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## 6.1 NATURAL NUMBERS

Let X and Y be two natural numbers with Y > 0. Define Q and R, respectively, as the quotient and the remainder of the division of X by Y, with an accuracy of p fractional base-B digits: where Q and R are natural numbers, and R < Y. In other words, so that the unit in the least significant position (ulp) of Q.B−p and R.B−p is equal to B−p. In the particular case where p = 0, that is, Q and R are the quotient and the remainder of the integer division of X by Y.

The basic algorithm applies to operands X and Y such that In the general case, to ensure that X < Y, a previous alignment step is necessary. Assume that X is an m-digit base-B number, that is, X < Bm; then

substitute Y by Y′ = Bm.Y, so that Y′Bm.1> X;

compute the quotient Q and the remainder R′ of the division of X by Y′, with an accuracy of p + m fractional base-B digits, that is, so that The next theorem ...

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