CS15: Number Bases
Understanding decimal (base 10), binary (base 2), and hexadecimal (base 16) number systems, and why computers use binary and humans use hex.
📋 What is a Number Base?
A number base (or radix) is the number of unique digits used to represent numbers in a positional numeral system. The position of each digit determines its value - this is called positional notation.
| Number Base | Name | Digits Used | Example |
|---|---|---|---|
| Base 10 | Decimal / Denary | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 | 255 |
| Base 2 | Binary | 0, 1 | 11111111 |
| Base 16 | Hexadecimal | 0-9, A, B, C, D, E, F | FF |
🔟 Decimal (Base 10)
Decimal is the number system we use every day. It has 10 digits (0-9). Each position represents a power of 10: units, tens, hundreds, thousands, etc.
Decimal Place Values:
Position: 4 3 2 1 0
Multiplier: 10000 1000 100 10 1
Example: 3754 = 3x1000 + 7x100 + 5x10 + 4x1
= 3000 + 700 + 50 + 4 = 3754
Position: 4 3 2 1 0
Multiplier: 10000 1000 100 10 1
Example: 3754 = 3x1000 + 7x100 + 5x10 + 4x1
= 3000 + 700 + 50 + 4 = 3754
Why do humans use base 10? Most likely because we have 10 fingers! There is nothing mathematically special about base 10 - it is just what we are used to.
01️⃣ Binary (Base 2)
Binary uses only two digits: 0 and 1. Each position represents a power of 2. Binary is the fundamental language of computers because electronic circuits have two states: on (1) and off (0).
Binary Place Values (8-bit):
Position: 7 6 5 4 3 2 1 0
Multiplier: 128 64 32 16 8 4 2 1
Example: 10110100
= 1x128 + 0x64 + 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 0x1
= 128 + 32 + 16 + 4 = 180
Position: 7 6 5 4 3 2 1 0
Multiplier: 128 64 32 16 8 4 2 1
Example: 10110100
= 1x128 + 0x64 + 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 0x1
= 128 + 32 + 16 + 4 = 180
Binary Values
| Binary | Decimal |
|---|---|
| 00000000 | 0 |
| 00000001 | 1 |
| 00001010 | 10 |
| 01100100 | 100 |
| 11111111 | 255 |
An 8-bit binary number can represent values from 0 to 255 (256 different values).
💻 Why Computers Use Binary
Computers use binary because their hardware is built from billions of transistors, which are electronic switches with exactly two states: ON (electricity flowing) and OFF (no electricity flowing). These two states map perfectly to 1 and 0.
Reasons for Binary
- Transistors have two states: on or off - this naturally maps to 1 and 0
- Reliability: With only two states, it is easy to distinguish between them even with electrical noise
- Simplicity: Logic gates (AND, OR, NOT) work naturally with binary inputs
- Cost effective: Two-state circuits are cheaper and simpler to manufacture
Key point: If we tried to use more states (e.g. base 10 with 10 voltage levels), it would be much harder to distinguish between similar values, especially with electrical interference. Binary's two clear states make it incredibly reliable.
🔟🔢 Hexadecimal (Base 16)
Hexadecimal uses 16 digits: 0-9 and A-F (where A=10, B=11, C=12, D=13, E=14, F=15). Each position represents a power of 16. Hex is used as a shorthand for binary because it is much easier for humans to read.
| Hex Digit | Decimal Value | Binary (4 bits) |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| A | 10 | 1010 |
| B | 11 | 1011 |
| C | 12 | 1100 |
| D | 13 | 1101 |
| E | 14 | 1110 |
| F | 15 | 1111 |
Hex Place Values:
Position: 2 1 0
Multiplier: 256 16 1
Example: 3C4
= 3 x 256 + 12 x 16 + 4 x 1
= 768 + 192 + 4 = 964
Position: 2 1 0
Multiplier: 256 16 1
Example: 3C4
= 3 x 256 + 12 x 16 + 4 x 1
= 768 + 192 + 4 = 964
🤔 Why Use Hexadecimal?
Hex is NOT used by the computer - the computer always uses binary internally. Hex is used by humans as a convenient shorthand for binary values. One hex digit represents exactly 4 binary digits (bits).
Hex as Binary Shorthand
Binary: 1101001111010110 (16 digits - hard to read!)
Hex: D3D6 (4 digits - much easier!)
Each hex digit maps to exactly 4 bits: D=1101, 3=0011, D=1101, 6=0110
Where Hex is Used
- Colour codes: #FF0000 = red (RR GG BB format)
- Memory addresses: 0x7FFF (shorter than 0111111111111111)
- MAC addresses: 4A:3B:FC:12:D5:77
- Error codes: 0x80070005 (easier to identify than the binary equivalent)
- Machine code: Programming at a low level uses hex for readability
| Representation | Value | Length | Readability |
|---|---|---|---|
| Binary | 1111111111111111 | 16 digits | Hard to read |
| Decimal | 65535 | 5 digits | Can't see bit patterns |
| Hex | FFFF | 4 digits | Short and bit-aligned |
⚠️ Common Mistakes
| Mistake | Why It's Wrong | How to Fix It |
|---|---|---|
| Saying computers use hex | Computers only use binary; hex is for humans | Computers use binary; hex is a human-readable shorthand |
| Counting binary positions from right as 1 | Positions start at 0, not 1 | Rightmost bit is position 0 (worth 1), next is position 1 (worth 2) |
| Forgetting hex digits go up to F | A=10 through F=15, not just 0-9 | Memorise A=10, B=11, C=12, D=13, E=14, F=15 |
| Confusing digit count with value | 8-bit binary is not the same as decimal 8 | 8 bits can store 0-255, which is 256 different values |
❓ Practice Questions
Q1: What is the decimal value of binary 10110?
