Real Life Application for Congruent Integers and Modulus
Real Life Application for Congruent Integers and modulus.
The modulus m = 12 is often used and applied in everyday life, for example, the most used and common of all ---"clock arithmetic" analogy, in which the day is divided into two 12-hour periods. Take for example, if it is 5:00 now, what time will it be in 25 hours? Since 25 ≡ 1 mod 12, we simply add 1 to 5:
5 + 25 ≡ 5 + 1 ≡ 6 mod 12.
Usual addition would suggest that the later time should be 5+25=30, however, this is not the answer because a clock time only “wraps around” every 12 hours and there is no such thing as 30 o’clock. Therefore, using arithmetic modulus, the clock will read 6:00 the next day instead of 30:00.
Using arithmetic modulo, we would create a number system with addition and multiplication in which only certain numbers exists. For example, with m = 12, there are only 12 numbers ("hours") we need to think about. We count from 1 to 12 and start over with 1 again. The numbers from 1 to 12 represent the twelve equivalence classes modulo 12: Every integer is congruent to exactly one of the numbers from 1 to 12, just as the hour on the clock always reads exactly one of the numbers from 1 to 12. These are given by
12n + 1, 12n + 2, 12n + 3, ..., 12n + 11, 12n as n ranges over the integers.
In a sense, a number system with addition and multiplication but in which the only numbers that exist are from 1 to 12 is created and this enables one to “count” time more easily.
Bibliography:
Eric S. Rowland. (No date) Modular Arithmetic (online)
URL: http://www.math.rutgers.edu/~erowland/modulararithmetic.html. Date of access: 20th June
The modulus m = 12 is often used and applied in everyday life, for example, the most used and common of all ---"clock arithmetic" analogy, in which the day is divided into two 12-hour periods. Take for example, if it is 5:00 now, what time will it be in 25 hours? Since 25 ≡ 1 mod 12, we simply add 1 to 5:
5 + 25 ≡ 5 + 1 ≡ 6 mod 12.
Usual addition would suggest that the later time should be 5+25=30, however, this is not the answer because a clock time only “wraps around” every 12 hours and there is no such thing as 30 o’clock. Therefore, using arithmetic modulus, the clock will read 6:00 the next day instead of 30:00.
Using arithmetic modulo, we would create a number system with addition and multiplication in which only certain numbers exists. For example, with m = 12, there are only 12 numbers ("hours") we need to think about. We count from 1 to 12 and start over with 1 again. The numbers from 1 to 12 represent the twelve equivalence classes modulo 12: Every integer is congruent to exactly one of the numbers from 1 to 12, just as the hour on the clock always reads exactly one of the numbers from 1 to 12. These are given by
12n + 1, 12n + 2, 12n + 3, ..., 12n + 11, 12n as n ranges over the integers.
In a sense, a number system with addition and multiplication but in which the only numbers that exist are from 1 to 12 is created and this enables one to “count” time more easily.
Bibliography:
Eric S. Rowland. (No date) Modular Arithmetic (online)
URL: http://www.math.rutgers.edu/~erowland/modulararithmetic.html. Date of access: 20th June
Bibliography: Eric S. Rowland. (No date) Modular Arithmetic (online) URL: http://www.math.rutgers.edu/~erowland/modulararithmetic.html. Date of access: 20th June 2012