what is the history of congruence and silmilarity conditions?

Answer 1

Congruence arithmetic is perhaps most familiar as a generalization of the arithmetic of the clock. Since there are 60 minutes in an hour, "minute arithmetic" uses a modulus of m==60. If one starts at 40 minutes past the hour and then waits another 35 minutes, 40+35=15 (mod 60), so the current time would be 15 minutes past the (next) hour.

Similarly, "hour arithmetic" on a 12-hour clock uses a modulus of m==12, so 10 o'clock (a.m.) plus five hours gives 10+5=3 (mod 12), or 3 o'clock (p.m.)

Answer 2

It's very long and goes back to antiquity. It's difficult not to notice that the 'remainders' produced by division form interesting patterns and the ancients probably started playing with it about the same time that Mathematics was emerging as a science in its own right.


Doug