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The integer n is called the exponent and the real number m is called the significand or mantissa. [1] The term "mantissa" can be ambiguous where logarithms are involved, because it is also the traditional name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m, as in ordinary decimal notation.
The same value can also be represented in scientific notation with the significand 1.2345 as a fractional coefficient, and +2 as the exponent (and 10 as the base): 123.45 = 1.2345 × 10 +2. Schmid, however, called this representation with a significand ranging between 1.0 and 10 a modified normalized form. [12] [13]
The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a unit in the last place (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22 hex and 1.45a70c24 hex , the ULP is 2×16 −8 , or 2 −31 .
Several earlier 16-bit floating point formats have existed including that of Hitachi's HD61810 DSP of 1982 (a 4-bit exponent and a 12-bit mantissa), [2] Thomas J. Scott's WIF of 1991 (5 exponent bits, 10 mantissa bits) [3] and the 3dfx Voodoo Graphics processor of 1995 (same as Hitachi). [4]
The part of the representation that contains the significant figures (1.30 or 1.23) is known as the significand or mantissa. The digits in the base and exponent ( 10 3 or 10 −2 ) are considered exact numbers so for these digits, significant figures are irrelevant.
A floating-point number is typically packed into a single bit-string, as the sign bit, the exponent field, and the significand or mantissa, from left to right. As an example, a IEEE 754 standard 32-bit float ("FP32", "float32", or "binary32") is packed as follows: The IEEE 754 binary floats are:
In a normal floating-point value, there are no leading zeros in the significand (also commonly called mantissa); rather, leading zeros are removed by adjusting the exponent (for example, the number 0.0123 would be written as 1.23 × 10 −2). Conversely, a denormalized floating point value has a significand with a leading digit of zero.
A 2-bit float with 1-bit exponent and 1-bit mantissa would only have 0, 1, Inf, NaN values. If the mantissa is allowed to be 0-bit, a 1-bit float format would have a 1-bit exponent, and the only two values would be 0 and Inf. The exponent must be at least 1 bit or else it no longer makes sense as a float (it would just be a signed number).