A complex number is of the form $$x+iy$$ and is usually represented by $$z$$. Every complex number indicates a point in the XY-plane.

For example, the complex number $$x+iy$$ represents the point $$(x,y)$$ in the XY-plane.

You can see this in the following illustration. Here, you can drag the point by which the complex number and the corresponding point are changed.

But, how to calculate complex numbers? Let's learn how to add complex numbers in this sectoin.

## Lesson Plan

 1 What Do You Mean by Addition of Complex Numbers? 2 Important Notes on Addition of Complex Numbers 3 Solved Examples on Addition of Complex Numbers 4 Tips and Tricks on Addition of Complex Numbers 5 Interactive Questions on Addition of Complex Numbers

## What Do You Mean by Addition of Complex Numbers?

Geometrically, the addition of two complex numbers is the addition of corresponding position vectors using the parallelogram law of addition of vectors.

Consider two complex numbers: $\begin{array}{l} z_{1}=a_{1}+i b_{1} \\[0.2cm] z_{2}=a_{2}+i b_{2} \end{array}$

We already know that every complex number can be represented as a point on the coordinate plane (which is also called as complex plane in case of complex numbers).

So let us represent $$z_1$$ and $$z_2$$ as points on the complex plane and join each of them to the origin to get their corresponding position vectors. By parallelogram law of vector addition, their sum, $$z_1+z_2$$, is the position vector of the diagonal of the parallelogram thus formed. For this,

• First, draw the parallelogram with $$z_1$$ and $$z_2$$ as opposite vertices.

• Draw the diagonal vector whose endpoints are NOT $$z_1$$ and $$z_2$$.

• The resultant vector is the sum $$z_1+z_2$$.

i.e., the sum is the tip of the diagonal that doesn't join $$z_1$$ and $$z_2$$. ### Example

Let us add two complex numbers:

$\begin{array}{l} z_{1}=3+3i\\[0.2cm] z_{2}=-3+i \end{array}$

We know that:

• $$z_1=3+3i$$ corresponds to the point (3, 3) and

• $$z_2=-3+i$$ corresponds to the point (-3, 1).

We just plot these on the complex plane and apply the parallelogram law of vector addition (by which, the tip of the diagonal represents the sum) to find their sum. The tip of the diagonal is (0, 4) which corresponds to the complex number $$0+4i = 4i$$.

Hence, $z_1+z_2 = 4i$

You can visualize the geometrical addition of complex numbers using the following illustration:

## What Are the Steps to Add Complex Numbers?

But isn't that process difficult?

Here is the easy process to add complex numbers.

### Example

Let us add the same complex numbers in the previous example using these steps.

$\begin{array}{l} z_{1}=3+3i\\[0.2cm] z_{2}=-3+i \end{array}$

Their sum is $z_1+z_2 = (3+3i)+(-3+i)$

 Group the real and imaginary numbers $$(3-3)+(3i+i)$$ $$0+4i=4i$$

Thus, the sum of the given two complex numbers is: $z_1+z_2= 4i$

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More Important Topics
Numbers
Algebra
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More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
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