How to convert positive denary whole numbers to binary numbers (up to and…

The idea

Denary is base 10, using digits 0–9. Binary is base 2, using only 0 and 1. An 8-bit binary number uses place values 128, 64, 32, 16, 8, 4, 2 and 1. To convert denary to binary, subtract the largest place value that fits, put 1 in that column, and repeat with the remainder until zero, filling unused columns with 0. For example, 157 = 128 + 16 + 8 + 4 + 1, so 157 in 8-bit binary is 10011101. To convert binary to denary, add the place values where the bit is 1. For example, 00101101 = 32 + 8 + 4 + 1 = 45. Always write exactly 8 bits when the question asks for 8-bit binary.

What earns the marks

• Identify the place values for 8-bit binary: 128, 64, 32, 16, 8, 4, 2, 1.
• Convert denary to binary by repeatedly subtracting the largest possible place value and recording 1s and 0s.
• Convert binary to denary by adding the place values of all bits that are 1.
• Produce an 8-bit result, including leading zeros where necessary, for example 45 as 00101101.
• Check the result by converting back to the original number or by ensuring the sum of place values equals the denary value.

Common mistakes and how to put them right

• Writing fewer than 8 bits when 8-bit binary is requested.

  Put it right: Pad with leading zeros on the left until there are exactly 8 bits.

• Using the wrong place values, such as starting at 1 instead of 128.

  Put it right: Write the place values 128 to 1 above the columns before converting.

• Adding all place values instead of only those with a 1.

  Put it right: For binary to denary, sum only the columns containing 1.

Exam tips

• Write the place values 128, 64, 32, 16, 8, 4, 2, 1 above your working to avoid column errors.
• For denary to binary, work left to right, placing 1 where the place value fits and 0 where it does not.
• Check your answer by converting back; the sum of the place values with 1s must equal the original denary number.