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Because modular exponentiation is an important operation in computer science, and there are efficient algorithms (see above) that are much faster than simply exponentiating and then taking the remainder, many programming languages and arbitrary-precision integer libraries have a dedicated function to perform modular exponentiation: Python's ...
Python supports normal floating point numbers, which are created when a dot is used in a literal (e.g. 1.1), when an integer and a floating point number are used in an expression, or as a result of some mathematical operations ("true division" via the / operator, or exponentiation with a negative exponent).
x 1 = x; x 2 = x 2 for i = k - 2 to 0 do if n i = 0 then x 2 = x 1 * x 2; x 1 = x 1 2 else x 1 = x 1 * x 2; x 2 = x 2 2 return x 1 The algorithm performs a fixed sequence of operations ( up to log n ): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value.
Numba is used from Python, as a tool (enabled by adding a decorator to relevant Python code), a JIT compiler that translates a subset of Python and NumPy code into fast machine code. Pythran compiles a subset of Python 3 to C++ . [165] RPython can be compiled to C, and is used to build the PyPy interpreter of Python.
In plain text and programming languages, a slash (also called a solidus) is used, e.g. 3 / (x + 1). Exponents are usually formatted using superscripts, as in x 2. In plain text, the TeX mark-up language, and some programming languages such as MATLAB and Julia, the caret symbol, ^, represents exponents, so x 2 is written as x ^ 2.
In the base ten number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 10 3 = 1000 and 10 −4 = 0.0001. Exponentiation with base 10 is used in scientific notation to denote large or small numbers.
The properties involving multiplication, division, and exponentiation generally require that a and n are integers. Identity: (a mod n) mod n = a mod n. n x mod n = 0 for all positive integer values of x. If p is a prime number which is not a divisor of b, then ab p−1 mod p = a mod p, due to Fermat's little theorem. Inverse: [(−a mod n) + (a ...
The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations.