Number systems

A positional number system gives each digit a weight that is a power of the base (radix). Decimal is base 10; computers work in base 2 because a wire is either on or off.

The common bases

Converting

• To decimal: multiply each digit by its positional weight and sum. 1011₂ = 8+0+2+1 = 11₁₀.
• From decimal: repeatedly divide by the base; the remainders, read bottom-up, are the digits.
• Binary ↔ hex: group binary into 4-bit *nibbles*; each nibble is one hex digit. 11111010₂ = FA₁₆.
• Binary ↔ octal: group into 3-bit sets; each set is one octal digit.

Units

• 1 bit = one binary digit (0 or 1).
• 1 nibble = 4 bits = one hex digit.
• 1 byte = 8 bits = two hex digits, holding values 0–255.

Two's complement (signed integers)

To store negatives, computers use two's complement: invert every bit of the magnitude and add 1. In *n* bits the range is −2ⁿ⁻¹ … 2ⁿ⁻¹−1 — for 8 bits that's −128…127. The most-significant bit acts as the sign.