Modular Arithmetic

One rat 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 deuce statements is understood well, one hasunderstood modular mathematics well.The conventional arithmetical is based on linear number system known as the number line. Modular Arithemetic was introduced by Carl Friedrich Gauss in 1801, in his set aside Disquisitiones Arithmeticae. (modular). It is based on circle. A circle can be dissever into any number of parts. Once change integrity, all(prenominal) part can benamed as a number, just like a clock, which consists of 12 divisions and eachdivision is numbered progressively. Usually, the starting transport is named as 0. So,the starting point of a set of numbers on a clock is 0 and not 1. Since thedivisions be 12, all integers, positive or negative, which are multiples of 12, leadalways be corresponding to 0, on the clock. Hence, number 18 on a clockcorresponds to 18/12 . Here the counterpoise is 6, so the root 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 up by 5.Some examples of addition and multiplication with mod (5)1) 6 + 5 = 11. straight 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 figuring 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 plain encyclopedia, 2006,Retrieved on 19-02-07 from http//en.wikipedia.org/wiki/Modular_arithmetic2. The entire explanation is based on a entanglement page available at , http//www.math.csub.edu/faculty/susan/number_bracelets/mod_arith.htmlAdditional information An automatonlike 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