AP Precalculus: Complex Numbers

Master operations with imaginary and complex numbers

🔢 Standard Form ➕ Operations 🔄 Conjugates ⚡ Powers of i

📚 Understanding Complex Numbers

Complex numbers extend the real number system to include solutions to equations like \(x^2 = -1\). The imaginary unit \(i = \sqrt{-1}\) allows us to work with square roots of negative numbers. Complex numbers are essential in engineering, physics, and advanced mathematics.

1 Definition & Standard Form

A complex number is written as \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i = \sqrt{-1}\) is the imaginary unit.

Standard Form \(z = a + bi\) where \(a, b \in \mathbb{R}\) and \(i^2 = -1\)

Real Part

\(\text{Re}(z) = a\)

Imaginary Part

\(\text{Im}(z) = b\)

Pure Real

\(b = 0\) (e.g., \(5\))

Pure Imaginary

\(a = 0\) (e.g., \(3i\))

📌 Examples

\(z = 3 + 4i\): Real part = 3, Imaginary part = 4

\(z = -2 - 5i\): Real part = -2, Imaginary part = -5

\(z = 7\): Real part = 7, Imaginary part = 0 (pure real)

\(z = 6i\): Real part = 0, Imaginary part = 6 (pure imaginary)

2 Powers of \(i\)

The powers of \(i\) follow a cyclic pattern that repeats every 4 powers.

\(i^1\)

\(i\)

\(i^2\)

\(-1\)

\(i^3\)

\(-i\)

\(i^4\)

\(1\)

Pattern Rule \(i^n = i^{n \mod 4}\) — Divide exponent by 4 and use remainder

📌 Example

Find: \(i^{27}\)

Divide: \(27 \div 4 = 6\) remainder \(3\)

Answer: \(i^{27} = i^3 = -i\)

💡 Quick Check

Remainder 0 → \(i^0 = 1\), Remainder 1 → \(i\), Remainder 2 → \(-1\), Remainder 3 → \(-i\)

3 Addition & Subtraction

Add or subtract complex numbers by combining like parts — real with real, imaginary with imaginary.

Addition/Subtraction Formula \((a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i\)

📌 Example: Addition

\((3 + 4i) + (2 - 5i)\)

= \((3 + 2) + (4 + (-5))i\)

= \(5 - i\)

📌 Example: Subtraction

\((7 - 2i) - (4 + 3i)\)

= \((7 - 4) + (-2 - 3)i\)

= \(3 - 5i\)

4 Multiplication

Multiply complex numbers using FOIL (distribute) and remember that \(i^2 = -1\).

Multiplication Formula \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)

Multiply using FOIL: First, Outer, Inner, Last
Combine like terms
Replace \(i^2\) with \(-1\)
Simplify to standard form \(a + bi\)

📌 Example

Calculate: \((2 + 3i)(4 - i)\)

FOIL: \(2(4) + 2(-i) + 3i(4) + 3i(-i)\)

= \(8 - 2i + 12i - 3i^2\)

= \(8 + 10i - 3(-1)\)

= \(8 + 10i + 3\)

= \(11 + 10i\)

5 Complex Conjugates

The complex conjugate of \(z = a + bi\) is \(\bar{z} = a - bi\). Just change the sign of the imaginary part.

Conjugate

\(\overline{a + bi} = a - bi\)

Product with Conjugate

\(z \cdot \bar{z} = a^2 + b^2\)

📌 Example

Given: \(z = 3 + 4i\)

Conjugate: \(\bar{z} = 3 - 4i\)

Product: \(z \cdot \bar{z} = (3 + 4i)(3 - 4i) = 9 - 16i^2 = 9 + 16 = 25\)

💡 Key Property

Multiplying a complex number by its conjugate always produces a real number: \(z \cdot \bar{z} = |z|^2\)

6 Division

To divide complex numbers, multiply both numerator and denominator by the conjugate of the denominator.

Division Formula \(\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\)

Identify the conjugate of the denominator
Multiply numerator and denominator by this conjugate
Simplify — denominator becomes \(c^2 + d^2\) (real)
Write in standard form \(a + bi\)

📌 Example

Calculate: \(\frac{2 + 3i}{1 - 2i}\)

Conjugate of denominator: \(1 + 2i\)

Multiply: \(\frac{(2 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)}\)

Numerator: \(2 + 4i + 3i + 6i^2 = 2 + 7i - 6 = -4 + 7i\)

Denominator: \(1 - 4i^2 = 1 + 4 = 5\)

Result: \(\frac{-4 + 7i}{5} = -\frac{4}{5} + \frac{7}{5}i\)

7 Absolute Value (Modulus)

The modulus (or absolute value) of a complex number is its distance from the origin in the complex plane.

Modulus Formula \(|z| = |a + bi| = \sqrt{a^2 + b^2}\)

Modulus

\(|z| = \sqrt{a^2 + b^2}\)

Product Rule

\(|z_1 \cdot z_2| = |z_1| \cdot |z_2|\)

Quotient Rule

\(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)

Conjugate Property

\(|z| = |\bar{z}|\)

📌 Example

Find: \(|3 - 4i|\)

Calculate: \(\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

8 The Complex Plane

Complex numbers can be plotted on the complex plane (Argand diagram), where the x-axis represents the real part and the y-axis represents the imaginary part.

Horizontal axis: Real axis (Re)
Vertical axis: Imaginary axis (Im)
Complex number \(a + bi\) is plotted at point \((a, b)\)
Modulus \(|z|\) = distance from origin to point
Conjugate \(\bar{z}\) = reflection across real axis

💡 Geometric Interpretation

Addition: Vector addition (parallelogram rule). Modulus: Length of vector from origin. Conjugate: Mirror image across x-axis.

📋 Quick Reference

Standard Form

\(z = a + bi\)

Powers of i

\(i, -1, -i, 1\) (repeats)

Conjugate

\(\bar{z} = a - bi\)

Modulus

\(|z| = \sqrt{a^2 + b^2}\)

Division

Multiply by conjugate of denominator

\(z \cdot \bar{z}\)

\(= a^2 + b^2 = |z|^2\)

📚 Understanding Complex Numbers

1 Definition & Standard Form

2 Powers of \(i\)

3 Addition & Subtraction

4 Multiplication

5 Complex Conjugates

6 Division

7 Absolute Value (Modulus)

8 The Complex Plane

📋 Quick Reference

Standard Form

Powers of i

Conjugate

Modulus

Division

\(z \cdot \bar{z}\)

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