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Converting Between Binary and Decimal

FoundationHigherAQA

Revise Converting Between Binary and Decimal for GCSE Computer Science with this free worksheet and full mark scheme — Foundation and Higher exam-style questions with worked answers for AQA GCSE Computer Science (8525). Binary is converted to decimal using place values that double: 1, 2, 4, 8, 16 and so on.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA specification. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

Converting between binary and denary uses place values, which double with each position moving left: 1, 2, 4, 8, 16, 32, 64, 128.

To convert binary to denary, write the place values above the digits and add up the values where a 1 appears. So 10110 has place values 16, 8, 4, 2, 1, and the 1s sit above 16, 4 and 2, giving 16 + 4 + 2 = 22.

To convert denary to binary, work from the largest place value downwards. Ask whether each fits into the remaining amount: if it does, write 1 and subtract it; if not, write 0. Working left to right this way is more reliable than repeated division, because the place values keep the arithmetic visible.

Revision notes

The place values

From the right: 1, 2, 4, 8, 16, 32, 64, 128 — each double the one before.

An 8-bit number has these eight place values, giving a maximum of 255 when every bit is 1.

Binary to denary

Write the place values above the digits, then add the values where a 1 appears.

For 10110: place values 16, 8, 4, 2, 1. The 1s are above 16, 4 and 2, so the total is 16 + 4 + 2 = 22.

Denary to binary

Start with the largest place value and work down.

For 37: does 32 fit? Yes — write 1, leaving 5. Does 16 fit into 5? No — write 0. Does 8? No — 0. Does 4? Yes — 1, leaving 1. Does 2? No — 0. Does 1? Yes — 1. Result: 100101.

Key points

  • Place values double from right to left.
  • The values are 1, 2, 4, 8, 16, 32, 64, 128.
  • Add place values where a 1 appears.
  • An 8-bit number has a maximum of 255.
  • Work from the largest place value downwards.
  • Subtract each value that fits.

Worked examples

Example 1

Convert the binary number 10110 to denary. [3 marks]

Working

Place values from the right are 1, 2, 4, 8, 16write out the place values
The 1s sit above 16, 4 and 2identify which place values have a 1
16 + 4 + 2 = 22add them together

Example 2

Convert the denary number 37 to binary. [3 marks]

Working

32 fits into 37, leaving 5. Write 1start with the largest place value
16 and 8 do not fit into 5, so write 0 and 0. 4 fits, leaving 1. Write 1work down through the place values
2 does not fit, write 0. 1 fits, write 1. Result: 100101complete the remaining place values

Example 3

State the largest denary number that can be stored in 8 bits. [2 marks]

Working

All eight bits set to 1 gives 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1add every place value
= 255work out the maximum

Common mistakes

  • Reading place values from the left.

    The rightmost digit is always the 1s column.

  • Forgetting a place value in the middle.

    Write out all the place values before starting.

  • Getting the 8-bit maximum wrong.

    It is 255, not 256 — 256 needs nine bits.

  • Not subtracting as you go.

    Subtracting keeps track of what remains to be represented.

Exam tips

  • Write out the place values before converting.
  • Work from the largest place value downwards.
  • Subtract each value that fits.
  • Check by converting your answer back.

Key terms

Place value
The value of each binary column, doubling leftwards.
Bit
A single binary digit.
Byte
Eight bits, holding values 0 to 255.
Most significant bit
The leftmost bit, with the largest place value.

Related topics

Written and reviewed against the current AQA specification. Spotted an error? Let us know.