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returns the nearest integer using current rounding mode with exception if the result differs Floating-point manipulation functions frexp: decomposes a number into significand and a power of 2 ldexp: multiplies a number by 2 raised to a power modf: decomposes a number into integer and fractional parts scalbn scalbln
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.
In the example from "Double rounding" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9. With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "round half to odd"). For example, when rounding to 1/4 (0.01 in binary), x = 2.0 ⇒ result is 2 (10.00 in binary)
For division to always yield one number rather than an integer quotient plus a remainder, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c / b means a × b = c, as long as b is not zero.
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.)
This rounding rule is biased because it always moves the result toward zero. Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also, the last digit, in binary form, is equal to 0) is used.
Integer division; Efficient integer division and calculating of the remainder when the divisor is known; Integer square and cube roots; Unusual number systems, including base −2; Transfer of values between floating-point and integer; Cyclic redundancy checks, error-correcting codes and Gray codes; Hilbert curves, including a discussion of ...
Rounding to the nearest integer will give the best approximation but can result in / being larger than /, which can cause underflows. Thus m = ⌊ 2 k / n ⌋ {\displaystyle m=\lfloor {2^{k}}/{n}\rfloor } is used for unsigned arithmetic.