Binary numbers explained: converting and adding

Write the place values over the digits and add the ones under each 1.

$128 + 64 + 8 = 200$. So 11001000 in binary is 200.

Smaller example: $1101 = 8 + 4 + 1 = 13$.

Subtract the largest place values that fit, from the left:

13: 8 fits (leaves 5), 4 fits (leaves 1), 2 doesn't, 1 fits. So $13 = 1101$.

Or keep dividing by 2 and read the remainders from the bottom up: $13 \div 2 = 6$ r 1, $6 \div 2 = 3$ r 0, $3 \div 2 = 1$ r 1, $1 \div 2 = 0$ r 1, giving 1101.

The rules: $0 + 0 = 0$, $0 + 1 = 1$, $1 + 1 = 10$ (write 0, carry 1), $1 + 1 + 1 = 11$ (write 1, carry 1).

$1011 + 0110 = 10001$ (that's $11 + 6 = 17$).

If the answer needs more bits than are available, an overflow error occurs: in 4 bits, 10001 doesn't fit.

• A bit is a single binary digit.
• A nibble is 4 bits; a byte is 8 bits.
• With $n$ bits you can represent $2^n$ different values: 8 bits give $2^8 = 256$ values, from 0 to 255.

File sizes build up from bytes: kilobytes, megabytes, gigabytes. In GCSE Computer Science, a kilobyte is 1,000 bytes and a kibibyte is 1,024.

Hexadecimal (base 16) uses 0 to 9 then A to F (A = 10, F = 15). Each hex digit represents exactly 4 bits, so hex is a short way to write binary.

• 200 in binary is 1100 1000. 1100 = 12 = C and 1000 = 8, so 200 = C8 in hex.
• FF in hex = $15 \times 16 + 15 = 255$.

Programmers use hex for colours (#FF0000 is red) and memory addresses.

• Writing place values in the wrong order (the largest is on the left).
• Forgetting to carry in addition.
• Thinking 8 bits store up to 256 (the largest is 255, because 0 is one of the 256 values).

For more computing help, see our guide to learning Python and our computer science tutor.

1. Convert 10110 to decimal.
2. Convert 45 to binary.
3. Add 0101 and 0011 in binary.
4. Convert the hex number 3A to decimal.

Answers: 1. $16 + 4 + 2 = 22$. 2. 101101 ($32 + 8 + 4 + 1$). 3. 1000 (that's $5 + 3 = 8$). 4. $3 \times 16 + 10 = 58$.

• Text: each character has a binary code. ASCII uses 7 bits, giving $2^7 = 128$ characters; Unicode uses more bits to cover characters from every writing system and emoji.
• Images: a picture is a grid of pixels, and each pixel's colour is stored as a binary number. With a colour depth of 1 bit there are 2 colours; with 8 bits, $2^8 = 256$. More bits mean more colours and a bigger file.