Understanding How Computers Store Floating-Point Numbers

IEEE 754 represents a floating-point value as:

V = (-1)^s × M × 2^E

s: sign bit; 0 means positive and 1 means negative
M: significand represented as a binary fraction
E: exponent that scales the significand by a power of two

Floating-point values are finite approximations of real numbers. Many decimal fractions cannot be represented exactly in binary, which explains rounding errors, overflow, underflow, and why financial systems often use decimal or fixed-point arithmetic.

Bit layout

Single precision
+---------+----------------+------------------------+
|    s    |       E        |           M            |
+---------+----------------+------------------------+
     1             8                   23

Double precision
+---------+----------------+------------------------+
|    s    |       E        |           M            |
+---------+----------------+------------------------+
     1            11                   52

Convert a decimal number to binary

Consider 4.25 in single precision. Convert the integer and fractional parts separately.

For the integer part, repeatedly divide by two and read the remainders in reverse:

4 / 2 = 2 remainder 0
2 / 2 = 1 remainder 0
1 / 2 = 0 remainder 1

4 = 100₂

For the fraction, repeatedly multiply by two and record each integer part:

0.25 × 2 = 0.5  -> 0
0.5  × 2 = 1.0  -> 1

0.25 = 0.01₂

Therefore:

4.25 = 100.01₂
     = 1.0001₂ × 2²

For a normalized single-precision value:

s = 0
E = 2 + 127 = 129 = 10000001₂
M = 00010000000000000000000

The complete representation is:

0 10000001 00010000000000000000000

Convert the bits back to decimal

Exponent bits: 10000001₂ = 129
Stored fraction: 0001... = 2^-4 = 0.0625

The implicit leading 1 makes the significand 1.0625, so:

(-1)^0 × 1.0625 × 2^(129 - 127) = 4.25

Special exponent patterns represent zero, subnormal values, infinity, and NaN. Normalized numbers use an implicit leading one; subnormal numbers do not.