Complex numbers are numbers that have a real part and an imaginary part. These numbers have the form a+bi, where a and b are real numbers and “i” is the imaginary unit, defined as the square root of negative one. We can perform various operations on these numbers, including addition, subtraction, multiplication, and division.

Here, we will learn how to solve addition of complex numbers. In addition, we will look at several examples with answers to fully master this topic.

ALGEBRA
addition of complex numbers

Relevant for

Learning to solve addition of complex numbers.

See examples

ALGEBRA
addition of complex numbers

Relevant for

Learning to solve addition of complex numbers.

See examples

How to solve addition of complex numbers?

To add two or more complex numbers, we simply have to add the real and imaginary parts separately. This is similar to adding polynomials, where we add like terms. Therefore, if we have the numbers z_{1}=a+bi and z_{2}=c+di, their addition is equal to:

z_{1}+z_{2}=(a+c)+(b+d)i

We see that the real part of the resulting number is the sum of the real parts of each complex number and the imaginary part of the resulting number is the sum of the imaginary parts of each complex number. That is, we have:

Re(z_{1}+z_{2})=Re(z_{1})+Re(z_{2})

Im(z_{1}+z_{2})=Im(z_{1})+Re(z_{2})

This applies to any number of complex numbers that we are adding.


Addition of complex numbers – Examples with answers

The process of solving additions of complex numbers mentioned above is used to solve the following examples. Each exercise has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the answer.

EXAMPLE 1

Add the numbers z_{1}=15+7i and z_{2}=4+8i.

We have to identify the real and imaginary parts of the numbers and add them separately. Therefore, we have:

z_{1}+z_{2}=15+7i+4+8i

=(15+4)+(7+8)i

=19+15i

EXAMPLE 2

Add the numbers z_{1}=-25+14i and z_{2}=13-15i.

We group the real and imaginary parts to add separately:

z_{1}+z_{2}=-25+14i+13-15i

=(-25+13)+(14-15)i

=-12-i

EXAMPLE 3

Add the numbers z_{1}=2+6i, z_{2}=-5-4i and z_{3}=4+2i.

Here, we have three complex numbers, but we have to follow the same procedure. We simply group the real and imaginary parts to add them separately:

z_{1}+z_{2}+z_{3}=2+6i-5-4i+4+2i

=(2-5+4)+(6-4+2)i

=1+4i

EXAMPLE 4

Add the numbers z_{1}=-5+3i, z_{2}=-12+11i and z_{3}=7-7i.

The process to follow is the same no matter how many complex numbers we have. We simply group the real and imaginary parts to add them separately:

z_{1}+z_{2}+z_{3}=-5+3i-12+11i+7-7i

=(-5-12+7)+(3+11-7)i

=-10+7i

EXAMPLE 5

If we have the numbers z_{1}=a+7i, z_{2}=-4+bi and z_{3}=4+2i, what is the value of a and b if we have z_{3}=z_{1}+z_{2}?

Adding the real and imaginary parts of the numbers z_{1} and z_{2} separately, we have:

4=a-4

⇒  a=8

2=7+b

⇒  b=-5

EXAMPLE 6

If we have the numbers z_{1}=a-5i, z_{2}=7+bi and z_{3}=-10-4i, what is the value of a and b if we have z_{3}=z_{1}+z_{2}?

Adding the real and imaginary parts of the numbers z_{1} and z_{2} separately, we have:

-10=a+7

⇒  a=-17

-4=-5+b

⇒  b=1


Addition of complex numbers – Practice problems

Test your knowledge of the addition of complex numbers to solve the following problems. Select an answer and click “Check” to verify that you chose the correct answer.

What is the result of the addition of z_{1}=6+4i and z_{2}=7+12i?

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What is the result of the addition of z_{1}=-7+11i y z_{2}=5-7i?

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If we have the numbers z_{1}=a-3i, z_{2}=5+bi and z_{3}=10+6i, what is the value of a and b if we have z_{3}=z_{1}+z_{2}?

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If we have the numbers z_{1}=a+13i, z_{2}=-8+bi and z_{3}=-1+5i, what is the value of a and b if we have z_{3}=z_{1}+z_{2}?

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See also

Interested in learning more about operations with complex numbers? Take a look at these pages:

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