Unit 11.1A · Term 1

Number Systems: Binary, Decimal, Hexadecimal

Computers process all data as electrical signals, interpreted as 1s and 0s. While humans naturally calculate in Base-10 (Decimal/Denary), machines rely on Base-2 (Binary). Computer scientists use Base-16 (Hexadecimal) as a compact, human-friendly way to represent long binary strings.

Learning Objectives

  • 11.1.1.1 Explain the differences between the binary, decimal and hexadecimal number systems
  • 11.1.1.2 Convert numbers between the binary, decimal and hexadecimal number systems
  • 11.1.1.3 Explain the advantages of using hexadecimal numbers in computer systems
  • 11.1.1.4 Perform arithmetic operations of addition and multiplication with binary numbers

Lesson Presentation

Number Systems & Conversions · Slides for classroom use

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Video Explanation

Watch a visual walkthrough of Base conversions (Binary, Decimal, Hexadecimal) and Binary Arithmetic.

Conceptual Anchor

The Language of Machines and Humans

Imagine number bases as different languages expressing the same exact quantity.
Decimal is our native tongue (10 fingers = 10 digits).
Binary is the hardware's native tongue (switches: ON = 1, OFF = 0).
Hexadecimal is the "translator's shorthand" — a way for programmers to read and write binary quickly without getting lost in a sea of 0s and 1s.


Conversion chart for Binary, Decimal, and Hexadecimal
Number Systems mapping: grouping 4 binary bits exactly matches 1 hexadecimal digit.

Rules & Theory

Core Number Systems

System Base Available Digits Example
Decimal (Denary) 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 25510
Binary 2 0, 1 111111112
Hexadecimal 16 0-9, then A (10), B (11), C (12), D (13), E (14), F (15) FF16

Why use Hexadecimal?

  • Definition: A Base-16 number system widely used in computing to represent binary data.
  • Advantages:
    • Shorter and more compact than binary (1 hex digit exactly represents 4 binary bits / a nibble).
    • Easier for humans to read, write, and debug without making transcription errors.
    • Faster conversion to and from binary compared to decimal.
  • Where it is applied: MAC addresses (e.g., 00:1A:2B:3C:4D:5E), IPv6 addresses, HTML color codes (e.g., #FF0000 for red), memory dumps, and assembly language programming.

Conversions

  • Binary to Decimal: Use the positional weights (128, 64, 32, 16, 8, 4, 2, 1). Multiply each binary digit by its column value and sum them up.
  • Decimal to Binary: Use the "Divide by 2" method (keep track of remainders reading from bottom to top) or subtract the largest possible power of 2.
  • Binary to Hexadecimal: Group the binary number into sets of 4 bits (nibbles) starting from the right. Convert each nibble to its hex equivalent (e.g., 1010 1111 = A F).
  • Hexadecimal to Binary: Convert each hex digit directly into a 4-bit binary sequence.
# Example: Converting Hexadecimal 2A to Binary and Decimal Hex: 2 A (10 in decimal) Binary: 0010 1010 Combined Binary: 00101010 # Converting 00101010 to Decimal: (0 * 128) + (0 * 64) + (1 * 32) + (0 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1) = 32 + 8 + 2 = 42 in Decimal

Binary Arithmetic

When working directly with hardware or low-level programming, we must add and multiply using only 0s and 1s.

Addition Rules Multiplication Rules
0 + 0 = 0 0 × 0 = 0
0 + 1 = 1 0 × 1 = 0
1 + 0 = 1 1 × 0 = 0
1 + 1 = 0 (with a carry of 1) 1 × 1 = 1
1 + 1 + 1 = 1 (with a carry of 1) Multiplication of larger numbers uses standard long multiplication (shift and add).
# Binary Addition Example: 1011 + 0110 1011 (11 in decimal) + 0110 (6 in decimal) ------ 10001 (17 in decimal) # Note the carry operations bubbling to the left!

Common Pitfalls

Hexadecimal is NOT executed by the CPU

A frequent misconception is that processors execute hexadecimal code. They do not. The CPU only processes pure binary. Hexadecimal is strictly a presentation format to make life easier for human programmers.

Tasks

Remember

List three practical computing applications where hexadecimal numbers are commonly used.

Understand

Explain why a computer scientist would prefer viewing a memory dump in hexadecimal rather than binary.

Apply

Convert the decimal number 156 into binary, and then convert that binary result into hexadecimal.

Apply

Perform the following binary addition: 1101101 + 101011. Show your carry bits.

Self-Check Quiz

Q1: How many bits are represented by exactly one hexadecimal digit?

Exactly 4 bits (also known as a nibble).

Q2: What is the rule for adding 1 + 1 in binary?

1 + 1 = 0, with a carry of 1 to the next most significant bit (making it 10 in binary).

Q3: Convert the hexadecimal value '1F' into decimal.

1F = (1 × 16) + (15 × 1) = 16 + 15 = 31 in decimal.