Numbers: How to convert positive denary whole numbers to binary numbers (up to and including 8 bits) and vice versa

Denary (decimal, base 10) numbers need to be converted to binary (base 2) for computer processing. You must be able to convert both ways for numbers 0-255 using up to 8 bits.

Key points

• Definition:Denary (Base 10): a number system that uses ten digits, from 0 to 9.
• Binary uses place value headers based on powers of 2: 128, 64, 32, 16, 8, 4, 2, 1.
• To convert binary to denary: add together the place values where there is a 1.
• To convert denary to binary: start from the largest place value, place a 1 if it fits (subtract it), 0 if not, then move right.
• Definition:Most Significant Bit (MSB): the leftmost 1 in a binary number — it has the highest value.
• Definition:Least Significant Bit (LSB): the rightmost bit — it has the lowest value (1).
• Adding leading zeros does not change the value: 11010 = 00011010.
• Exam Tip:Always write out the place value headers (128, 64, 32, 16, 8, 4, 2, 1) when doing conversions — it reduces mistakes.
• Exam Tip:If the answer must be 8-bit binary, you MUST include leading zeros. '111' must be written as 00000111.
• Common Mistake:Writing the binary header numbers the wrong way around (1, 2, 4, 8... left-to-right). The LEFTMOST bit must be the HIGHEST value (128).
• Common Mistake:Saying the smallest number in 4-bit binary is 1. It is 0 (0000). The smallest is always 0.
• Exam Tip:The largest denary number in n bits = 2^n - 1. E.g. 6 bits = 63 (not 64). 8 bits = 255 (not 256).