Algebra 2 · Unit 2 · Topic 2.3
Complex Arithmetic and Conjugates
Add complex numbers by combining real and imaginary parts.
Definition and explanation
Add complex numbers by combining real and imaginary parts. Multiply by distribution and replace i² with −1. For division by a nonzero complex number, multiplying by its conjugate makes the denominator real.
Worked example
Compute (2 + i)/(3 − i). Multiply by (3 + i)/(3 + i): the numerator is 6 + 5i + i² = 5 + 5i and the denominator is 9 − i² = 10. The result is 1/2 + i/2.
Common mistake
Multiplying by a conjugate changes both numerator and denominator; changing just one alters the value.
Key takeaway
Add complex numbers by combining real and imaginary parts.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.