Skip to main content
Mathovia

Algebra / Preliminaries

Complex Number Arithmetic

Every operation on complex numbers reduces to ordinary algebra with one extra fact in hand: i squared equals negative 1. This lesson works through addition, subtraction, multiplication, and division in turn, treating a complex number's real and imaginary parts the same careful way Adding Polynomials and Subtracting Polynomials treat like terms.

Practice Problems
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Chapter 1 · Lesson 21 of 21Preliminaries
  1. 1.Algebra Preliminaries
  2. 2.Integer Exponents
  3. 3.Negative Exponents
  4. 4.Exponent Rules
  5. 5.Rational Exponents
  6. 6.Radicals
  7. 7.Simplifying Radicals
  8. 8.Rationalizing Denominators
  9. 9.Polynomials
  10. 10.Adding Polynomials
  11. 11.Subtracting Polynomials
  12. 12.Multiplying Polynomials
  13. 13.Factoring Polynomials
  14. 14.Greatest Common Factor
  15. 15.Factoring by Grouping
  16. 16.Factoring Quadratics
  17. 17.Factoring Higher-Degree Polynomials
  18. 18.Rational Expressions
  19. 19.Simplifying Rational Expressions
  20. 20.Complex Numbers
  21. 21.Complex Number Arithmetic
See all algebra chapters →

Every operation on complex numbers is ordinary algebra with exactly one extra fact available: . This lesson works through addition, subtraction, multiplication, and division on complex numbers of the form , treating the real part and the imaginary part the way Adding Polynomials and Subtracting Polynomials treat like terms — because that’s really what they are.

Complex Number Arithmetic — key formula
Key formula

Adding and Subtracting Complex Numbers

Combine the real parts together and the imaginary parts together, exactly as like terms combine in a polynomial — a real number and an imaginary number are never like terms with each other.

Subtraction distributes the negative sign across both parts of the second complex number, exactly the same way it distributes across every term of a polynomial.

Multiplying Complex Numbers

A complex number is a binomial with standing in for a variable, so multiplying two of them uses ordinary FOIL — with one extra simplification step at the end, since .

The term is the one genuinely new thing that happens compared to multiplying two ordinary binomials — every that appears gets replaced with before the real and imaginary parts are combined.

Powers of i

The powers of repeat in a cycle of four:

PowerValue

For any power, divide the exponent by and match the remainder to the table: a remainder of behaves like , a remainder of like , and so on, with a remainder of behaving like .

Dividing Complex Numbers

Since can’t be left in a denominator, dividing by a complex number multiplies the numerator and denominator by the denominator’s conjugate — the same complex number with the sign of its imaginary part flipped.

The denominator always simplifies to — a real number — because it’s a difference of squares where the “squared” term becomes and flips the subtraction into addition.

Worked Example A: Adding and Subtracting

Simplify .

Step 1 — combine the first two complex numbers:

Step 2 — subtract the third, distributing the negative sign to both of its parts:

Answer:

Worked Example A: adding complex numbers adds their vectors tip-to-tail.
Worked Example A: adding complex numbers adds their vectors tip-to-tail.

Worked Example B: Multiplying Two Complex Numbers

Multiply .

Step 1 — apply FOIL:

Step 2 — replace with :

Step 3 — combine the real parts:

Answer:

Worked Example B: multiplying scales and rotates in the plane.
Worked Example B: multiplying scales and rotates in the plane.

Worked Example C: A Higher Power of i

Simplify .

Step 1 — divide the exponent by and find the remainder:

Step 2 — match the remainder to the power-of- table:

Answer:

Worked Example C: the powers of i repeat every four steps.
Worked Example C: the powers of i repeat every four steps.

Worked Example D: Dividing Complex Numbers

Divide .

