Information AboutRemainder |
| CATEGORIES ABOUT REMAINDER | |
| elementary arithmetic | |
| number theory | |
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In mathematics, the result of the Division of two Integer s usually cannot be expressed with an integer Quotient , unless a remainder —an amount "left over"— is also acknowledged. THE REMAINDER FOR NATURAL NUMBERS If ''a'' and ''d'' are Natural Number s, with ''d'' non-zero, it can be proved that there exist unique integers ''q'' and ''r'', such that ''a'' = ''qd'' + ''r'' and 0 ≤ ''r'' < ''d''. The number ''q'' is called the ''quotient'', while ''r'' is called the ''remainder''. The Division Algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder. Examples
THE CASE OF GENERAL INTEGERS | ||
| When ''a'' And ''d'' Are | "http://wwwinformationdelightinfo/encyclopedia/entry/real_number" class="copylinks">Real Number s, with ''d'' non-zero, ''a'' can be divided by ''d'' without remainder, with the quotient being another real number If the quotient is constrained to being an integer however, the concept of remainder is still necessary It can be proved that there exists a unique integer quotient ''q'' and a unique real remainder ''r'' such that ''a''=''qd''+''r'' with 0&le''r'' < ''d'' As in the case of division of integers, the remainder could be required to be negative, that is, -''d'' < ''r'' &le 0 |
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| The Way Remainder Was Defined, In Addition To The Equality ''a'' | ''qd''+''r'' an inequality was also imposed, which was either 0&le ''r'' < ''d'' or -''d'' < ''r'' &le 0 Such an inequality is necessary in order for the remainder to be unique &mdash that is, for it to be well-defined The choice of such an inequality is somewhat arbitrary Any condition of the form ''x'' < ''r'' &le ''x''+''d'' (or ''x'' &le ''r'' < ''x''+''d''), where ''x'' is a constant, is enough to guarantee the uniqueness of the remainder |
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