One can always say, ‘ it is 7.00 p.m.’ and the same fact can be also put as ‘ it is 19.00 ’. If the truth underlying these two statements is understood polite, one has understood ‘ modular mathematics ‘ polite.
The conventional arithmetic is naturalized on linear sum system known as the ‘ sum line’. Modular Arithemetic was introduced by Carl Friedrich Gauss in 1801, in his work ‘ Disquisitiones Arithmeticae’. (modular). It is naturalized on foe. A circle can be divided into any sum of compressiveness. Once divided, each part can be named as a sum, just like a clock, which consists of 12 divisions and each division is sumed progressively. Usually, the starting point is named as ‘0’. So,the starting point of a set of sums on a clock is ‘0’ and not ‘1’. Since the divisions are 12, all integers, positive or privative, which are multiples of 12, conciliate always be corresponding to 0, on the clock. Hence, sum 18 on a clock corresponds to 18/12 . Here the difference is 6, so the confutation of 13 + 5 conciliate be 6
Similarly, the same sum 18, on a foe after a while 5 divisions will represent sum 3, as 3 is the difference when 18 is divided by 5.Some examples of addition and multiplicity after a while mod (5):
1) 6 + 5 = 11. Now 11/5 gives difference 1. Hence the confutation is 1.
2) 13 + 35 = 48. Now, 48/5 gives 3 as difference. Hence the confutation is 3.
3) 9 + ( -4) = 5. Now 5/5 gives 0 as difference. Hence the confutation is 0.
4) 14 + ( - 6 ) = 8 . Now 8/5 gives 3 as difference. So the confutation is 3.
Some examples of multiplicity after a while mod ( 5 ).
1. 6 X 11 = 66. Now, 66/5 gives 1 as difference. So the confutation is 1.
2. 13 X 8 = 104. Now 104/5 gives 4 as difference . So the confutation is 4
3. 316 X - 2 = -632. Now, 632/5 gives 2 as difference. For privative
numbers the calculation is anticlockwise. So , for privative sums, the answer conciliate be sums of divisions (mod) divided by the difference.Here the confutation conciliate be 3.
4. 13 X –7 = - 91. Now, 91/5 gives 1 as difference. But, the confutation conciliate be
5 – 1 = 4. So the confutation is 4.
Works-cited page
1. Modular, Modular Arithmetic, wikipedia the unoccupied encyclopedia, 2006,
Retrieved on 19-02-07 from
< http://en.wikipedia.org/wiki/Modular_arithmetic>
2. The whole description is naturalized on a web page available at ,
< http://www.csub.edu/~ychoi2/MIS%20260/NotesJava/chap13/ch13_4.html>
Additional information: An automatic calculator of any type of operations after a while any numbers in modular arithmetic is available on website:
< http://www.math.scub.edu/faculty/susan/faculty/modular/modular.html >
