Online College Courses for Credit

3 Tutorials that teach Add and Subtract Complex Numbers
Take your pick:
Add and Subtract Complex Numbers

Add and Subtract Complex Numbers

Author: Sophia Tutorial

This lesson covers adding and subtracting complex numbers. 

See More

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to many different colleges and universities.*

Begin Free Trial
No credit card required

29 Sophia partners guarantee credit transfer.

311 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 27 of Sophia’s online courses. Many different colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.


What's Covered

  • Complex Numbers
  • Adding Complex Numbers
  • Subtracting Complex Numbers

Add and Subtract Complex Numbers

Complex Numbers

A complex number is a number in the form a plus b i, containing both a real and imaginary part.  The imaginary part is followed by i, which is the imaginary unit, root index blank of negative 1 end root

We can combine complex numbers through addition and subtraction, just like we can add or subtract real numbers.  The biggest distinction is that the real numbers and imaginary numbers remain separated, as is the case when combining like terms.  This is the case with both addition and subtraction, where the only difference in the processes is the operation. 

Adding Complex Numbers

We can think of a complex number addition problem as containing two addition sets for us.  First, we'll add the real numbers together, which will constitute the first half of our solution.  Secondly, we'll add the imaginary numbers together, and express that as the second half to the solution.  This is illustrated in the example below:

Sometimes, the addition has negative numbers in them.  In these cases, we follow the same process when adding negative numbers: we can think of adding a negative number as subtracting a positive number.


Subtracting Complex Numbers

When subtracting complex numbers, we again can break the problem down into two sets of subtraction: one set for all real numbers, and another set for the imaginary numbers.  The trickiest part with subtraction problems is paying attention to the sign of the numbers and the differences.  This will be a particular concern when the subtraction problem contains negative numbers.  Below are some examples of complex number subtraction:


Be sure to still combine the real number and the imaginary number with addition.  Although we are performing subtraction, remember that our general complex number is in the form a plus b i.  We would only see a minus sign between the two terms if the imaginary part was negative. 


There are two things in particular to note in the previous example:

  • When subtracting the real numbers, we subtracted a negative number.  This can be thought of as adding a positive number. 
  • When subtracting the imaginary numbers, the sum was negative.  This is why we see a minus sign in our solution. 
Formulas to Know
Imaginary Number

i equals square root of negative 1 end root

i squared equals negative 1