Lesson 2, Topic 1
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Binary Arithmetic Operations

Basic arithmetic operations of addition, subtraction, multiplication, and division are carried out in their bases similar to base 10.

To add in base two note that;

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (which is 0 carry 1)

Remember that in base 10, 1 + 1 = 2, But 2 in base 2 is 10. therefore, in binary 1 + 1 = 10.

Examples

Solve the following;

b. Add 1101two + 10001two + 1011two
c. 1.1011twoÂ  + 10.1001twoÂ  + 10.01two
d. 1011two + 1101two
e. 10011two + 1101two + 111two

Solution

Alternative Explanation

1st Column (from right to left) = 1 + 1 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 2.

2nd Column = 1 + 1 + 0 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 3.

3rd Column = 1 + 1 + 1 = 3. How many twos are there in 3? $$\frac{3}{2}$$ = 1 remainder 1. Write 1 and carry 1 to column 4.

4th column = 1…Write 1.

b. Add 1101two + 10001two + 1011two

Alternative Explanation

1st Column (from right to left) = 1 + 1 + 1 = 3. How many twos are there in 3? $$\frac{3}{2}$$ = 1 remainder 1. Write 1 and carry 1 to column 2.

2nd Column = 1 + 0 + 0 + 1 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 3.

3rd Column = 1 + 1 + 0 + 0 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 4.

4th Column = 1 + 1 + 0 + 1 = 3. How many twos are there in 3? $$\frac{3}{2}$$ = 1 remainder 1. Write 1 and carry 1 to column 5.

5th Column = 1 + 1 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 6.

6th Column = +1 …. Write 1.

c. 1.1011twoÂ  + 10.1001twoÂ  + 10.01two

Alternative Explanation

1st Column (from right to left) = 1 + 1 + 0 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 2.

2nd Column = 1 + 1 + 0 + 0 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 3.

3rd Column = 1 + 0 + 0 + 1 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 4.

4th Column = 1 + 1 + 1 + 0 = 3. How many twos are there in 3? $$\frac{3}{2}$$ = 1 remainder 1. Write 1 and carry 1 to column 5.

5th Column = 1 + 1 + 0 + 0 = 2. How many twos are there in 2? $$\frac{2}{2}$$ = 1 remainder 0. Write 0 and carry 1 to column 6.

6th Column = 1 +1 + 1 = 3. How many twos are there in 3? $$\frac{3}{2}$$ = 1 remainder 1. Write 1 and carry 1 to column 7.

7th column = 1…Write down 1

d. 1011two + 1101two

e. 10011two + 1101two + 111two

Work To Do

1. 10011 + 1101

2. 1011 + 1101 + 101