# Perform Operations with Complex Numbers

In this section we will perform operations with complex numbers such as addition,subtraction, multiplication and division of complex numbers.**Addition/subtraction of complex numbers :**The two or more complex numbers are to be added or subtracted like algebraic expression. That means like terms are to be added or subtracted.

**The terms with ‘i’ are like terms**

Let the two complex numbers z

_{1}= a

_{1}+ ib

_{1}and z

_{2}= a

_{2}+ ib

_{2}

So, z

_{1}+ z

_{2}= a

_{1}+ ib

_{1}+ a

_{2}+ ib

_{2}

= (a

_{1}+ a

_{2}) + (b

_{1}+ b

_{2})i

Similarly for subtraction

z

_{1}- z

_{2}= (a

_{1}- a

_{2}) + (b

_{1}- b

_{2})i

**Example 1 :**Add (3 + 7i) and (4 + 5i)

**Solution :**(3 + 7i) + (4 + 5i)

= 3 + 4 + 7i + 5i

= 7 + 12 i

**Example 2 :**Find the difference of z

_{1}= -3 + 2i and z

_{2}= 13 – i

**Solution :**z

_{2}- z

_{1}= 13 – i –(-3 + 2i)

= 13 – i + 3 – 2i

= 16 – 3i

**Multiplication of complex numbers**

The multiplication of complex numbers is same as the product of two binomial. Here you can use a FOIL method.

z

_{1}= a

_{1}+ ib

_{1}and z

_{2}= a

_{2}+ ib

_{2}z

_{1}. z

_{2}= (a

_{1}+ ib

_{1})(a

_{2}+ ib

_{2})

= a

_{1}a

_{2}+ a

_{1}b

_{2}i + a

_{2}b

_{1}i + b

_{1}b

_{2 }i

^{2}

= a

_{1}a

_{2}+ a

_{1}b

_{2}i + a

_{2}b

_{1}i - b

_{1}b

_{2}(since i

^{2}= -1)

**Example 1:**Multiply 2 + 3i and 5 – 4i

**Solution :**(2 + 3i)(5 – 4i)

= 10

__– 8i + 15i__– 12 i

^{2}

= 10 + 7i + 12 ( i

^{2}= -1)

= 22 + 7i

**Example 2 :**Show that : i

^{12}+ i

^{13}+ i

^{14}+ i

^{15}= 0

Solution :We have ,

i

^{12}+ i

^{13}+ i

^{14}+ i

^{15}

= 1 + i – 1 – i

= 0

**Division of complex numbers**z = z

_{1}/ z

_{2}, z

_{2}≠ 0 ,

In division always multiply the numerator and denominator by conjugates of denominator.

**Example :**z

_{1}= 3 + i and z

_{2}= 1 + i

**Solution :**

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