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Number Systems

Data Representation · 4 question types

Practise

Exam Frequency Analysis

Past paper frequency (2018 to 2024)

This topic accounts for approximately 12% of your exam marks.

stable
High
Stable12%

Binary/hex conversion and binary arithmetic appear in every Paper 1. Consistently 8 to 15 marks.

Denary to binary

The standard method is the place-value table:

  1. Write a row of place-value headings (1, 2, 4, 8, 16, 32, 64, 128) from right to left.
  2. Working from the left of the table, ask: does the largest column heading fit into the denary number? If yes, write a 1 under it and subtract that value from the denary number. If no, write a 0.
  3. Move to the next column heading and repeat with the remainder.
  4. Continue until you reach the rightmost column.
  5. The string of 1s and 0s along the bottom is the binary value.

Binary to denary

The reverse is simpler:

  1. Write the binary number under a row of place-value headings.
  2. For every column that has a 1, write down the column heading.
  3. Add them all up.
Worked example

Converting denary to binary

Convert the denary number 175 to 8-bit binary.

Solution:

  • Write the 8-bit place-value headings: 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1
  • 175 ≥ 128? Yes → write 1, remainder = 175 − 128 = 47
  • 47 ≥ 64? No → write 0
  • 47 ≥ 32? Yes → write 1, remainder = 47 − 32 = 15
  • 15 ≥ 16? No → write 0
  • 15 ≥ 8? Yes → write 1, remainder = 15 − 8 = 7
  • 7 ≥ 4? Yes → write 1, remainder = 7 − 4 = 3
  • 3 ≥ 2? Yes → write 1, remainder = 3 − 2 = 1
  • 1 ≥ 1? Yes → write 1
  • Result: 10101111