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Study Guides > Intermediate Algebra

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.

[latex](6x+8)+(4x+2)[/latex]

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.

[latex](6x+8)+(4x+2)=10x+10[/latex]

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.

Example

Add. [latex](−3+3i)+(7–2i)[/latex]

Answer: Rearrange the sums to put like terms together.

[latex]−3+3i+7–2i=−3+7+3i–2i[/latex]

Combine like terms.

[latex]−3+7=4[/latex]

and

[latex]3i–2i=(3–2)i=i[/latex]

The answer is [latex]4+i[/latex].

Example

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

Answer: Be sure to distribute the subtraction sign to all terms in the set of parentheses that follows.

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

Rearrange the terms to put like terms together.

[latex]−3–7+3i+2i[/latex]

Combine like terms.

[latex]−3–7=−10[/latex]

and

[latex]3i+2i=(3+2)i=5i[/latex]

The answer is [latex]-10+5i[/latex].

In the following video, we show more examples of how to add and subtract complex numbers. https://youtu.be/SGhTjioGqqA

Licenses & Attributions

CC licensed content, Original

CC licensed content, Shared previously

  • Ex 1: Adding and Subtracting Complex Numbers. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download fro free at: http://cnx.org/contents/[email protected]:1/Preface.