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# Add and Subtract Complex Numbers

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Author: Sophia Tutorial
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This lesson covers adding and subtracting complex numbers.

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Tutorial

• Complex Numbers
• Subtracting Complex Numbers

Complex Numbers

A complex number is a number in the form , containing both a real and imaginary part.  The imaginary part is followed by i, which is the imaginary unit,

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.

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 .  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