The big idea: Reading a binary number works just like reading a decimal one. That is the only difference: the column values.
Decimal
- Ten digits
- Each column is worth ten times the one to its right
- Columns: 1, 10, 100
Binary
- Two digits
- Each column is worth twice the one to its right
- Columns: 1, 2, 4, 8, 16
Everything else: Everything else about reading a number is the same.
Why a computer uses two: A wire is either carrying a voltage or it is not. Two states are easy to build and hard to mistake, so computers count with two digits rather than ten.
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One byte — 10110101 — read as 181 in decimal and as B5 in hex. Step through it.
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Binary to decimal: Write the column values above the bits, then add the columns with a 1 under them.
10110101 → 128 + 32 + 16 + 4 + 1 = 181.
Decimal to binary
Take the biggest column value that fits, subtract it, repeat.
181: 128 fits, 53 left. 64 does not. 32 fits, 21 left. 16 fits, 5 left. 8 does not. 4 fits, 1 left. 2 does not. 1 fits, 0 left.
Columns used: 128, 32, 16, 4, 1 → 10110101.
Check it
Add your columns back up. If they do not come to the number you started with, you dropped a zero.
128 + 32 + 16 + 4 + 1 = 181. Correct.
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Why hex exists: Four bits hold 0 to 15 — exactly the range of one hex digit.
So any byte splits neatly into two halves, and each half becomes one character. 10110101 becomes B5: two characters instead of eight.
The sixteen digits
- 0-9 are themselves
- A = 10, B = 11, C = 12
- D = 13, E = 14, F = 15
- F is the largest single hex digit
Binary → hex
- Split the bits into groups of four, from the right
- Pad the left group with zeros if it is short
- Turn each group into its digit
- 1011 0101 → 11 and 5 → B5
Hex → binary
- Turn each hex digit into four bits
- Keep the leading zeros — the groups must stay four wide
- 2F → 0010 and 1111 → 00101111
- Never convert through decimal; it is slower and invites errors
Going through decimal for hex: For hex to decimal, multiply the left digit by 16 and add the right.
B5 → 11 × 16 + 5 = 176 + 5 = 181. The same number as before, which is the point: three notations, one value.
How this is tested — you must convert accurately and show the column values you used. It comes up two ways:
Paper 1 Section A
- Convert between binary, decimal and hex, 1-2 marks each
- State why hex is used for binary data
- Often the first, quickest marks on the paper
Paper 1 Section B — case study
- Read or write values in the case study's context
- Colour codes, memory addresses, file identifiers
The classic trap: Grouping bits from the left when converting to hex. Always group in fours from the right, and pad the leftmost group with zeros — 101101 is 0010 1101, which is 2D, not B4.
Convert the binary number 11010110 to decimal and to hexadecimal, showing your working.
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