Binary numbers: convert and add them by hand
Binary works like the decimal system, with only two digits. Each position is worth twice the one to its right instead of ten times. Once that clicks, conversions become simple additions.
Reading a binary number
From right to left, positions are worth 1, 2, 4, 8, 16, 32 and so on. Add the values of the positions holding a 1.
Example: 1011
- 8 + 0 + 2 + 1
1011 in binary is 11 in decimal
Converting decimal to binary
Divide by 2 repeatedly and note the remainders. Read them from last to first.
Example: 13
- 13 ÷ 2 = 6, remainder 1
- 6 ÷ 2 = 3, remainder 0
- 3 ÷ 2 = 1, remainder 1
- 1 ÷ 2 = 0, remainder 1
13 is 1101 in binary
Adding in binary
The rules are 0 + 0 = 0, 0 + 1 = 1 and 1 + 1 = 10, which means writing 0 and carrying 1.
Example: 101 + 011
- Rightmost column: 1 + 1 = 10, write 0, carry 1
- Middle column: 0 + 1 + 1 = 10, write 0, carry 1
- Left column: 1 + 0 + 1 = 10, write 10
101 + 011 = 1000, or 5 + 3 = 8
Bytes and hexadecimal
A byte holds 8 bits, giving values from 0 to 255. Hexadecimal groups bits by four: each group becomes one digit from 0 to F. So 1111 is F, and 1010 is A.
Common pitfalls
- Reading position values from the left: the smallest is always on the right.
- Reading remainders in the order they were produced.
- Forgetting the final carry when adding.
Worked exercises
Try to find the answer first, then open the solution.
Convert 10010 to decimal.
Show the solution
- 16 + 2
Answer: 18
Convert 25 to binary.
Show the solution
- 25 = 16 + 8 + 1
Answer: 11001
Calculate 110 + 101 in binary.
Show the solution
- 6 + 5 = 11 in decimal
- 11 = 8 + 2 + 1
Answer: 1011