# Number Theory

## Gauss And Congruence

Karl Friedrich Gauss (1777-1855) has been called the "Prince of Mathematicians" for his many contributions to pure and applied mathematics. He was born to poor parents in Germany. His high intelligence was noticed early and nurtured by his mother and uncle, but his father never encouraged Gauss in his education.

One of Gauss's most important contributions to number theory involved the invention of the idea of congruence (or agreement) in numbers and the use of what he called "modulos" or small measures or sets of numbers. In effect, his theory of congruence allows people to break up the infinite series of whole numbers into smaller, more manageable chunks of numbers and perform computations upon them. This arrangement makes the everyday arithmetic involved in such things as telling time much easier to program into computers.

Gauss said that if one number is subtracted from another (a - b), and the remainder of the subtraction can be divided by another number, m, then a and b are congruent to each other by the number m. Gauss's formula is as follows: a is congruent to b modulo c. For example, 720 - 480 = 240. The remainder, 240, can be divided by 60, 20, 10, and other numbers. However, for our purposes, we will only focus on 60. Using Gauss's expression, 720 is congruent to 480 by modulo 60. That is, both 720 and 480 are related to a third number (the remainder after 480 is subtracted from 720), which can be multiplied by 60.

In an abstract sense, this computation is related to such everyday arithmetic functions as telling the time of day on a digital watch. When the watch tells the time, it does not say "240 minutes past noon." It says "4 o'clock" or "4:00." To express the time of day, the digital watch uses several kinds of modulos (or small measures) which have been used for centuries: 60 minutes in an hour, 12 hours in the a.m. or p.m. of a day, and so on. If the watch says it is 4:00 in the afternoon, then, from one frame of reference, it has subtracted 480 minutes from the 720 minute period between 12 noon to midnight. What remains is 240 minutes past noon. That is, 720 - 480 = 240. The remainder, 240, can be divided evenly by the modulo 60 (and by other numbers which we will ignore).

When we tell the time everyday, however, we do not use Gauss's terminology. Our clocks are already divided into modulos and we simply note the hour and how many minutes come before or after the hour. The importance of Gauss's congruence theory is that he created the formulas that allowed an immense variety of arithmetic actions to be performed based on different sets of numbers.