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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.
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).
sign = 0 ; e = 1 ; s = 110010010000111111011011 (including the hidden bit) The sum of the exponent bias (127) and the exponent (1) is 128, so this is represented in the single-precision format as 0 10000000 10010010000111111011011 (excluding the hidden bit) = 40490FDB [27] as a hexadecimal number. An example of a layout for 32-bit floating point is
The sign bit determines the sign of the number, which is the sign of the significand as well. "1" stands for negative. The exponent field is an 8-bit unsigned integer from 0 to 255, in biased form: a value of 127 represents the actual exponent zero.
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:
Sign bit: 1 bit; 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 ...
For example, a significand of 8 000 000 is encoded as binary 0111 1010000100 1000000000, with the leading 4 bits encoding 7; the first significand which requires a 24th bit (and thus the second encoding form) is 2 23 = 8 388 608. In the above cases, the value represented is: (−1) sign × 10 exponent−101 × significand
IEEE 754 adds a bias to the exponent so that numbers can in many cases be compared conveniently by the same hardware that compares signed 2's-complement integers. Using a biased exponent, the lesser of two positive floating-point numbers will come out "less than" the greater following the same ordering as for sign and magnitude integers.