking prose all his life. I was recently reminded (by Mikael Johanssons’ blog) that all my life I was calculating 2-cocycles. Indeed, a carry in elementary arithmetic, a digit that is transferred from one column of digits to another column of more significant digits during addition of two decimals, 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 ->….
Of course, what else it could be?
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