Group Laws Satisfying 1+1=11 and 2+2=22

Abstract

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. ‘Classroom’ is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science.

Group Laws Satisfying 1+1=11 and 2+2=22*

In a recent election campaign, Prime Minister Modi claimed that if one works hard in a correct political climate, one can make 1 + 1 = 11. In a similar speech, Mr Sitaram Yachuri mentioned that if we all work together then our combined strength will make 2 + 2 = 22. Here, we address the question: Is there a field k with a polynomially defined binary law of composition + such that the two equations 1 + 1 = 11 and 2 + 2 = 22 are both valid in k. In this note, we show that there are infinitely many such fields and characterize them all.