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Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.
Horner's method is a fast, code-efficient method for multiplication and division of binary numbers on a microcontroller with no hardware multiplier. One of the binary numbers to be multiplied is represented as a trivial polynomial, where (using the above notation) a i = 1 {\displaystyle a_{i}=1} , and x = 2 {\displaystyle x=2} .
As with any calculation that divides a binary data word into short blocks and treats the blocks as numbers, any two systems expecting to get the same result should preserve the ordering of bits in the data word. In this respect, the Fletcher checksum is not different from other checksum and CRC algorithms and needs no special explanation.
In binary arithmetic, division by two can be performed by a bit shift operation that shifts the number one place to the right. This is a form of strength reduction optimization. For example, 1101001 in binary (the decimal number 105), shifted one place to the right, is 110100 (the decimal number 52): the lowest order bit, a 1, is removed.
Logical shifts can be useful as efficient ways to perform multiplication or division of unsigned integers by powers of two. Shifting left by n bits on a signed or unsigned binary number has the effect of multiplying it by 2 n. Shifting right by n bits on an unsigned binary number has the effect of dividing it by 2 n (rounding towards 0).
Shifting left by n bits on a signed or unsigned binary number has the effect of multiplying it by 2 n. Shifting right by n bits on a two's complement signed binary number has the effect of dividing it by 2 n, but it always rounds down (towards negative infinity). This is different from the way rounding is usually done in signed integer division ...
In the offset binary representation, also called excess-K or biased, a signed number is represented by the bit pattern corresponding to the unsigned number plus K, with K being the biasing value or offset. Thus 0 is represented by K, and −K is represented by an all-zero bit pattern.
In modular arithmetic, Barrett reduction is an algorithm designed to optimize the calculation of [1] without needing a fast division algorithm. It replaces divisions with multiplications, and can be used when n {\displaystyle n} is constant and a < n 2 {\displaystyle a<n^{2}} .