GCSE · Computer Science · AQA · Spec 8525

Convert binary ↔ decimal

Every file, message, and pixel your computer handles is secretly a string of 1s and 0s.

What you need to know

  • For an 8-bit number the place values are 128, 64, 32, 16, 8, 4, 2 and 1, read from left to right.
  • To convert binary → denary: add up all the place values where a 1 appears.
  • To convert denary → binary: find the largest place value that fits, write a 1, subtract, repeat.
  • The maximum 8-bit denary value is 255 (all eight bits set to 1).

The big picture

Binary is the language computers use internally — every number is stored using only the digits 0 and 1. Each binary digit (bit) represents a power of 2, doubling in value as you move left. Converting between binary and denary (decimal) means reading or building those place values, in both directions.

TONIGHT'S REVISION

Convert Binary ↔ Decimal

Master the 8-bit place values and you can translate any number the computer stores.

Computer Science · Data representation

Binary place value

Toggle the bit columns to build the denary number 109 — watch the total update as each place value switches on or off.

Challenge

Build the number 109

0
128+128
64+64
32+32
16+16
8+8
4+4
2+2
1+1
Denary total00000000

0

All bits are 0 — every column is switched off.

How to convert denary → binary

Follow these steps in order for any 8-bit conversion — the place values are always the same.

Computer Science · Number representation

8-bit unsigned integer

Tap a bit to toggle. Decimal, binary and hex are three names for the same number.

128
64
32
16
8
4
2
1
64+32+8+4+1=109
Exam line: Practise both directions — denary to binary and binary to denary — since the specification expects you to convert either way.
Watch out: Always check your answer by adding the place values back up. A single wrong bit flips the whole answer.

Binary → Denary vs Denary → Binary

Binary → DenaryvsDenary → Binary

Both directions use the same 8 place values — the method just runs in opposite ways.

Focus

Starting point

Binary → Denary

Given an 8-bit pattern (e.g. 01101101)

Denary → Binary

Given a denary number (e.g. 109)

The insight

The direction of the problem decides which method to use — the place values are identical either way.

Method

Binary → Denary

Read each column: wherever there is a 1, note its place value

Denary → Binary

Work left to right: fit in the largest place value that does not exceed what remains

Calculation used

Binary → Denary

Addition: sum all place values that sit under a 1

Denary → Binary

Subtraction: subtract each fitted place value from the running remainder

Common error

Binary → Denary

Adding a place value where the bit is 0

Denary → Binary

Forgetting to write all 8 bits — omitting leading zeros

Answer format

Binary → Denary

A denary whole number between 0 and 255

Denary → Binary

A binary string at the length the question asks for — for an 8-bit answer, show all 8 bits

AQA 8525 — Data Representation

AQA Exam Practice — Binary Conversion

Read the question, write your answer, then check it against the mark scheme.

Question

Convert the binary number 10110100 into denary. Show your working.

Student answer

128 + 32 + 16 + 4 = 180

Method marks0/2

Key points

1Place values double each time you move one column to the left: 1, 2, 4, 8, 16, 32, 64, 128.
2A 1 in a column means 'include that place value'; a 0 means 'ignore it'.
3Binary → denary: list the place values under the 1s and add them up.
4Denary → binary: work from the largest place value (128) downward, fitting it in or not.
58 bits make one byte — enough to store any denary number from 0 to 255.

Worked example

Problem

Convert the denary number 109 into 8-bit binary.

🧠

Memory hook

Think of each bit as a light switch: ON (1) adds its place value to the total, OFF (0) adds nothing. Read the 'on' switches left to right: 128, 64, 32, 16, 8, 4, 2, 1.

★ Exam tip

Read what the question asks for. If it asks for an 8-bit answer, write all eight digits including any leading zeros; 109 becomes 01101101. Whatever the length, check your answer by adding the place values back up.

⚠ Watch out

Dropping a leading 0 when the question asks for an 8-bit answer — 109 as an 8-bit number is 01101101, not 1101101. Check how many bits the question asks for and write that many.

✓

Check yourself

Without looking — what are the eight binary place values, left to right? Say them in order before you scroll back up.

Flashcards

(26)
What is the binary place value of the rightmost bit (bit 0)?
1 (2⁰)
What is the binary place value of bit 1 (second from right)?
2 (2¹)
What is the binary place value of bit 7 (the leftmost in an 8-bit number)?
128 (2⁷)
Write out all 8 place values for an 8-bit binary number, left to right.
128, 64, 32, 16, 8, 4, 2, 1
What is the maximum denary value that can be stored in 8 bits?
255 (all eight bits set to 1: 11111111)
Convert 00001010 from binary to denary.
8 + 2 = 10
Convert 11111111 from binary to denary.
128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255
Convert 10000001 from binary to denary.
128 + 1 = 129
Convert denary 15 into 8-bit binary.
00001111 (8+4+2+1 = 15)
Convert denary 200 into 8-bit binary.
11001000 (128+64+8 = 200)
Convert denary 1 into 8-bit binary.
00000001
Convert denary 128 into 8-bit binary.
10000000
Convert 01100100 from binary to denary.
64 + 32 + 4 = 100
Convert denary 255 into 8-bit binary.
11111111
What does a 0 in a binary column mean?
Do not include that place value — it contributes 0 to the total.
What does a 1 in a binary column mean?
Include that place value — add it to the running total.
How many bits make one byte?
8 bits
Why does each binary place value double moving left?
Binary is base 2 — each position is 2 times the one to its right.
Convert 11010010 from binary to denary.
128 + 64 + 16 + 2 = 210
Convert denary 37 into 8-bit binary.
00100101 (32 + 4 + 1 = 37)
Convert 00110011 from binary to denary.
32 + 16 + 2 + 1 = 51
Convert denary 64 into 8-bit binary.
01000000
What is the first step when converting denary to binary?
Find the largest place value (up to 128) that is less than or equal to your number, write a 1, then subtract it.
Convert denary 0 into 8-bit binary.
00000000
How many different values can be represented using 8 bits?
256 (0 to 255 inclusive, because 2⁸ = 256)
What is the denary value of binary 01010101?
64 + 16 + 4 + 1 = 85

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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