Number bases revision notes

Number bases

A number base determines the values represented by the positions in a number. The three bases required here are decimal (base 10), binary (base 2) and hexadecimal (base 16). A base can be shown using a subscript: 67₁₀, 10011011₂ and AE₁₆.

Decimal and binary

Decimal uses the digits 0 to 9. Each position represents a power of 10. Binary uses only 0 and 1. Each position represents a power of 2, starting with 2⁰ at the right-hand end.

For example, to convert 101101₂ to decimal, identify the place values:

text Binary digit: 1 0 1 1 0 1 Place value: 32 16 8 4 2 1 Contribution: 32 0 8 4 0 1

Add the contributions: 32 + 8 + 4 + 1 = 45, so 101101₂ = 45₁₀.

To convert 45₁₀ to binary, use powers of 2. The largest required power is 32. Subtracting successively gives 45 - 32 = 13, 13 - 8 = 5, 5 - 4 = 1, and 1 - 1 = 0. Therefore the bits for 32, 16, 8, 4, 2 and 1 are 1 0 1 1 0 1, giving 101101₂.

Hexadecimal

Hexadecimal is base 16. It uses the digits 0 to 9 and the letters A to F, where A represents 10, B represents 11, C represents 12, D represents 13, E represents 14 and F represents 15. Each position represents a power of 16.

For example, 2F₁₆ is (2 × 16) + 15 = 47₁₀.

Hexadecimal as a shorthand for binary

One hexadecimal digit represents four binary bits because 16 possible hexadecimal values correspond to the 16 possible patterns of four bits. The conversion is direct:

text Hex: 0 1 2 3 4 5 6 7 8 9 A B C D E F Binary:0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

For AE₁₆, convert each hexadecimal digit separately: A becomes 1010 and E becomes 1110. Thus AE₁₆ = 10101110₂. When converting binary to hexadecimal, group bits from the right into groups of four and add leading zeroes if the leftmost group is incomplete. For example, 1101011₂ becomes 0110 1011, which is 6B₁₆.

Common errors

• Writing a digit outside the permitted range for the base, such as 2 in binary.
• Reversing the powers of 2 or powers of 16.
• Forgetting that A to F represent 10 to 15.
• Grouping binary digits incorrectly when converting to hexadecimal.
• Omitting leading zeroes when a four-bit hexadecimal group is needed.
• Adding hexadecimal digits as if they were decimal digits without using their place values.