Bit width

Also called: bits needed, number of bits, data width

The number of bits a value or register uses. It fixes how many patterns are available, so it decides which values fit and when overflow happens.

A value's bit width is how many bits it's stored in: a 4-bit value, an 8-bit register, a 32-bit bus. In hardware it's fixed when the circuit is built, and every result has to fit in it.

The width decides:

the range: n bits make 2ⁿ patterns
the MSB: bit n − 1, which is also the sign bit for signed values
when overflow happens: a result outside the range wraps around

How many bits does a value need? Find the smallest n whose range contains it.

Unsigned value v: the smallest n with 2ⁿ − 1 ≥ v.
Two's complement value v: the smallest n with −2ⁿ⁻¹ ≤ v ≤ 2ⁿ⁻¹ − 1.

Watch exact powers of 2. 256 = 2⁸ needs 9 unsigned bits, because 8 bits stop at 255. And signed ranges are lopsided: −64 fits in 7 bits, but +64 needs 8.

To widen a value, add zeros on the left for unsigned numbers and use sign extension for two's complement.

Worked examples

Example

Bits needed for 100, unsigned and signed

Compare the value against each width's range.

1.
Unsigned: 6 bits reach 63, too small. 7 bits reach 127. So 7 bits: 1100100.
2.
Signed +100: 7 bits reach only +63. 8 bits reach +127. So 8 bits.
3.
Signed −100: 7 bits reach only −64. 8 bits reach −128. So 8 bits.

Example

A reading from −300 to +300

Two's complement with 9 bits covers −256 to 255, which is too small. 10 bits cover −512 to 511, which contains the whole span. So 10 bits.

Common mistakes

Counting decimal digits instead of bits.
Forgetting that an exact power of 2 needs one more bit than you might expect.
Using the unsigned bit count for a signed value.

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Bit width is taught in Number Systems.

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