Question

How many different normalized double-precision numbers are there? Express your answer using powers of 2.

Answer 1

Based on the IEEE-754 Single Precision standard,

1 bit for the Sign

8 bits for the exponent

23 bits for the mantissa,

2^1 = 2 possibilities for the sign

2^8 = 256 possibilities for the exponent bits, of which two are reserved for non-normals: an exponent field of all zeros is used for floating-point zeros and subnormals, while an exponent field of all ones is used for infinities and nans

2^23 = 8388608 possibilities for the mantissa (note that 'significand' is generally the preferred term).

Multiplying, that gives 2 * 2^23 * (2^8 - 2), or equivalently 2^32 - 2^25, possibilities altogether. So there are 2^32 - 2^25 =2^7.

2⁷ different normalized double-precision numbers are there.

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