Posted by: Alexandre Borovik | December 25, 2008

## Primary School Arithmetic and Group Cohomology

[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 -> 0.
Of course, what else it could be?

And here is some Molier:

PHILOSOPHY MASTER: Without doubt. Is it verse that you wish to write her?
MONSIEUR JOURDAIN: No, no. No verse.
PHILOSOPHY MASTER: Do you want only prose?
MONSIEUR JOURDAIN: No, I don’t want either prose or verse.
PHILOSOPHY MASTER: It must be one or the other.
MONSIEUR JOURDAIN: Why?
PHILOSOPHY MASTER: Because, sir, there is no other way to express oneself than with prose or verse.
MONSIEUR JOURDAIN: There is nothing but prose or verse?
PHILOSOPHY MASTER: No, sir, everything that is not prose is verse, and everything that is not verse is prose.
MONSIEUR JOURDAIN: And when one speaks, what is that then?
PHILOSOPHY MASTER: Prose.
MONSIEUR JOURDAIN: What! When I say, “Nicole, bring me my slippers, and give me my nightcap,” that’s prose?
PHILOSOPHY MASTER: Yes, Sir.
MONSIEUR JOURDAIN: By my faith! For more than forty years I have been speaking prose without knowing anything about it, and I am much obliged to you for having taught me that.

More on carries – in my draft book Shadows of the Truth.