Q2: What is the decimal value of hex 2F?
Q3: Why do computers use binary instead of decimal?
Q4: Why is hexadecimal used instead of binary when humans read computer data?
Q5: What decimal number does the hex digit C represent?
✅ Answers
- 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = 16 + 4 + 2 = 22
- 2 x 16 + 15 x 1 = 32 + 15 = 47
- Computers use binary because their hardware consists of transistors that have only two states: on (1) and off (0). This makes binary the natural representation for electronic circuits. Using only two states also makes it reliable and easy to distinguish between values even with electrical noise.
- Hexadecimal is used because one hex digit represents exactly 4 binary bits, making it a compact and easy-to-read shorthand for binary. A 16-digit binary number like 1111111111111111 becomes just FFFF in hex. It is much easier for humans to read and less error-prone than long binary strings.
- C = 12 in decimal.
🎯 Exam Tips
- Remember: computers ONLY use binary internally - hex is for human convenience
- Binary place values double each position: 1, 2, 4, 8, 16, 32, 64, 128
- Hex place values multiply by 16: 1, 16, 256, 4096
- Each hex digit = exactly 4 binary bits
- Know the hex digits: A=10, B=11, C=12, D=13, E=14, F=15
- When explaining why binary: mention transistors, on/off states, reliability
⚠️ Common Errors
✗ Thinking binary digits can be 0, 1, or 2 ✓ Binary (base 2) only uses digits 0 and 1. The digit 2 is invalid in binary — 2 in decimal is 10 in binary.
✗ Confusing the place values of different number bases ✓ Binary place values are powers of 2 (1, 2, 4, 8, 16...), hexadecimal uses powers of 16 (1, 16, 256...), and decimal uses powers of 10 (1, 10, 100...).
✗ Thinking hexadecimal is a different number system from binary in terms of storage ✓ Hexadecimal is a shorthand representation of binary for human readability. Each hex digit represents 4 binary bits; the computer still stores binary.
✗ Believing that 10 in binary equals 10 in decimal ✓ 10 in binary equals 2 in decimal. 10 in hexadecimal equals 16 in decimal. Always check which base is being used.
✍️ Model Answer
Full-Mark Response
Convert the decimal number 156 to binary and to hexadecimal. Show your working for each conversion. [4 marks]
Decimal to Binary (divide by 2, record remainders): 156 ÷ 2 = 78 r 0 78 ÷ 2 = 39 r 0 39 ÷ 2 = 19 r 1 19 ÷ 2 = 9 r 1 9 ÷ 2 = 4 r 1 4 ÷ 2 = 2 r 0 2 ÷ 2 = 1 r 0 1 ÷ 2 = 0 r 1 Read remainders bottom-up: 10011100 Decimal to Hexadecimal (divide by 16): 156 ÷ 16 = 9 remainder 12 9 ÷ 16 = 0 remainder 9 12 = C in hex Read remainders bottom-up: 9C So 156₁₀ = 10011100₂ = 9C₁₆
📊 AO Deep Dive
Assessment Objective Analysis
AO1 (Computational Thinking — 40%): Demonstrate knowledge and understanding of the principles and concepts of computer science, including number bases: binary, decimal and hexadecimal for AQA 8525, OCR J277 & Edexcel 1CP2.
AO2 (Application — 40%): Apply knowledge and understanding of computer science, including number bases: binary, decimal and hexadecimal to analyse problems in computational terms and to design, write and evaluate solutions.
AO3 (Evaluation — 20%): Evaluate the effectiveness, correctness and efficiency of computational solutions, including number bases: binary, decimal and hexadecimal, and make reasoned judgements about trade-offs.
📝 Exam Technique
GCSE Computer Science Exam Tips:
Binary → Decimal: multiply each bit by its place value (powers of 2) and sum. Decimal → Binary: repeated division by 2, read remainders bottom-up. Hex → Binary: convert each hex digit to 4 binary bits. Binary → Hex: group bits in fours from right, convert each group. Always show your working step by step. Label the base of your answer (₂, ₁₀, ₁₆).
Binary → Decimal: multiply each bit by its place value (powers of 2) and sum. Decimal → Binary: repeated division by 2, read remainders bottom-up. Hex → Binary: convert each hex digit to 4 binary bits. Binary → Hex: group bits in fours from right, convert each group. Always show your working step by step. Label the base of your answer (₂, ₁₀, ₁₆).
📝 Exam Questions by Topic
- AQA Past Papers - Topic Questions -- AQA
- Edexcel Past Papers - Topic Questions -- Edexcel
- OCR Past Papers - Topic Questions -- OCR
- Eduqas Past Papers - Topic Questions -- Eduqas
🎬 Video Resources
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