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On a typical computer system, a double-precision (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2 10 = 1024, the complete range of the positive normal floating-point numbers in this format is from 2 −1022 ≈ 2 × 10 −308 to approximately 2 1024 ≈ ...
In 1946, Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks [11] et al.) by analogy with the then-prevalent common logarithm tables: the characteristic is the integer part of the logarithm (i.e. the exponent), and the mantissa is the fractional part.
[3] [4] The word mantissa was introduced by Henry Briggs. [5] For a positive number written in a conventional positional numeral system (such as binary or decimal), its fractional part hence corresponds to the digits appearing after the radix point, such as the decimal point in English. The result is a real number in the half-open interval [0, 1
MBF numbers consist of an 8-bit base-2 exponent, a sign bit (positive mantissa: s = 0; negative mantissa: s = 1) and a 23-, [43] [8] 31-[8] or 55-bit [43] mantissa of the significand. There is always a 1-bit implied to the left of the explicit mantissa, and the radix point is located before this assumed bit.
Notice that for a binary radix, the leading binary digit is always 1. In a subnormal number, since the exponent is the least that it can be, zero is the leading significant digit (0.m 1 m 2 m 3...m p−2 m p−1), allowing the representation of numbers closer to zero than the smallest normal number. A floating-point number may be recognized as ...
Exponent: 11 bits; Significand precision: 53 bits (52 explicitly stored) The sign bit determines the sign of the number (including when this number is zero, which is signed). 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 ...
The half-precision binary floating-point exponent is encoded using an offset-binary representation, with the zero offset being 15; also known as exponent bias in the IEEE 754 standard. [9] E min = 00001 2 − 01111 2 = −14; E max = 11110 2 − 01111 2 = 15; Exponent bias = 01111 2 = 15
Numbers can be stored in a fixed-point format, or in a floating-point format as a significand multiplied by an arbitrary exponent. However, since division almost immediately introduces infinitely repeating sequences of digits (such as 4/7 in decimal, or 1/10 in binary), should this possibility arise then either the representation would be ...