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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 exponential of a variable is denoted or , with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent ...
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). Python also supports complex numbers ...
A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number (with the smaller exponent) is shifted right by three digits, and one then proceeds with the usual addition method: 123456.7 = 1.234567 × 10^5 101.7654 = 1.017654 × 10^2 = 0.001017654 × 10^5
This logarithmic number of operations is to be compared with the trivial algorithm which requires n − 1 multiplications. This algorithm is not tail-recursive. This implies that it requires an amount of auxiliary memory that is roughly proportional to the number of recursive calls -- or perhaps higher if the amount of data per iteration is ...
The exponent field is an 11-bit unsigned integer from 0 to 2047, in biased form: an exponent value of 1023 represents the actual zero. Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved for special numbers.
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.