Number systems and conversions

Why computers use binary 二进制

• A computer only has two states: on and off, written 1 and 0.
• A two-digit system is binary (base 基数 2) — and all data must become binary first.
• We use three number systems to work with it.

The three number systems

• The base is how many digits a system uses. Hex's A–F stand for 10–15.
• A bit 位 is one 0 or 1; 8 bits = 1 byte 字节; 4 bits = 1 nibble 半字节.

Adding two 8-bit numbers column by column; carries ripple to the left. 118 + 48 = 166.

Place value 位值 and binary ↔ denary

• In binary the place values double right to left. For 8 bits: 128 64 32 16 8 4 2 1.
• Denary → binary: put a 1 under each value you need so they sum to your number. $150 = 128+16+4+2$ → 10010110.
• Binary → denary: add the place values where there's a 1.

A microprocessor holds millions of tiny transistors, each a switch that is on (1) or off (0).

Hexadecimal and why we use it

• Hex → binary: each hex digit becomes its own nibble (4 bits). F08 → 1111 0000 1000.
• Binary → hex: group bits into nibbles from the right; each nibble is one hex digit.
• Why hex? it is shorter than binary and easier to read, so fewer mistakes — used for MAC/IPv6 addresses, HTML colours (#FF0000), and memory addresses.

A bitmap image is a grid of pixels; the resolution is the number of pixels (here 8 by 8).

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