Number Systems

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand how and why computers use binary to represent all forms of data
  2. Understand the denary, binary and hexadecimal number systems, and convert between them
  3. Understand how and why hexadecimal is used as a beneficial method of data representation
  4. Add two positive 8-bit binary integers and understand overflow errors
  5. Perform logical binary shifts on positive 8-bit binary integers
  6. Use two's complement to represent positive and negative 8-bit binary integers

1. Why Computers Use Binary

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What is Binary?

Binary is a number system that uses only two digits: 1 and 0. It is called a base-2 system because everything is built using powers of 2.

Why Do Computers Use Binary?

Computers use binary because of how they are physically built. Inside a computer, there are millions of tiny electronic components called logic gates. These logic gates can only exist in two states:

  • ON (represented by 1)
  • OFF (represented by 0)

Think of it like a light switch — it's either on or off, nothing in between.

All data must be converted to binary before a computer can process it. This includes:

  • Numbers
  • Text
  • Images
  • Sounds
  • Videos

How Data is Processed and Stored

  1. Processing with Logic Gates: When you perform any operation on a computer (like adding two numbers), the data is processed using logic gates. These gates only understand two states (1 and 0), so binary is the perfect match.

  2. Storage in Registers: A register is a small storage area inside the computer's processor that holds binary data temporarily while it's being processed. For example, an 8-bit register can hold 8 binary digits (bits) at once.

Real-World Example: Storage Devices

  • Hard drives use magnetic fields with North and South poles — one pole represents 1, the other represents 0
  • CDs and DVDs use flat areas (land) and bumps (pits) — light reflecting differently from these creates 1s and 0s

2. Number Systems: Denary, Binary, and Hexadecimal

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Denary (Base-10)

Denary is the number system we use every day. It is called base-10 because it uses 10 different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Each position in a denary number represents a power of 10:

10³10²10¹10⁰
1000100101

Example: The number 3268 means:

  • (3 × 1000) + (2 × 100) + (6 × 10) + (8 × 1)
  • = 3000 + 200 + 60 + 8
  • = 3268

Binary (Base-2)

Binary is a base-2 number system using only two digits: 0 and 1.

Each position represents a power of 2:

2⁷2⁶2⁵2⁴2³2²2¹2⁰
1286432168421

This is an 8-bit binary number (8 digits). A single binary digit is called a bit. 8 bits together make a byte.

Important: In your exam, binary numbers will be a maximum of 16 bits long.

The largest 8-bit binary number is 11111111 which equals 255 in denary. The largest 16-bit binary number is 1111111111111111 which equals 65,535 in denary.

Hexadecimal (Base-16)

Hexadecimal (often shortened to "hex") is a base-16 number system. It uses 16 different symbols:

  • The digits 0-9 (same as denary)
  • The letters A-F (where A=10, B=11, C=12, D=13, E=14, F=15)
DenaryHexadecimal
00
11
22
33
44
55
66
77
88
99
10A
11B
12C
13D
14E
15F

Each hex digit represents exactly 4 bits (called a nibble).

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