Edexcel 4CP0 · Topic 3 · 3.1a ★ FREE LESSON

Binary
Numbers

Base 2 · Place values · Denary ↔ Binary conversion

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Why Binary?

Computers use base 2

Computers use transistors — tiny switches that are either ON or OFF

ON = 1 OFF = 0 — only two states

Binary (base 2) uses only digits 0 and 1, called bits

Every number, letter, image and sound is stored as binary

NUMBER BASES COMPARED

Base	Name	Digits used
Base 10	Denary	0–9
Base 2	Binary	0, 1
Base 16	Hexadecimal	0–9, A–F

Binary Place Values

8-bit place value table

128

2⁷

64

2⁶

32

2⁵

16

2⁴

8

2³

4

2²

2

2¹

1

2⁰

KEY FACTS

Each place doubles: 1, 2, 4, 8, 16, 32, 64, 128 Maximum 8-bit value: 128+64+32+16+8+4+2+1 = 255 8 bits = 1 byte

Binary → Denary

Converting: 10110101

128

1

64

0

32

1

16

1

8

0

4

1

2

0

1

Add the values where bit = 1: 128 + 32 + 16 + 4 + 1 = 181

Denary → Binary

Converting: 75 to binary

SUBTRACTION METHOD

        128 → too big, bit = 0

        64 → fits! 75 − 64 = 11, bit = 1

        32 → too big for 11, bit = 0

        16 → too big for 11, bit = 0

        8 → fits! 11 − 8 = 3, bit = 1

        4 → too big for 3, bit = 0

        2 → fits! 3 − 2 = 1, bit = 1

        1 → fits! 1 − 1 = 0, bit = 1

128

0

64

1

32

0

16

0

8

1

4

0

2

1

75₁₀ = 01001011₂

Key Terms

Bit, Nibble, Byte

Term	Size	Example	Range (unsigned)
Bit	1 digit	1	0 or 1
Nibble	4 bits	1010	0–15
Byte	8 bits	10110101	0–255

With n bits you can represent 2ⁿ different values. 8 bits = 2⁸ = 256 values (0 to 255).

Exam Practice

Question 1

Convert the binary number 11010110 to denary. Show your working.

2 marks

✓ MARK SCHEME

128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214
(Mark for working shown + correct answer)

Question 2

Convert 109 to an 8-bit binary number.

2 marks

64+32+8+4+1 = 109 → 01101101

Common Mistakes

Binary conversion pitfalls

1

Writing binary MSB (most significant bit) on the right. Always write the largest place value (128) on the LEFT, smallest (1) on the right — same direction as denary.

2

Forgetting to write leading zeros. 75 in 8-bit binary is 01001011 not 1001011. The question usually specifies 8-bit — always fill all 8 places.

3

Adding all place values regardless of whether the bit is 1 or 0. Only add a column's value if its bit is 1 — if the bit is 0, that place contributes nothing.

Topic Summary — 3.1a

Key Takeaways

PLACE VALUES (8-BIT)

128 · 64 · 32 · 16 · 8 · 4 · 2 · 1

CONVERSION METHODS

Binary→Denary: add place values where bit=1

Denary→Binary: subtract largest fitting power of 2

2ⁿ values representable with n bits

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Next Lesson

3.1b

Hexadecimal

Base 16 · Hex ↔ Binary · Hex ↔ Denary

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