How to Add and Subtract Complex Numbers? (+FREE Worksheet!)

How to add and subtract complex numbers, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Learn more about complex numbers and how to add and subtract them using the following step-by-step guide.

How to Add and Subtract Complex Numbers? (+FREE Worksheet!)

Tutor-style math help

Add and Subtract Complex Numbers: what to notice and how to work it

Complex skill

Complex numbers have a real part and an imaginary part. Keeping those parts organized makes operations feel much more predictable.

What to notice first

Group real terms with real terms and imaginary terms with imaginary terms. The special fact (i^2=-1) drives multiplication and division.

Common student mistake

Do not leave (i^2) unchanged. Replacing it with -1 is the key simplification step.

Key formulas and cues

(i^2=-1)

((a+bi)+(c+di)=(a+c)+(b+d)i)

((a+bi)(c+di)=ac+adi+bci+bd i^2)

(|a+bi|=sqrt{a^2+b^2})

A reliable path

Separate partsKeep real and imaginary terms in their own lanes.
Use i squaredReplace (i^2) with -1 whenever it appears.
Use conjugatesFor division, multiply by the conjugate to make the denominator real.

Worked examples

Add complex numbers

Example: ((4+3i)+(2-5i))

Add real parts: 4 + 2.
Add imaginary parts: 3i – 5i.
Write both parts together.

Answer: (6-2i)

Use i squared

Example: (i(5i))

Multiply coefficients to get 5.
i times i is i squared.
Replace i squared with -1.

Answer: (-5)

Open pop-up practice

Try one before moving on

Try: Simplify ((5-2i)+(1+6i)).

Answer: (6+4i).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Related Topics

Step by step guide to add and subtract the complex numbers

A complex number is expressed in the form (a+bi), where (a) and (b) are real numbers, and (i), which is called an imaginary number, is (a ) solution of the equation (x^2=-1)
For adding complex numbers: ((a+bi)+(c+di)=(a+c)+(b+d)i)
For subtracting complex numbers: ((a+bi)-(c+di)=(a-c)+(b-d)i)

For education statistics and research

Adding and Subtracting Complex Numbers, Example 1:

Solve: (10+(-5-3i)-2)

Solution:

Remove parentheses: (10+(-5-3i)-2=10-5-3i-2)
Combine like terms: (10-5-2=10-7=3)

Then: (10-5-3i-2=3-3i)

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Adding and Subtracting Complex Numbers, Example 2:

Solve: (-3+(4i)+(9-2i))

Solution:

Remove parentheses: (-3+(4i)+(9-2i)=-3+4i+9-2i)
Combine like terms: (-3+9=6, 4i- 2i=2i)

Then: (-3+4i+9-2i=6+2i)

Adding and Subtracting Complex Numbers, Example 3:

Solve: ((-8+2i)+(-8+6i))

Solution:

Remove parentheses: ((-8+2i)+(-8+6i)=-8+2i-8+6i)
Combine like terms: (-8-8=-16, 2i+6i=8i)
Then: (-8+2i-8+6i=-16+8i)

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Adding and Subtracting Complex Numbers, Example 4:

Solve: ((6-3i)-(-8-7i))

Solution:

Remove parentheses by multiplying the negative sign to the second parenthesis: ((6-3i)-(-8-7i)=6-3i+8+7i)
Combine like terms: (6+8=14, -3i+7i=4i)

Then: (6-3i+8+7i= 14+ 4i)

Exercises for Adding and Subtracting Complex Numbers

Simplify.

(color{blue}{, 8 + (2i) + (, 8 + 6i)})
(color{blue}{12, (5i) + (4 to 14i)})
(color{blue}{, 2 + (, 8 to 7i), 9})
(color{blue}{(, 18 to 3i) + (11 + 5i)})
(color{blue}{(3 + 5i), (8 + 3i)})
(color{blue}{(8 to 3i), (4 + i)})

Download Adding and Subtracting Complex Numbers Worksheet

(color{blue}{- 16 + 8i})
(color{blue}{16 – 19i})
(color{blue}{-19 – 7i})
(color{blue}{-7 + 2i})
(color{blue}{-5 + 2i})
(color{blue}{4 – 4i})

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Effortless Math Team

Effortless Math Team

3 days ago

Effortless Math Team

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