Algebra 2
Complex numbers
Operations with complex numbers
20 practice questions
0 video lessons
Theory + worked examples
Theory
Complex numbers obey the usual algebra with \(i^2=-1\):
- Add / subtract: combine real parts and imaginary parts.
- Multiply: FOIL, then replace \(i^2\) with \(-1\).
- Divide: multiply by the conjugate \(a-bi\) to clear \(i\) from the denominator.
The product of conjugates \((a+bi)(a-bi)=a^2+b^2\) is real.
The four operations on \(a+bi\).
Addition adds the parts, like vectors.
Multiplication and conjugates:
\[(a+bi)(c+di)=(ac-bd)+(ad+bc)i,\qquad (a+bi)(a-bi)=a^2+b^2\]
To divide, multiply by the conjugate of the denominator.
How to operate
- Add/subtract by combining like parts.
- Multiply with FOIL, then simplify \(i^2=-1\).
- Divide by multiplying by the denominator's conjugate.
- Write the result as \(a+bi\).
Example 1 — Add
Add \((3+2i)+(1-4i)\).
Solution
Combine real and imaginary parts.
| \((3+1)+(2-4)i\) | \(=\) | \(4-2i\) |
Example 2 — Subtract
Subtract \((5+i)-(2+3i)\).
Solution
Distribute the minus sign.
| \((5-2)+(1-3)i\) | \(=\) | \(3-2i\) |
Example 3 — Multiply
Multiply \((2+3i)(1-i)\).
Solution
FOIL, then replace \(i^2\) with \(-1\).
| \(2-2i+3i-3i^2\) | ||
| \(=\) | \(2+i+3\) | |
| \(=\) | \(5+i\) |
Example 4 — Divide
Simplify \(\dfrac{4}{1+i}\).
Solution
Multiply top and bottom by the conjugate \(1-i\).
| \(\dfrac{4(1-i)}{(1+i)(1-i)}\) | \(=\) | \(\dfrac{4-4i}{2}\) |
| \(=\) | \(2-2i\) |
Common pitfalls
Replace \(i^2\) with \(-1\) after multiplying.
Use the conjugate to divide — never leave \(i\) in a denominator.
Combine only like parts: reals with reals, \(i\) with \(i\).
Frequently asked questions
How do you add complex numbers?
Add the real parts and the imaginary parts separately.
How do you multiply complex numbers?
FOIL, then replace \(i^2\) with \(-1\).
What is a complex conjugate?
\(a-bi\) is the conjugate of \(a+bi\); their product is real.
How do you divide complex numbers?
Multiply numerator and denominator by the conjugate of the denominator.
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Introduction to complex numbers (i, a + bi form)
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Complex solutions to quadratics
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