**Binary Arithmetic Operations**

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

### Addition & Subtraction:

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;**a.** Add 111_{two} and 101_{two}**b.** Add 1101_{two} + 10001_{two} + 1011_{two}**c.** 1.1011_{two}Â + 10.1001_{two}Â + 10.01_{two}**d.** 1011_{two} + 1101_{two}**e.** 10011_{two} + 1101_{two} + 111_{two}

**Solution**

**a.** Add 111_{two} and 101_{two}

**Answer** = 1100_{two}

**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**.

Answer = 1100_{2}

**b.** Add 1101_{two} + 10001_{two} + 1011_{two}

**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**.

**Answer** = 101001_{2}

**c. ** 1.1011_{two}Â + 10.1001_{two}Â + 10.01_{two}

**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

**Answer** = 110.1000_{2}

**d.** 1011_{two} + 1101_{two}

**e.** 10011_{two} + 1101_{two} + 111_{two}

**Work To Do**

**1.** 10011 + 1101

**2.** 1011 + 1101 + 101

**Answers**

**1.** 100000_{two}

**2.** 11101_{two}

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