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Modular arithmetic (sometimes called '''modulo arithmetic''') is a system of Arithmetic for Integer s, where numbers "wrap around" after they reach a certain value — the '''modulus'''. Modular arithmetic was introduced by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae '' published in 1801. One way to understand modular arithmetic is to consider "clock arithmetic": the arithmetic of hours on the clock face. If the time is noted at 7 o'clock in the evening and then again 8 hours later, then rather than ending at 15 o'clock (as in usual addition), the time will actually be 3 o'clock, albeit the next day. Likewise, if the clock starts at noon and 7 hours elapses three times (3 × 7), then the time will be 9 o'clock the next morning (rather than 21). Since the time starts over at 1 after passing 12, this is similar to arithmetic ''modulo'' 12, except that normal modular arithmetic would begin at 0 and "roll over" after 11. The 24 hour clock, which goes from 00:00 to 23:59, is a closer approximation to modular arithmetic, using a ''modulus'' of 24. THE CONGRUENCE RELATION Two integers ''a'' and ''b'' are said to be congruent '''modulo''' ''n'', if ''a'' and ''b'' have the same remainder when divided by n, or equivalently, that their difference (a−b) is a Multiple of ''n''. In this case, it is expressed as a For instance, :38 ≡ 14 (mod 12) because 38 − 14 = 24 which is a multiple of 12. For positive numbers, congruence can also be thought of as asserting that two numbers have the same Remainder after dividing by the modulus ''n''. So, :38 ≡ 14 (mod 12) because when divided by 12 both numbers give 2 as remainder. Congruence is an Equivalence Relation , and the equivalence class of the integer ''a'' is denoted by {Link without Title} ''n'' = { ..., ''a'' − 2''n'', ''a'' − ''n'', ''a'', ''a'' + ''n'', ''a'' + 2''n'', ''a'' + 3''n'', ...}. This set of all integers congruent to ''a'' modulo ''n'' is called the congruence class or '''residue class''' of ''a'' modulo ''n'', and is also denoted by . If a and a then a and a This observation underpins modular arithmetic. This is the prototypical example of a Congruence Relation . In simpler terms: Given any positive integer ''n'' and any non-negative integer ''a'', if ''a'' is divided by ''n'', the result can be expressed as an integer quotient ''q'' and an integer remainder ''r''. Modular arithmetic is only interested in the remainder (or residue) after division by some modulus, and results with the same remainder are regarded as equivalent, or ''congruent''. Two integers ''a'' and ''b'' are said to be ''congruent modulo n'' if (''a'' mod ''n'') = (''b'' mod ''n''). THE RING OF CONGRUENCE CLASSES One can then define formally an addition and multiplication on the set | ||
| In This Way, '''Z'''/''n'''''Z''' Becomes A Commutative | "http://wwwinformationdelightinfo/encyclopedia/entry/Ring_(mathematics)" class="copylinks">Ring with ''n'' elements For instance, in the ring '''Z'''/12'''Z''', we have |
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