Number Systems

• Why binary? Logic gates only have two states (on/off), so binary (1/0) is the natural fit for computer hardware.
• Three number systems: denary (base-10, for humans), binary (base-2, for computers), hexadecimal (base-16, a human-friendly shorthand for binary).
• Conversions: denary ↔ binary, denary ↔ hexadecimal, and binary ↔ hexadecimal (using nibbles).
• Why hex exists: it's shorter to write and easier for humans to read/copy accurately than long strings of binary — used in MAC addresses, colour codes, and URLs.
• Binary addition: adding two binary numbers together, including how carrying works and what happens when you run out of bits (overflow).
• Binary shifts: moving bits left or right to multiply or divide by powers of 2.
• Two's complement: how computers represent negative numbers using only 1s and 0s.

The core reason

Inside every computer, data is processed using logic gates — tiny electronic switches that can only be in one of two states: on or off, high voltage or low voltage, current flowing or not flowing. There's no "half on."

Binary is the number system that's made up of exactly two digits — 1 and 0. This lines up perfectly: 1 can represent "on" and 0 can represent "off." Because of this natural match, all data — text, images, sound, video — must be converted into binary before a computer can understand and process it.

Where you can actually see this happening

Secondary storage devices physically store binary using two-state properties:

• Magnetic hard drives use North and South magnetic polarity to represent a 1 or a 0.
• Optical disks (CDs/DVDs) use a laser: light hitting a flat area (land) is read as a 1, and light hitting a bump (pit) is read as a 0.

2.1 Denary (base-10)

10 digits (0–9) weight

Reading this: (3×1000) + (2×100) + (6×10) + (8×1) = 3268. To represent bigger numbers, you simply add more digits (more columns) to the left.

2.2 Binary (base-2)