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Given real numbers x and y, integers m and n and the set of integers, floor and ceiling may be defined by the equations ⌊ ⌋ = {}, ⌈ ⌉ = {}. Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
In Python 3, x / y performs "true division", meaning that it always returns a float, even if both x and y are integers that divide evenly. >>> 4 / 2 2.0 and // performs integer division or floor division , returning the floor of the quotient as an integer.
Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation (assuming x is a positive integer, or using a non-truncating definition):
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.
An example is the topological closure operator; in Kuratowski's characterization, axioms K2, K3, K4' correspond to the above defining properties. An example not operating on subsets is the ceiling function , which maps every real number x to the smallest integer that is not smaller than x .
The floor and ceiling functions are usually typeset with left and right square brackets where only the lower (for floor function) or upper (for ceiling function) horizontal bars are displayed, as in ⌊π⌋ = 3 or ⌈π⌉ = 4.
Traditional savings account rates. The Federal Deposit Insurance Corporation tracks monthly average interest rates paid on savings and other deposit accounts, like certificates of deposit, that ...
Many programming languages provide functions that can be used to divide a floating point number by a power of two. For example, the Java programming language provides the method java.lang.Math.scalb for scaling by a power of two, [7] and the C programming language provides the function ldexp for the same purpose. [8]