Complex Number Properties

Why This Matters

Complex numbers aren't just an abstract mathematical curiosity—they're the key to unlocking problems that real numbers simply can't handle. In Honors Pre-Calc, you're being tested on your ability to manipulate, visualize, and convert between forms of complex numbers. These skills directly prepare you for calculus, physics, and engineering, where complex numbers model everything from electrical circuits to wave behavior.

The real power here lies in understanding how different representations connect. You'll need to know when rectangular form is most efficient, when polar form saves you time, and how conjugates and moduli help simplify messy expressions. Don't just memorize formulas—know why each property works and when to apply it. That's what separates a good score from a great one.

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Foundations: Building Blocks of Complex Numbers

Every complex number operation builds on these core definitions. Master these first, and everything else clicks into place.

Definition of a Complex Number

• Standard form is $a + bi$—where $a$ is the real part and $b$ is the imaginary part
• The imaginary unit $i$ satisfies $i^2 = -1$—this single property drives all complex arithmetic
• Each complex number maps to a unique point $(a, b)$ in the complex plane, bridging algebra and geometry

Real and Imaginary Parts

• $\text{Re}(z) = a$ extracts the real component—think of it as the horizontal coordinate
• $\text{Im}(z) = b$ extracts the imaginary component—the vertical coordinate (note: it's just $b$, not $bi$)
• Separating parts is essential for addition, subtraction, and equality comparisons—you'll use this constantly

Equality of Complex Numbers

• $z_1 = z_2$ requires both $a = c$ AND $b = d$—matching real and imaginary parts separately
• This creates a system of two equations—one equation from real parts, one from imaginary
• Equality problems often appear in equation-solving contexts—set coefficients equal on both sides

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Arithmetic Operations: Rectangular Form

When working in standard $a + bi$ form, these operations follow predictable patterns. Think of complex numbers as binomials with the special rule $i^2 = -1$.

Addition of Complex Numbers

• Add like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$—real with real, imaginary with imaginary
• Geometrically, this is vector addition—place arrows tip-to-tail in the complex plane
• Commutative and associative properties hold—order and grouping don't affect the result

Subtraction of Complex Numbers

• Subtract componentwise: $(a + bi) - (c + di) = (a - c) + (b - d)i$—same pattern as addition
• Represents the vector from $z_2$ to $z_1$—useful for finding distances and directions
• Watch your signs carefully—sign errors are the most common mistake on complex arithmetic problems

Multiplication of Complex Numbers

• FOIL and simplify: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$—the $-bd$ comes from $i^2 = -1$
• Multiplication rotates and scales in the complex plane—this geometric interpretation matters for polar form
• Commutative and associative—though the geometric effect (rotation + scaling) may not feel intuitive at first

Division of Complex Numbers

• Multiply by the conjugate: $\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$—this rationalizes the denominator
• The denominator becomes real: $c^2 + d^2 = |z_2|^2$—always a positive real number
• Final answer must be in $x + yi$ form—separate the real and imaginary parts after simplifying

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Complex Conjugates and Modulus

These tools transform complex expressions into simpler, often real-valued results. The conjugate "flips" across the real axis; the modulus measures distance from the origin.

Complex Conjugates

• The conjugate of $z = a + bi$ is $\overline{z} = a - bi$—flip the sign of the imaginary part only
• $z \cdot \overline{z} = a^2 + b^2$—always a non-negative real number (this is why conjugates simplify division)
• Conjugates distribute over operations: $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$—useful for simplifying expressions

Modulus (Absolute Value)

• $|z| = \sqrt{a^2 + b^2}$—the distance from the origin to point $(a, b)$
• $|z|^2 = z \cdot \overline{z}$—connects modulus to conjugates (memorize this relationship)
• Modulus is always non-negative and real—it behaves like absolute value for real numbers

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Polar Form and Geometric Representation

Polar form reveals the geometry behind complex numbers: every complex number has a magnitude (how far) and an argument (which direction). This form makes multiplication and powers dramatically easier.

Graphing on the Complex Plane

• The complex plane has a real axis (horizontal) and imaginary axis (vertical)—point $(a, b)$ represents $a + bi$
• Each complex number is a vector from the origin—length is the modulus, angle is the argument
• Operations have geometric meanings—addition is vector addition, multiplication involves rotation and scaling

Argument (Angle)

• $\theta = \tan^{-1}\left(\frac{b}{a}\right)$—but watch the quadrant! (adjust based on signs of $a$ and $b$)
• Measured from the positive real axis, counterclockwise—typically in radians for calculus preparation
• Principal argument is usually $-\pi < \theta \leq \pi$ or $0 \leq \theta < 2\pi$—know which convention your course uses

Polar Form

• $z = r(\cos\theta + i\sin\theta)$ where $r = |z|$—often abbreviated as $r\text{ cis }\theta$
• Multiplication becomes elegant: $z_1 z_2 = r_1 r_2 \text{ cis}(\theta_1 + \theta_2)$—multiply moduli, add angles
• Division follows the pattern: $\frac{z_1}{z_2} = \frac{r_1}{r_2}\text{ cis}(\theta_1 - \theta_2)$—divide moduli, subtract angles

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Powers, Roots, and Advanced Formulas

These formulas unlock the most challenging problems: finding high powers and extracting roots of complex numbers. De Moivre's Theorem is your best friend here.

De Moivre's Theorem

• $(r\text{ cis }\theta)^n = r^n \text{ cis}(n\theta)$—raise the modulus to the power, multiply the angle
• Works for integer powers, including negative exponents—$z^{-1} = \frac{1}{r}\text{ cis}(-\theta)$
• Essential for computing roots—to find $n$-th roots, use $r^{1/n}\text{ cis}\left(\frac{\theta + 2\pi k}{n}\right)$

Euler's Formula

• $e^{i\theta} = \cos\theta + i\sin\theta$—the most beautiful equation in mathematics
• Polar form becomes $z = re^{i\theta}$—exponential notation simplifies many calculations
• Special case: $e^{i\pi} + 1 = 0$—connects five fundamental constants (a famous result, sometimes tested)

Complex Roots of Unity

• The $n$-th roots of unity solve $z^n = 1$—there are exactly $n$ solutions
• Evenly spaced on the unit circle: $e^{2\pi i k/n}$ for $k = 0, 1, \ldots, n-1$—separated by angles of $\frac{2\pi}{n}$
• Sum of all $n$-th roots of unity equals zero—a useful property for simplification problems

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Quick Reference Table

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Self-Check Questions

1. What two conditions must be satisfied for $z_1 = z_2$? Why does this create a system of equations when solving for unknowns?

2. Compare rectangular and polar form: which would you choose to compute $(1 + i)^{10}$, and why? What theorem would you apply?

3. If $z = 3 - 4i$, find $|z|$ and $z \cdot \overline{z}$. How are these two values related?

4. The 4th roots of unity divide the unit circle into equal parts. What angle separates consecutive roots, and what are all four roots in exponential form?

5. (FRQ-style) Given $z_1 = 2\text{ cis }\frac{\pi}{6}$ and $z_2 = 3\text{ cis }\frac{\pi}{3}$, find $z_1 \cdot z_2$ and $\frac{z_1}{z_2}$ in polar form. Then convert $z_1 \cdot z_2$ to rectangular form.