Number systems and IEEE 754

Updated 11 Oct 2026

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Number systems
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  Binary -> positional values use powers of $2$ .
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  Hexadecimal -> positional values use powers of $16$ .
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  Hex digit -> corresponds to four binary bits.
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  Integer conversion -> repeated division by the base.
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  Fraction conversion -> repeated multiplication by the base.
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  Check -> evaluate the resulting place values.
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Normalized floating-point value
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  IEEE 754 normalized binary value -> $(-1)^s\times(1.f)_2\times2^{e-\mathrm{bias}}$ .
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  Binary32
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    Sign -> 1 bit.
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    Exponent -> 8 bits; bias $127$ .
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    Stored fraction -> 23 bits.
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    Significand precision -> 24 bits including the implicit leading $1$ .
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  Binary64
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    Sign -> 1 bit.
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    Exponent -> 11 bits; bias $1023$ .
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    Stored fraction -> 52 bits.
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    Significand precision -> 53 bits.
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  Special exponent patterns
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    All ones -> infinities and NaNs.
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    All zeros -> signed zero and subnormal values.
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  Add bit-field diagram for binary32 and binary64: sign | biased exponent | stored fraction; mark the implicit leading 1 for normalized values.
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Encoding example: -26.625
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  $-26.625_{10}=-11010.101_2=-1.1010101_2\times2^4$ .
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  Sign bit -> $1$ .
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  Biased exponent -> $4+127=131=10000011_2$ .
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  Stored fraction -> $10101010000000000000000$ .
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  Binary32 encoding -> $1\ 10000011\ 10101010000000000000000_2$ , hexadecimal $\texttt{0xC1D50000}$ .
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Decoding example: 0x41440000
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  Sign -> positive.
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  Exponent bits -> $10000010_2$ ; unbiased exponent $130-127=3$ .
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  Significand -> $1+2^{-1}+2^{-5}=1.53125$ .
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  Value -> $1.53125\times2^3=12.25$ .
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Machine precision and underflow
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  Machine epsilon near $1$
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    Binary32 -> $2^{-23}\approx1.1921\times10^{-7}$ .
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    Binary64 -> $2^{-52}\approx2.2204\times10^{-16}$ .
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  Unit roundoff (round-to-nearest) -> half the spacing near $1$ .
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  Smallest positive normal binary32 -> $2^{-126}$ .
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  Smallest positive subnormal binary32 -> $2^{-149}$ ; reduced precision.
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  Decimal representation -> many fractions, including $0.1$ , have no finite binary representation; arithmetic uses approximations.

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Number systems and IEEE 754

Number systems

Normalized floating-point value

Encoding example: -26.625

Decoding example: 0x41440000

Machine precision and underflow

Linked references 1