Modular Arithmetic

One can always say, it is 7.00 p.m. and the same fact can be also put as itis 19.00 . If the truth underlying these two statements is understood well, one hasunderstood advancedular mathematics well.The conventional arithmetic is found on linear payoff system known as the number line. Modular Arithemetic was introduced by Carl Friedrich Gauss in 1801, in his book Disquisitiones Arithmeticae. (modular). It is based on rotary. A circle can be divided into any number of parts. Once divided, every(prenominal)(prenominal) part can benamed as a number, just care 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, testamentalways be equal to 0, on the clock. and then, number 18 on a clockcorresponds to 18/12 . Here the remainder is 6, so the se rvice of 13 + 5 will be 6Similarly, the same number 18, on a circle with 5 divisions will represent number3, as 3 is the remainder when 18 is divided by 5. whatever examples of addition and multiplication with mod (5)1) 6 + 5 = 11. instantly 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. Modular, Modular Arithmetic, wikipedia the free encyclopedia, 2006,Retrieved on 19-02-07 from< http//en.wikipedia.org/wiki/Modular_arithmetic>2. The entire explanation is based on a web page available at ,< http//www.math.csub.edu/ staff/susan/number_bracelets/mod_arith.html>Additional 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 >