3.1a Binary numbers — Edexcel 4CP0

Why Binary?

Computers use the binary (base-2) number system because electronic components have two states: on (1) and off (0). These are called bits. Every piece of data in a computer — numbers, text, images, sound — is ultimately stored as binary.

Binary Place Values

Binary uses powers of 2. Each column represents a power of 2, doubling in value from right to left:

Bit position	7	6	5	4	3	2	1	0
Place value (2ⁿ)	128	64	32	16	8	4	2	1

Binary to Denary (Decimal)

To convert binary to denary, write the binary digits under their place values and add up the values where there is a 1.

Example: Convert 10110101 to denary

128	64	32	16	8	4	2	1
1	0	1	1	0	1	0	1

128 + 32 + 16 + 4 + 1 = 181

Denary to Binary

To convert denary to binary, subtract the largest possible power of 2 from the number, write 1 in that column, and repeat until the number reaches 0. Write 0 in all unused columns.

Example: Convert 147 to binary

128 fits in 147 → write 1, remainder 19
64 does not fit in 19 → write 0
32 does not fit in 19 → write 0
16 fits in 19 → write 1, remainder 3
8 does not fit in 3 → write 0
4 does not fit in 3 → write 0
2 fits in 3 → write 1, remainder 1
1 fits in 1 → write 1, remainder 0

Result: 10010011

Check: 128 + 16 + 2 + 1 = 147 ✓

Bits, Nibbles and Bytes

Term	Number of bits	Example
Bit	1	0 or 1
Nibble	4	1010
Byte	8	10110101

An 8-bit (1 byte) unsigned binary number can represent values from 0 to 255 (2⁸ − 1).

📝 Exam Tip: Always write binary numbers with 8 bits (pad with leading zeros if needed). So the number 5 in binary should be written as 00000101, not just 101. Edexcel mark schemes expect 8-bit answers where an 8-bit system is specified.

⚠️ Common Mistakes

Adding up the wrong columns — double-check which bits are 1 before summing
Writing the place values in the wrong order — highest value (128) is always on the LEFT
Forgetting leading zeros — 5 in 8-bit binary is 00000101, not 101

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Q1. Convert the binary number 11001010 to denary (decimal). Show your working. [2 marks]

✅ Mark Scheme:

Award 1 mark for correct place values identified (128, 64, 8, 2 are set bits); award 1 mark for correct total: 128 + 64 + 8 + 2 = 202.

Q2. Convert the denary number 175 to 8-bit binary. Show your working. [2 marks]

✅ Mark Scheme:

Award 1 mark for correct method (subtracting powers of 2); award 1 mark for correct answer: 10101111 (128+32+8+4+2+1=175).

Q3. A computer uses 8-bit binary to store integers. What is the maximum denary value that can be stored? Explain why. [2 marks]

✅ Mark Scheme:

Award 1 mark: maximum value is 255; award 1 mark: because all 8 bits are 1 (11111111), giving 128+64+32+16+8+4+2+1 = 255, or 2⁸−1.

Q4. A student claims that the binary number 1001 is the same as the denary number 9. Are they correct? Justify your answer. [2 marks]

✅ Mark Scheme:

Award 1 mark: Yes, they are correct; award 1 mark: 1001 in binary = 8 + 1 = 9 in denary, so the claim is correct.

Q5. Explain what is meant by the term "bit" and why computers use binary rather than denary. [3 marks]

✅ Mark Scheme:

Award 1 mark: a bit is the smallest unit of data with a value of 0 or 1; award 1 mark: computers use binary because electronic components have two states (on/off or high/low voltage); award 1 mark: these two states are easily represented as 1 and 0, making binary natural for digital circuits.

Binary Numbers Quiz

5 questions · Edexcel 4CP0 specific

out of 5 correct

Click card to reveal definition

Bit

The smallest unit of data in a computer system; can only be 0 or 1.

Nibble

A group of 4 bits.

Byte

A group of 8 bits; can represent values from 0 to 255 in unsigned binary.

Binary (base-2)

A number system using only two digits: 0 and 1. Each place value is a power of 2.

Denary (base-10)

The standard number system humans use, with digits 0–9 and place values as powers of 10.

Place value

The value of a digit based on its position in a number; in binary: 1, 2, 4, 8, 16, 32, 64, 128 from right to left.

Most significant bit (MSB)

The leftmost bit in a binary number; has the highest place value.

Least significant bit (LSB)

The rightmost bit in a binary number; has the lowest place value (1).

10:00

Timed test · 4 questions · Edexcel mark scheme

Binary Numbers — Mini Test10:00

Question 1 · Short Answer

Convert the binary number 01110110 to denary. Show your working.

Question 2 · Short Answer

Convert the denary number 219 to 8-bit binary.

Question 3 · Multiple Choice

How many different values can be represented using 4 bits?

Question 4 · Short Answer

A student writes the binary number 101101 and claims it equals 45 in denary. Are they correct? Justify your answer.

Check mark schemes below

Q1: Convert 01110110 to denary.

Your answer:

Mark scheme: 64+32+16+4+2 = 118. Award 1 mark for correct place values; 1 mark for answer 118.

Q2: Convert 219 to 8-bit binary.

Your answer:

Mark scheme: 11011011. (128+64+16+8+2+1=219). Award 1 mark for correct method; 1 mark for correct 8-bit answer.

Q3: Values using 4 bits.

Your selection:

Correct answer: C — 16. (2⁴ = 16 different values, 0000 to 1111).

Q4: Is 101101 = 45?

Your answer:

Mark scheme: Yes, correct (1 mark). 101101 = 32+8+4+1 = 45. (1 mark)