Modular Arithmetic

Arithmetic of the Clock & Calendar

One surprise question with which a teacher can start a class is to ask for a situation where 10 + 6 = 4! Most students would think that it is a mistake in writing the problem.

The answer is simple and it happens in many schools! School starts at 10 a.m. If it lasts for 6 hours, when does school end? Any primary school student can say the answer is 4 p.m. So does that not make 10 + 6 = 4?

This arithmetic can be called “clock arithmetic”. It is how we count time as per the clock. The highest number is 12 & 1 and 13 will show the clock hands in the same position.

This kind of arithmetic happens in the calendar also. If we represent each day starting with Sunday with numbers starting with 1, then Saturday would be 7. The following Sunday would be 8 and the Sunday after that would be 15! If now we want to find what day is a day with number 45, then all we need to do is to divide by 7 and find the remainder. For day 45, the remainder would be 3 and hence it would be Tuesday!

Modular Arithmetic

The idea discussed in the paragraphs above can be used in any situation where there are regular repetitions. In the clock the numbers repeat after 12, in calendar days repeat after 7 and in a year days repeat after 365 or 366 days!

They have been developed into a separate topic called “modular arithmetic” by mathematicians starting with Carl Friedrich Gauss in a book written in 1801.

Modular arithmetic has its own grammar and syntax.

35 divided by 12 leaves a remainder of 11. This is written as 35 = 11 (Mod 12). This is pronounced as 35 equals 11 modulo 12.

Both 35 and 47 leave a remainder of 11 when divided by 12. This is written as 35=47 (mod 12). This is pronounced as 37 is congruent to 47 modulo 12.

Starting with these ideas, modulo arithmetic has developed a special arithmetic of its own.

The method of “casting out nines” to check numerical calculations, uses properties of modulo 9 arithmetic, which is 10 = 1 (mod 9)

Uses of Modular Arithmetic

Though this idea seems simple, it has many applications in a large number of fields. In life, many events are cyclical. The day of the week repeats every 7 days, the month repeats every 12 months etc. Hence modulo arithmetic can keep track of days & months.

Computer software use modular arithmetic to work with clock time, days & dates. They also use it to check the accuracy of huge amounts of transmitted data.

Many properties of numbers in Number Theory are expressed in terms of modular functions.