How to Add and Subtract Complex Numbers? (+FREE Worksheet!)
Learn more about complex numbers and how to add and subtract them using the following step-by-step guide.
Add and Subtract Complex Numbers: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
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
- Add real parts: 4 + 2.
- Add imaginary parts: 3i – 5i.
- Write both parts together.
Use i squared
- Multiply coefficients to get 5.
- i times i is i squared.
- Replace i squared with -1.
Try one before moving on
Add and Subtract Complex Numbers: pop-up practice
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.
- \(\color{blue}{– 8 + (2i) + (– 8 + 6i)}\)
- \(\color{blue}{12 – (5i) + (4 – 14i)}\)
- \(\color{blue}{– 2 + (– 8 – 7i) – 9}\)
- \(\color{blue}{(– 18 – 3i) + (11 + 5i)}\)
- \(\color{blue}{(3 + 5i) – (8 + 3i)}\)
- \(\color{blue}{(8 – 3i) – (4 + i)}\)
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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