AQA · GCSE · Computer Science
Course version: 8525 · Version from Sept 2020
Understand how binary can be used to represent whole numbers.
Full specification wording
Converting between number bases: Understand how binary can be used to represent whole numbers.
Binary is a base-2 place-value system: each column is a power of two, so the columns from the right are 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16 and so on.
Checked against the AQA 8525 specification (opens in a new tab) on .
- Point 1 of 10
Study notes
From the checked teaching notes for this specification point.
The idea
Binary is a base-2 place-value system: each column is a power of two, so the columns from the right are 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16 and so on. A whole number is represented by placing a 1 in each column that is needed and a 0 where it is not, then adding the values of the 1 columns. For example, 13 = 8 + 4 + 1, so it is written 1101. To convert back, multiply each bit by its column value and add the results. Because each column has only two possible digits, binary maps directly onto the two states of a digital circuit, which is why computers store whole numbers this way. An 8-bit pattern can represent 256 different values, from 0 to 255.
What earns the marks
- Recognise that binary is base-2 and each column is a power of two, with the rightmost column worth 2⁰ = 1.
- Convert a denary whole number to binary by selecting the powers of two that sum to it and placing 1s in those columns, e.g. 13 = 8 + 4 + 1 = 1101.
- Convert a binary pattern to denary by adding the column values where a 1 appears, e.g. 10110 = 16 + 4 + 2 = 22.
- Explain that two digits (0 and 1) correspond to two circuit states, so binary is used to represent whole numbers in a computer.
- State that an 8-bit unsigned pattern gives 2⁸ = 256 values, ranging from 0 to 255.
Common mistakes and how to put them right
Reading binary columns left to right as increasing powers of two.
Put it right: The rightmost column is the lowest value (1) and values double moving left.
Forgetting to include the 1 column when converting, so 13 is written as 1100.
Put it right: Check the sum of the 1 columns equals the target number.
Treating binary as base-10 with digits 0 and 1 only.
Put it right: Each column is a power of two, not a power of ten.
Exam tips
- Write the column values (128, 64, 32, 16, 8, 4, 2, 1) above the bits before converting.
- Check your answer by adding the values of the 1 columns back to the original number.
- For an 8-bit range question, remember the maximum is 2⁸ − 1 = 255, not 256.
Try the lesson
The start of the Converting between number bases: Understand how binary can be used to represent whole numbers. lesson, made from checked teaching notes for this specification point.
Have a go first
A computer stores the whole number 13 as an 8-bit binary pattern. Which 8-bit pattern represents 13?
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