Add and Subtract Complex Numbers

Learning Outcomes

  • Add complex numbers
  • Subtract complex numbers

Any time new kinds of numbers are introduced, one of the first questions that needs to be addressed is, “How do you add them?” In this section, you will learn how to add and subtract complex numbers.

First, consider the following expression.


To simplify this expression, you combine the like terms, [latex]6x[/latex] and [latex]4x[/latex]. These are like terms because they have the same variable with the same exponents. Similarly, 8 and 2 are like terms because they are both constants, with no variables.


In the same way, you can simplify expressions with radicals.

[latex] (6\sqrt{3}+8)+(4\sqrt{3}+2)=10\sqrt{3}+10[/latex]

You can add [latex] 6\sqrt{3}[/latex] to [latex] 4\sqrt{3}[/latex] because the two terms have the same radical, [latex] \sqrt{3}[/latex], just as [latex]6[/latex]x and [latex]4[/latex]x have the same variable and exponent.

The number i looks like a variable, but remember that it is equal to [latex]\sqrt{-1}[/latex]. The great thing is you have no new rules to worry about—whether you treat it as a variable or a radical, the exact same rules apply to adding and subtracting complex numbers. You combine the imaginary parts (the terms with i), and you combine the real parts.


Add. [latex](−3+3i)+(7–2i)[/latex]


Subtract. [latex](−3+3i)–(7–2i)[/latex]

In the following video, we show more examples of how to add and subtract complex numbers.