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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.
The long division may begin with a non-zero remainder. The remainder is generally computed using an -bit shift register holding the current remainder, while message bits are added and reduction modulo () is performed. Normal division initializes the shift register to zero, but it may instead be initialized to a non-zero value.
The reciprocal function y = 1 / x . As x approaches zero from the right, y tends to positive infinity. As x approaches zero from the left, y tends to negative infinity. In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case.
Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2 24 +1, and this value rounds to 2 24 in round to nearest, ties to even.
To maximise computation speed, an intermediate remainder can be calculated by first computing the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x 123 + x 111 + x 92 + x 84 + x 64 + x 46 + x 23 + 1 has the property that its terms (feedback taps) are at least 8 positions apart ...
Given an integer a and a non-zero integer d, it can be shown 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. (For a proof of this result, see Euclidean division. For algorithms describing how to calculate the remainder, see Division algorithm.)
The Zen of Python is a collection of 19 "guiding principles" for writing computer programs that influence the design of the Python programming language. [1] Python code that aligns with these principles is often referred to as "Pythonic". [2] Software engineer Tim Peters wrote this set of principles and posted it on the Python mailing list in ...
Each quotient polynomial is chosen such that each remainder is either zero or has a degree that is smaller than the degree of its predecessor: deg[r k (x)] < deg[r k−1 (x)]. Since the degree is a nonnegative integer, and since it decreases with every step, the Euclidean algorithm concludes in a finite number of steps.