Binary numbers

We count in denary (base 10), with digits 0–9 and place values of 1, 10, 100 and so on. Computers use binary (base 2), with only the digits 0 and 1. Each place is worth double the one to its right:

With 8 bits, the smallest number is 00000000 (0) and the biggest is 11111111 (255). The left-hand bit, worth 128, is the most significant bit (MSB); the right-hand bit, worth 1, is the least significant bit (LSB).

A binary number can have fewer than 8 bits: 11010 is the same as 00011010, just as 042 is the same as 42.

Write the number under the place values, then add up the place values that have a 1.

Example: convert 10110101 to denary.

128 + 32 + 16 + 4 + 1 = 181

Work from the left. For each place value, ask: does it fit into what's left? If it does, write 1 and take it away; if not, write 0.

Example: convert 99 to binary.

1. 128 doesn't fit into 99: 0
2. 64 fits: 1, leaving 35
3. 32 fits: 1, leaving 3
4. 16, 8 and 4 don't fit into 3: 0, 0, 0
5. 2 fits: 1, leaving 1
6. 1 fits: 1, leaving 0

So 99 is 01100011. Check: 64 + 32 + 2 + 1 = 99.

Add from the right, column by column, just like denary addition, using these rules:

Example: 01011010 (90) + 00111001 (57)

The answer is 10010011, which is 147. Check in denary: 90 + 57 = 147.

An 8-bit number can't be bigger than 255. If the answer to an addition needs a 9th bit, there's nowhere to store it, so it's lost. This is an overflow error, and it gives a wrong answer.

Example: 11001000 (200) + 01100100 (100)

The answer, 300, needs 9 bits: 100101100. There is no 256 place to store the 9th bit, so only the last 8 bits are kept, 00101100, which is 44, not 300.

A binary shift moves every bit left or right by a number of places, filling the gaps with 0s.

• Left shift: each place multiplies the number by 2. A left shift of 2 places multiplies by 4.
• Right shift: each place divides the number by 2. Bits that fall off the right-hand end are lost, so the answer is rounded down.

Example: 00010110 is 22.