Step 1 — multiply the numerator and denominator by the conjugate of the denominator:

Step 2 — expand the denominator using the difference of squares pattern:

Step 3 — expand the numerator with FOIL:

Answer:

Common Mistakes to Avoid

  • Combining real and imaginary parts as if they were like terms with each other. cannot be simplified to or to — the two parts stay separate unless another real or imaginary term is being added to the matching part.
  • Forgetting to replace with after multiplying. Leaving a term like unsimplified is not a finished answer; it must become .
  • Losing a sign when turns a subtraction into an addition. becomes , not — the double negative is easy to drop by accident.
  • Multiplying only the denominator by the conjugate, not the numerator. Both parts of the fraction need the same multiplier, since it has to equal to avoid changing the fraction’s value.
  • Writing the conjugate with the wrong sign. The conjugate of is — only the imaginary part’s sign flips, never the real part’s.
  • Miscounting the cycle when evaluating a power of . The cycle length is , not or ; dividing the exponent by and using the remainder is the reliable way to avoid a miscount.
Worked Example D: the conjugate makes the denominator real.
Worked Example D: the conjugate makes the denominator real.

Where This Shows Up Later

  • The quadratic formula. A negative discriminant produces two complex solutions, and reporting them in form uses exactly this lesson’s simplification rules.
  • Rationalizing Denominators. The conjugate trick used here to clear from a denominator is the identical technique used there to clear a radical.
  • Polar and trigonometric form of complex numbers. Multiplying and dividing complex numbers in polar form later gives a much faster alternative to FOIL, once angles and magnitudes are introduced.
  • Electrical engineering and signal processing. Complex number arithmetic, exactly as covered here, is the standard tool for combining alternating-current quantities in those fields.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Simplify .

Show answer

Answer:

Problem 2. Simplify .

Show answer

Answer:

Problem 3. Multiply .

Show answer

Answer:

Problem 4. Simplify .

Show answer

Answer:

Problem 5. Multiply .

Show answer

Difference of squares pattern:

Answer: (a real number, since these two complex numbers are conjugates of each other)

Problem 6. Divide .

Show answer

Step 1 — multiply by the conjugate:

Step 2 — expand the denominator:

Answer:

Problem 7. Simplify .

Show answer

Step 1 — combine the first two:

Step 2 — subtract the third:

Answer:

Problem 8. Multiply .

Show answer

Step 1 — apply the perfect-square pattern (or FOIL directly):

Step 2 — replace with :

Answer:

Problem 9. Divide .

Show answer

Step 1 — multiply by the conjugate:

Step 2 — expand the denominator:

Step 3 — expand the numerator:

Answer:

Problem 10. Simplify .

Show answer

Answer:

Quick Reference

OperationMove
Add or subtractCombine real parts together and imaginary parts together, separately
MultiplyFOIL, then replace every with and combine like parts
DivideMultiply top and bottom by the denominator’s conjugate
Conjugate of
Complex number times its conjugateAlways , a real number
Any power of Divide the exponent by ; match the remainder to

Nothing here is really new — Adding Polynomials, Multiplying Polynomials, and the conjugate trick from Rationalizing Denominators do all the heavy lifting, with as the one added fact. For where comes from in the first place, see Complex Numbers. Practice these operations with the Complex Number Arithmetic Practice Problems, or keep exploring the Algebra lessons.

Frequently Asked Questions

Why do I add or subtract the real and imaginary parts separately?+

A real number and an imaginary number are not like terms, the same way and aren't in an ordinary polynomial — and describe fundamentally different kinds of quantity, so they can only be combined with a part of the same kind.

Why does multiplying two complex numbers use FOIL?+

A complex number is a binomial in disguise, with playing the role an ordinary variable would. Multiplying two of them is a genuine binomial-times-binomial multiplication, which is exactly what FOIL is for — the only extra step is simplifying to afterward.

What happens to i squared, i cubed, and higher powers?+

The powers of cycle every four steps: , , , , and then the pattern repeats. Any power of can be evaluated by dividing the exponent by and using the remainder to look up which of those four values applies.

How do I divide by a complex number if i can't be in a denominator?+

Multiply the numerator and denominator by the conjugate of the denominator, exactly the same idea used to rationalize a radical denominator. A complex number times its own conjugate always produces a real number, which clears the imaginary part from the denominator completely.

What is the conjugate of a complex number?+

The conjugate of is — same real part, opposite sign on the imaginary part. Multiplying a complex number by its conjugate always gives , a real (non-negative) number, which is exactly why it clears an imaginary part from a denominator.

Is a complex number's answer always supposed to be in the form a + bi?+

Yes — every arithmetic result in this lesson should be reported with the real part first, the imaginary part second, and both fully simplified, even if one part turns out to be (in which case that part is usually just omitted).

Related lessons

Complex NumbersRationalizing DenominatorsRadicals