Math AA HL: Complex Numbers and De Moivre's Theorem

Complex numbers form a cornerstone of the IB Math AA HL syllabus, appearing heavily in both Paper 1 and Paper 2. Mastering their polar representation and De Moivre's theorem unlocks powerful algebraic and geometric techniques for solving equations, modeling periodic systems, and understanding rotations in the plane—skills essential for both exam success and higher mathematics.

Foundations: Cartesian and Polar Forms

A complex number $z$ is expressed in Cartesian form as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with $i^2 = -1$. On an Argand diagram, this represents the point $(a, b)$. From this geometric viewpoint, we derive the modulus and argument. The modulus, denoted $|z|$ or $r$, is the distance from the origin: $|z| = \sqrt{a^2 + b^2}$. The argument, denoted $\arg(z)$ or $\theta$, is the angle measured anticlockwise from the positive real axis. For a complex number in the first quadrant, $\theta = \arctan(b/a)$, but care must be taken to place the angle in the correct quadrant based on the signs of $a$ and $b$.

This leads to the polar form (or modulus-argument form): $z = r(\cos \theta + i\sin \theta)$. The principal argument is conventionally restricted to $-\pi < \theta \leq \pi$. Converting between forms is a fundamental skill. Given $a$ and $b$, you find $r$ and $\theta$. Conversely, given $r$ and $\theta$, you find $a = r\cos\theta$ and $b = r\sin\theta$.

Operations in Polar Form: Multiplication and Division

The true power of polar form becomes evident when multiplying and dividing complex numbers. Let $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$.

• Multiplication: The product is found by multiplying moduli and adding arguments.

$$z_1 z_2 = r_1 r_2 \left[ \cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2) \right]$$ Geometrically, multiplication by $z_2$ scales the point $z_1$ by $r_2$ and rotates it anticlockwise by $\theta_2$.

• Division: The quotient is found by dividing moduli and subtracting arguments.

$$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left[ \cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2) \right], \quad z_2 \neq 0$$ Geometrically, division by $z_2$ scales by $1/r_2$ and rotates clockwise by $\theta_2$.

These rules are vastly simpler than using Cartesian form, especially for successive operations.

De Moivre's Theorem: Powers and Roots

De Moivre's theorem is a logical extension of the multiplication rule. It states that for any integer $n$ and complex number $z = r(\cos\theta + i\sin\theta)$: $$[r(\cos\theta + i\sin\theta)]^n = r^{\,n}(\cos(n\theta) + i\sin(n\theta))$$

Finding Powers: The theorem provides a direct method. For example, to compute $(1 + i)^6$, first convert to polar form: $1+i = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))$. Then apply De Moivre: $$(\sqrt{2})^6 \left[ \cos(6 \cdot \pi/4) + i\sin(6 \cdot \pi/4) \right] = 8 (\cos(3\pi/2) + i\sin(3\pi/2)) = 8(0 - i) = -8i.$$

Finding Roots: Solving equations of the form $z^{\,n} = w$ is a primary application. If $w = R(\cos\phi + i\sin\phi)$, then the $n$ distinct nth roots of $w$ are given by: $$z_k = \sqrt[n]{R} \left[ \cos\left( \frac{\phi + 2\pi k}{n} \right) + i\sin\left( \frac{\phi + 2\pi k}{n} \right) \right]$$ for $k = 0, 1, 2, ..., n-1$. Crucially, you must use the general argument $\phi + 2\pi k$ to find all roots, not just the principal argument of $w$. The modulus of each root is the real $n$th root $\sqrt[n]{R}$.

Plotting on the Argand Diagram: These $n$ roots have a profound geometric property: they are equally spaced around a circle of radius $\sqrt[n]{R}$ centered at the origin. For example, the cube roots of unity ($z^3 = 1$) lie on the unit circle at angles $0$, $2\pi/3$, and $4\pi/3$, forming an equilateral triangle. Recognizing these geometric patterns is key to visualizing solutions.

Solving Equations and Geometric Applications

Beyond simple power equations, De Moivre's theorem facilitates solving more complex problems. You can express $\cos(n\theta)$ and $\sin(n\theta)$ as polynomials in $\cos\theta$ and $\sin\theta$, which is useful in calculus and trigonometry. Furthermore, the geometric interpretation of multiplication as a rotation-and-scaling transformation makes complex numbers ideal for modeling periodic phenomena or planar movements.

A common exam question involves solving a polynomial equation whose roots have a specific geometric relationship. For instance, "Find the vertices of a square centered at the origin if one vertex is at $2+2i$." This is solved by recognizing that multiplying by $i$ rotates by $\pi/2$. Thus, the vertices are $2+2i$, $i(2+2i) = -2+2i$, $i(-2+2i) = -2-2i$, and $i(-2-2i)=2-2i$.

Common Pitfalls

1. Incorrect Argument in Root Calculations: Using only the principal argument $\phi$ of $w$ when finding roots yields only one root ($k=0$). You must use the general form $\phi + 2\pi k$ for $k=0,1,...,n-1$ to find all $n$ distinct roots. Forgetting the $+2\pi k$ term is the most frequent error in this topic.

1. Argument Addition/Subtraction Errors: When using De Moivre's theorem for powers, ensure you correctly simplify $n\theta$. For $n=6$ and $\theta = \pi/4$, $n\theta = 3\pi/2$, not $3\pi/12$. For roots, carefully calculate $\frac{\phi + 2\pi k}{n}$.

1. Misapplying the Theorem to Non-Integer $n$: De Moivre's theorem, as presented in IB, applies for integer $n$. Be cautious not to apply it directly to fractional exponents (like $z^{1/3}$) without framing it as a root-finding problem $z^3 = w$. The root formula provided is the correct extension.

1. Polar Form Ambiguity: When converting from polar to Cartesian form, ensure your calculator is in radian mode, as arguments in IB are almost always in radians. Also, remember that $\cos(3\pi/2)=0$ and $\sin(3\pi/2)=-1$, not numerical approximations, unless a decimal answer is requested.

Summary

• A complex number $z = a+bi$ can be written in polar form as $z = r(\cos\theta + i\sin\theta)$, where $r=|z|$ is the modulus and $\theta = \arg(z)$ is the argument.
• Multiplication and division are elegant in polar form: multiply/dividing moduli and adding/subtracting arguments, which corresponds to geometric scaling and rotation.
• De Moivre's theorem, $[r(\cos\theta + i\sin\theta)]^n = r^{\,n}(\cos n\theta + i\sin n\theta)$, provides an efficient method for finding powers of complex numbers.
• To find all $n$ nth roots of a complex number $w = R(\cos\phi + i\sin\phi)$, use the formula involving $\frac{\phi + 2\pi k}{n}$ for $k = 0, 1, ..., n-1$. These roots are equally spaced on a circle in the Argand diagram.
• The geometric interpretation of complex operations makes them a powerful tool for solving equations related to polygons and rotations, frequently tested in IB exams.