3.1Introduction to Complex Numbers
Define i = √(−1), write complex numbers in a + bi form, perform addition, subtraction, multiplication, and division, and find complex conjugates.
Complex numbers extend the real number system to solve equations like x² = −1. They are essential in electrical engineering, quantum mechanics, signal processing, and the study of polynomial roots.
Essential Question
Why do mathematicians extend the real number system to include imaginary numbers, and how do complex numbers behave under arithmetic operations?
Lesson Overview
The real numbers cannot solve every polynomial equation. For example, x² = −1 has no real solution because no real number squared gives a negative result. To fill this gap, mathematicians defined the imaginary unit i = √(−1), giving us the complex number system. Every complex number has the standard form a + bi, where a is the real part and b is the imaginary part. In this lesson you will learn to add, subtract, multiply, and divide complex numbers, find conjugates, compute the modulus, and solve equations with complex solutions.
Key Definitions
- Imaginary unit: i = √(−1), so i² = −1
- Complex number: a + bi (a, b ∈ ℝ)
- Real part: a; Imaginary part: b
- Complex conjugate of a+bi is a−bi
- Modulus: |a+bi| = √(a²+b²)
Learning Targets
- Write complex numbers in a + bi form
- Add, subtract, multiply, and divide complex numbers
- Simplify powers of i using the cycle i, −1, −i, 1
- Find the conjugate and modulus of a complex number
- Solve quadratic equations with complex solutions
The Complex Plane
Points plotted as (Re, Im)
Powers of i
Cycle repeats every 4 steps
Conjugate Product
Middle terms cancel
Worked Examples
Write √(−25) in terms of i and simplify.
√(−25) = √(25 · −1)
= √25 · √(−1)
= 5 · i
Add (3 + 4i) + (−1 + 2i).
Group real parts: 3 + (−1) = 2
Group imaginary parts: 4i + 2i = 6i
Multiply (2 + 3i)(1 − 4i).
FOIL: 2·1 + 2·(−4i) + 3i·1 + 3i·(−4i)
= 2 − 8i + 3i − 12i²
Replace i² = −1: = 2 − 8i + 3i − 12(−1)
= 2 + 12 + (−8 + 3)i
= 14 − 5i
Divide (3 + 2i) ÷ (1 − i). Write in a + bi form.
Multiply numerator and denominator by the conjugate (1 + i):
(3 + 2i)(1 + i) / [(1 − i)(1 + i)]
Denominator: 1² + 1² = 2
Numerator: 3 + 3i + 2i + 2i² = 3 + 5i + 2(−1) = 1 + 5i
= (1 + 5i) / 2
Solve x² + 9 = 0 over the complex numbers.
x² = −9
x = ±√(−9)
= ±√9 · √(−1)
= ±3i
Guided Practice
Simplify √(−49).
Hint: Factor out −1 under the radical, then use √(−1) = i.
Subtract (5 − 3i) − (2 + 7i).
Hint: Distribute the minus sign, then combine real parts and imaginary parts separately.
Multiply (4 − i)(4 + i).
Hint: This is a conjugate pair. Use (a − bi)(a + bi) = a² + b².
Divide (2 + i) ÷ (3 − 2i).
Hint: Multiply top and bottom by the conjugate of the denominator: (3 + 2i).
Simplify i¹⁵.
Hint: Divide the exponent by 4 and use the remainder: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
Key Vocabulary
Imaginary unit (i)
The number defined by i = √(−1), so i² = −1. It is the foundation of the complex number system.
Example: i² = −1; i³ = −i; i⁴ = 1
Complex number
A number of the form a + bi where a and b are real numbers. a is the real part and b is the imaginary part.
Example: 3 + 4i, −2 − i, 0 + 5i = 5i
Complex conjugate
The conjugate of a + bi is a − bi. Conjugates have the same real part but opposite imaginary parts.
Example: Conjugate of 3 + 2i is 3 − 2i
Modulus (absolute value)
The distance from the origin to a complex number in the complex plane. |a + bi| = √(a² + b²).
Example: |3 + 4i| = √(9 + 16) = 5
Standard form
A complex number written as a + bi with the real part first and the imaginary part second.
Example: Write 2i + 5 in standard form: 5 + 2i
Argand diagram
A coordinate plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.
Example: 3 + 2i is plotted at the point (3, 2)
Check Your Understanding
Interactive Practice — 5 Questions
What is the value of i²?
What is (2 + 3i) + (4 − i)?
What is the product (3 + i)(3 − i)?
Which is the complex conjugate of −5 + 2i?
What is i²³?
Independent Practice
Independent Practice
Simplify √(−64).
Write 7 − √(−16) in standard form a + bi.
Add (−3 + 5i) + (6 − 2i).
Subtract (4 + i) − (−1 + 3i).
Multiply (2 + 5i)(3 − i).
Common Mistakes
Writing √(−9) = −3 (treating the negative as a sign, not under the radical).
√(−9) = √9 · √(−1) = 3i. Always factor out i from a negative radicand.
Forgetting to replace i² with −1 after multiplying: (1 + i)² = 1 + 2i + i² = 1 + 2i + i².
Always substitute i² = −1: 1 + 2i + (−1) = 2i.
Dividing by a complex number without multiplying by the conjugate: (2 + i)/(1 + i) ≠ 2.
Multiply numerator and denominator by the conjugate of the denominator to get a real denominator.
Confusing the modulus with the imaginary part: |3 + 4i| = 4.
|3 + 4i| = √(3² + 4²) = √(9 + 16) = √25 = 5. Use the distance formula.
Math Tips
The powers of i cycle with period 4: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. To simplify iⁿ, find n mod 4.
Multiplying a complex number by its conjugate always gives a non-negative real number: (a+bi)(a−bi) = a² + b².
When dividing complex numbers, the conjugate trick turns the denominator into a real number, making simplification straightforward.
The modulus |a + bi| is the Pythagorean distance from the origin to the point (a, b) in the Argand diagram.
If the discriminant b² − 4ac is negative, the quadratic has two complex conjugate roots: x = (−b ± i√|Δ|) / (2a).