Modular Arithmetic

One roll in the hay always say, it is 7.00 p.m. and the same fact can be also raise as itis 19.00 . If the truth underlying these two statements is understood well, one hasunderstood modular mathematics well.The conventional arithmetic is based on linear turning system known as the sum line. Modular Arithemetic was introduced by Carl Friedrich Gauss in 1801, in his book Disquisitiones Arithmeticae. (modular). It is based on circle. A circle can be divided into any number of parts. Once divided, each part can benamed as a number, just like a clock, which consists of 12 divisions and eachdivision is numbered progressively. Usually, the starting point is named as 0. So,the starting point of a set of numbers on a clock is 0 and not 1. Since thedivisions are 12, all integers, positive or negative, which are multiples of 12, willalways be corresponding to 0, on the clock. Hence, number 18 on a clockcorresponds to 18/12 . Here the remainder is 6, so the answer of 13 + 5 will be 6Si milarly, the same number 18, on a circle with 5 divisions will represent number3, as 3 is the remainder when 18 is divided by 5.Some examples of addition and multiplication with mod (5)1) 6 + 5 = 11. Now 11/5 gives remainder 1. Hence the answer is 1.2) 13 + 35 = 48. Now, 48/5 gives 3 as remainder. Hence the answer is 3.3) 9 + ( -4) = 5. Now 5/5 gives 0 as remainder. Hence the answer is 0.4) 14 + ( 6 ) = 8 . Now 8/5 gives 3 as remainder. So the answer is 3.Some examples of multiplication with mod ( 5 ).1. 6 X 11 = 66. Now, 66/5 gives 1 as remainder. So the answer is 1.2. 13 X 8 = 104. Now 104/5 gives 4 as remainder . So the answer is 43. 316 X 2 = -632. Now, 632/5 gives 2 as remainder. For negativenumbers the calculation is anticlockwise. So , for negative numbers, theanswer will be numbers of divisions (mod) divided by the remainder.Here the answer will be 3.4. 13 X 7 = 91. Now, 91/5 gives 1 as remainder. But, the answer will be5 1 = 4. So the answer is 4.Works-cited page1. Mod ular, Modular Arithmetic, wikipedia the free encyclopedia, 2006,Retrieved on 19-02-07 from http//en.wikipedia.org/wiki/Modular_arithmetic2. The entire explanation is based on a web page available at , http//www.math.csub.edu/faculty/susan/number_bracelets/mod_arith.htmlAdditional information An automatic calculator of any type of operations with anynumbers in modular arithmetic is available on website http//www.math.scub.edu/faculty/susan/faculty/modular/modular.html