Notes

Number Base Conversions

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Number Base Conversions

The 8-Bit Column Header

Always write this out for denary ↔ binary conversions:

1286432168421
________

Denary → Binary

Method: Work left to right across the column header.

  1. Ask: does this column value fit into the remaining number? , If yes → write 1, subtract the column value , If no → write 0
  2. Repeat for every column
  3. Pad with leading zeros to fill 8 bits

Example: Convert 75 to binary

1286432168421
01001011

75, 64 = 11 → 11, 8 = 3 → 3, 2 = 1 → 1, 1 = 0 ✓ Answer: 01001011

Binary → Denary

Method: Add up the column values wherever there is a 1.

Example: Convert 00110100 to denary

  • Columns with 1: 32 + 16 + 4 = 52

Denary → Hexadecimal

Method: Convert to binary first, then split into nibbles.

Example: Convert 140 to hex

  1. 140 in binary = 10001100
  2. Split into nibbles: 1000 | 1100
  3. 1000 = 8, 1100 = 12 = C
  4. Answer: 8C

Hexadecimal → Binary

Method: Convert each hex digit to its 4-bit nibble.

Example: Convert 4F to binary

  • 4 = 0100
  • F = 15 = 1111
  • Answer: 01001111

Hexadecimal → Denary

Method: Convert hex → binary → denary (or multiply column values).

Example: Convert 4F to denary

  • 01001111 → 64 + 8 + 4 + 2 + 1 = 79

Worked Examples Summary

DenaryBinary (8-bit)Hex
12000011000C
240001100018
75010010114B
1000110010064
140100011008C
17110101011AB
23011100110E6

Exam Questions

Q: Convert denary 75 into 8-bit binary.

01001011

Q: Convert binary 00110100 into denary.

32 + 16 + 4 = 52

Q: Convert denary 140 into hexadecimal.

8C

Q: Convert hexadecimal 4F into denary.

64 + 8 + 4 + 2 + 1 = 79

Exam Tip

Always show your working. Write out the column header, show which columns have 1s, and show any subtraction steps. Even a wrong final answer can score method marks.

Hex Shortcut

8C in hex: 8 × 16 + 12 = 128 + 12 = 140. Each hex digit’s value = digit × 16^(position from right, starting at 0).

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