Complex Numbers
Grade 11 · Math · Study Guide
Explore complex numbers of the form a + bi, learn arithmetic operations, and understand the complex plane.
A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i = sqrt(-1). The set of complex numbers extends the reals to include solutions to equations like x^2 = -1.
You add/subtract complex numbers by combining real and imaginary parts separately. Multiply using FOIL and replace i^2 with -1. To divide, multiply numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of a + bi is a - bi. The modulus is |a + bi| = sqrt(a^2 + b^2), which is the distance from the origin in the complex plane.
| Power | Value |
|---|---|
| i^1 | i |
| i^2 | -1 |
| i^3 | -i |
| i^4 | 1 |
| i^5 | i (cycle repeats) |
| Operation | Rule |
|---|---|
| Add | (a+bi) + (c+di) = (a+c) + (b+d)i |
| Subtract | (a+bi) - (c+di) = (a-c) + (b-d)i |
| Multiply | (a+bi)(c+di) = (ac-bd) + (ad+bc)i |
| Divide | Multiply numerator and denominator by conjugate of denominator |
Key terms
- Imaginary unit
- i = sqrt(-1), so i^2 = -1.
- Complex number
- A number of the form a + bi where a and b are real numbers.
- Complex conjugate
- The conjugate of a + bi is a - bi; their product is a^2 + b^2.
- Modulus
- |a + bi| = sqrt(a^2 + b^2); the distance from the origin in the complex plane.
Practice questions
What is i^2?
- 1
- i
- -1
- -i
Show answer
By definition, i = sqrt(-1), so i^2 = -1.
Add: (3 + 4i) + (1 - 2i).
Show answer
Real parts: 3+1=4; imaginary parts: 4+(-2)=2. Result: 4 + 2i.
What is (2 + 3i)(2 - 3i)?
- 4 + 9i
- 13
- 4 - 9
- -5
Show answer
(a+bi)(a-bi) = a^2 + b^2 = 4 + 9 = 13.
Simplify i^7.
Show answer
Powers of i cycle with period 4. 7 mod 4 = 3, so i^7 = i^3 = -i.
What is the modulus of 3 + 4i?
- 7
- sqrt(7)
- 5
- 25
Show answer
|3 + 4i| = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Fun facts
- Complex numbers are essential in electrical engineering — AC circuit analysis uses i (written j by engineers) to represent phase shifts.
- The Mandelbrot set, one of the most famous fractals, is defined entirely using iterations of complex number arithmetic.
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