Number bases: binary (base 2)

The idea

Binary is a positional number system with base 2, so each place value is a power of two and only the digits 0 and 1 are allowed. Reading from the right, the place values are 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16 and so on, doubling each time. To convert a denary value such as 13, subtract the largest power of two that fits (8), then 4, then 1, giving 1101. To convert back, add the place values where a 1 appears: 8 + 4 + 1 = 13. A common method is to divide repeatedly by 2 and read the remainders upwards. Recognising that base 2 means two digit symbols and powers of two place values is the key idea tested.

What earns the marks

• States that binary is base 2, so it uses only the digits 0 and 1.
• Explains that each position represents a power of 2, doubling from right to left (1, 2, 4, 8, 16...).
• Converts a denary number to binary by subtracting the largest fitting powers of two, or by repeated division by 2 reading remainders upwards.
• Converts a binary number to denary by adding the place values where a 1 appears.
• Recognises that n binary digits can represent 2ⁿ different values.

Common mistakes and how to put them right

• Thinking binary uses the digits 0 to 9 like denary.

  Put it right by stressing base 2 allows only 0 and 1.

• Reading place values left to right as increasing.

  Put it right by showing the rightmost column is 2⁰ = 1 and values double leftwards.

• Writing remainders from repeated division in the wrong order.

  Put it right by reading remainders from the last division upwards.

Exam tips

• Write the place-value headings (16, 8, 4, 2, 1) above the columns before converting.
• Check a conversion by adding the place values back to recover the original denary number.
• For multiple-choice questions, eliminate options containing any digit other than 0 or 1.