Number Representation

A computer stores every number as a fixed-width string of bits, so before the bits mean anything you need an agreed *encoding*. Whole numbers use two's complement; numbers with a fractional part use IEEE 754 floating point. Both are just rules for reading a bit pattern as a value.

Two's complement — one adder for both signs

In an n-bit two's complement number the bits keep their normal place values except the top (most-significant) bit, which counts as negative. For 8 bits the weights are -128, 64, 32, 16, 8, 4, 2, 1. That single change is the whole trick: because the sign bit carries a real negative weight, the *same* binary adder that adds unsigned numbers also adds signed ones correctly — no separate subtract hardware.

An n-bit two's complement value ranges from -2^(n-1) to 2^(n-1) - 1 — for 8 bits that is -128 to +127. The asymmetry (one more negative than positive) exists because there is a single zero, unlike older schemes.

IEEE 754 — binary scientific notation

A 32-bit float splits into three fields: 1 sign bit, an 8-bit exponent stored with a bias of 127, and a 23-bit mantissa (fraction). The value is (-1)^sign × 1.mantissa × 2^(exponent - 127). The leading 1. is implicit — it is never stored, which buys one extra bit of precision for free.

Encode -6.5 as a 32-bit float:
  6.5 in binary  = 110.1
  normalize      = 1.101 × 2^2
  sign           = 1            (negative)
  exponent       = 2 + 127 = 129 = 10000001
  mantissa       = 101 0000... (23 bits, leading 1 dropped)

  1 10000001 10100000000000000000000   =  0xC0D00000

1. 1Write the magnitude in binary, including the fractional part.
2. 2Normalize to 1.f × 2^e so there is a single 1 before the point.
3. 3Set the sign bit from whether the number is negative.
4. 4Store e + 127 (the bias) as the 8-bit exponent.
5. 5Store the fraction f in the 23-bit mantissa; the leading 1 is dropped.