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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.
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One method, more obscure than most, is to alternate direction when rounding a number with 0.5 fractional part. All others are rounded to the closest integer. Whenever the fractional part is 0.5, alternate rounding up or down: for the first occurrence of a 0.5 fractional part, round up, for the second occurrence, round down, and so on.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.
However, in some languages, division of signed binary numbers round towards 0 (which, if the result is negative, means it rounds up). For example, Java is one such language: in Java, -3 / 2 evaluates to -1, whereas -3 >> 1 evaluates to -2. So in this case, the compiler cannot optimize division by two by replacing it by a bit shift, when the ...
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 division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of ...
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 ...