Converting between number bases: binary and hexadecimal

The idea

This row targets conversion between binary and hexadecimal, the two bases most used in computing. Because 16 = 2⁴, each hexadecimal digit maps exactly onto four binary bits, so conversion is a grouping skill rather than arithmetic. To go from binary to hexadecimal, split the bit string into groups of four starting at the right-hand end, pad the leftmost group with leading zeros if needed, then replace each group with its hex digit: 1011 1110 becomes BE. To go the other way, expand each hex digit into its four-bit pattern and join them: 2F becomes 0010 1111, or 101111 once leading zeros are dropped. Practise both directions until the sixteen digit-to-nibble pairs are automatic, and always check group alignment from the right.

What earns the marks

• Recognises that one hexadecimal digit corresponds to exactly four binary bits because 16 = 2⁴.
• Converts binary to hexadecimal by grouping bits in fours from the right-hand end and padding the leftmost group with leading zeros.
• Converts hexadecimal to binary by expanding each hex digit into its four-bit pattern and concatenating the results.
• Uses the correct digit values A = 10, B = 11, C = 12, D = 13, E = 14, F = 15.
• Drops or retains leading zeros appropriately without changing the value of the number.

Common mistakes and how to put them right

• Grouping bits from the left instead of the right, which misaligns every nibble.

  Put it right: Start grouping at the least significant (right-hand) bit.

• Treating a hex digit as worth three or five bits.

  Put it right: Every hex digit is exactly four bits, so pad with leading zeros where necessary.

• Writing decimal digits for values above nine instead of letters.

  Put it right: Use A to F for ten to fifteen.

Exam tips

• Write the bit string out with spaces every four bits before converting, so alignment errors are visible.
• Learn the sixteen nibble-to-hex pairs as a lookup table and test yourself in both directions.
• Check your answer by converting back to the original base; a mismatch reveals a grouping slip.