bit kido.

Binary and addition

Binary Addition with Carry, Step by Step

Each binary digit can only be 0 or 1. Adding 1 and 1 makes 2, which does not fit in one digit, so you write 0 in this column and send a 1 to the next column on the left. You write it 10₂, and the 1 you sent is the carry.

The four basic rules

Adding one bit to one bit has only four cases. The sum is the digit that stays in this column; the carry is the digit sent to the next higher column.

One-bit binary addition
ProblemSumCarryBinary result
0+0000₂
0+1101₂
1+0101₂
1+10110₂

1+1+1 is 11₂

When you add numbers with several digits, a column can get a carry from the right on top of its two digits. Then three 1s are added in one column.

1+1+1 is 3, and 3 in binary is 11₂: write 1 in this column and send a carry of 1 to the next column.

3+1, column by column

Work out 0011₂ + 0001₂, which is 3+1. Start at the rightmost column. In each column, write the column's result first, then pass any carry to the left.

  • Column 1 (the 1s place): 1+1, write 0 and carry 1 to column 2.
  • Column 2 (the 2s place): 1+0 plus the carry 1 makes 2, write 0 and carry 1 to column 3.
  • Column 3 (the 4s place): 0+0 plus the carry 1, write 1 with no carry.
  • Column 4 (the 8s place): 0+0, write 0.
  • The answer is 0100₂, which is 4, matching 3+1.

0011₂ + 0001₂ = 0100₂

Place value8421
A0011
B0001
Carry in Cin0110
Sum S0100
Carry out Cout0011

Carry: 8 ← 4 ← 2 ← 1

The top row is 0011₂ and the bottom row is 0001₂. Rightmost column: 1+1, write 0, carry 1. Second column: 1+0+1, write 0, carry 1. Third column: 0+0+1, write 1, no carry. Fourth column: write 0. The result 0100₂ is 4.

Don't read binary results as decimal

The 10 in 1+1=10₂ is read "one zero". It means one 2 and no 1s, which is decimal 2, not ten.

Adding the subscript ₂ or writing the word binary keeps the two systems from getting mixed up.

Check yourself

  1. What decimal number is the 10₂ in 1₂ + 1₂ = 10₂?

    Answer

    It is 2. The 1 is in the 2s place and the 0 is in the 1s place, so 2+0 is 2, not ten.

  2. A column has 1+1 and also receives a carry of 1. What do you write, and is there a carry?

    Answer

    1+1+1 is 3, which is 11₂, so write 1 in this column and carry 1 to the next column.

  3. Work out 0010₂ + 0010₂.

    Answer

    Column 1: 0+0, write 0. Column 2: 1+1, write 0, carry 1. Column 3: 0+0+1, write 1. Column 4: write 0. The answer is 0100₂, and in decimal 2+2=4.