IB AA: Complex Numbers Introduction

Complex numbers are not just an abstract mathematical curiosity; they are a powerful and necessary extension of the real number system. They allow you to solve equations like $x^2 + 1 = 0$ that have no real solutions, and they form the foundational language for advanced fields in engineering, physics, and computer science. In the IB Analysis & Approaches course, mastering complex numbers is essential for understanding oscillatory motion, electrical circuits, and ultimately, the elegant power of de Moivre's theorem.

The Imaginary Unit and Cartesian Form

The journey begins with the imaginary unit $i$, which is defined by the property $i^2 = -1$. This single definition unlocks a new dimension of numbers. A complex number in Cartesian form (or rectangular form) is written as $z = a + bi$, where $a$ and $b$ are real numbers. Here, $a$ is called the real part, denoted $\Re(z)$, and $b$ is called the imaginary part, denoted $\Im(z)$.

Think of this as giving a location on a map: $a$ tells you how far east/west to go (the real axis), and $b$ tells you how far north/south to go (the imaginary axis). A purely real number like $5$ is simply $5 + 0i$, and a purely imaginary number like $3i$ is $0 + 3i$. The set of all complex numbers is denoted by $\mathbb{C}$.

Arithmetic Operations and Conjugates

You can perform arithmetic with complex numbers in Cartesian form by treating $i$ as a variable, but always remembering that $i^2 = -1$.

• Addition/Subtraction: Combine real parts with real parts, and imaginary parts with imaginary parts.

$$(2 + 3i) + (5 - i) = (2+5) + (3-1)i = 7 + 2i$$

• Multiplication: Use the distributive property (like FOIL for binomials).

$$(2 + 3i)(5 - i) = 10 - 2i + 15i - 3i^2 = 10 + 13i - 3(-1) = 13 + 13i$$ Notice how the $i^2$ term became $-1$, converting it back into a real number.

• Division: This requires the concept of a complex conjugate. The conjugate of $z = a + bi$ is $\overline{z} = a - bi$. To divide, multiply the numerator and denominator by the conjugate of the denominator to create a real number in the denominator.

$$\frac{2 + 3i}{5 - i} = \frac{(2 + 3i)}{(5 - i)} \cdot \frac{(5 + i)}{(5 + i)} = \frac{10 + 2i + 15i + 3i^2}{25 - i^2} = \frac{10 + 17i - 3}{25 + 1} = \frac{7 + 17i}{26} = \frac{7}{26} + \frac{17}{26}i$$

The conjugate is a crucial tool. When you multiply a complex number by its conjugate, the result is always a non-negative real number: $(a+bi)(a-bi) = a^2 + b^2$.

The Argand Diagram, Modulus, and Argument

An Argand diagram is the coordinate plane for complex numbers. The horizontal axis is the real axis, and the vertical axis is the imaginary axis. The complex number $z = a + bi$ corresponds to the point $(a, b)$.

This geometric view introduces two new, powerful concepts:

1. Modulus: The modulus of $z = a+bi$, written $|z|$, is its distance from the origin. By the Pythagorean theorem, $|z| = \sqrt{a^2 + b^2}$. It is always a real number and represents the magnitude or "size" of the complex number.
2. Argument: The argument of $z$, written $\arg(z)$, is the angle measured anti-clockwise from the positive real axis to the line segment connecting the origin to $z$. It is usually given in radians. We define the principal argument, $\Arg(z)$, to be in the interval $(-\pi, \pi]$.

For $z = 3 + 4i$, the modulus is $|z| = \sqrt{3^2 + 4^2} = 5$. Its argument, $\theta$, satisfies $\tan \theta = 4/3$, so $\theta \approx 0.927$ radians (in the first quadrant).

Polar Form and Conversion

The Cartesian form $a+bi$ uses "grid coordinates." The polar form uses "distance and direction" coordinates: the modulus $r$ and the argument $\theta$. From trigonometry on the Argand diagram, we know $a = r\cos\theta$ and $b = r\sin\theta$.

Therefore, any complex number can be written in polar form as: $$z = r(\cos\theta + i\sin\theta)$$ This is often abbreviated as $z = r\,\text{cis}\,\theta$.

Converting from Cartesian to Polar:

1. Calculate the modulus: $r = \sqrt{a^2 + b^2}$.
2. Find the argument $\theta$ using $\tan\theta = b/a$, but you must check the quadrant of the point $(a, b)$ to get the correct angle.

Example: Convert $z = -1 + i$ to polar form.

1. $r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$.
2. $\tan\theta = 1/(-1) = -1$. The point $(-1, 1)$ is in the second quadrant. The reference angle is $\pi/4$, so the principal argument is $\theta = \pi - \pi/4 = 3\pi/4$.

Thus, $z = \sqrt{2}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})$ or $\sqrt{2}\,\text{cis}\,\frac{3\pi}{4}$.

Converting from Polar to Cartesian: This is straightforward: simply calculate $a = r\cos\theta$ and $b = r\sin\theta$. Example: For $z = 4\,\text{cis}\,(-\pi/3)$, $a = 4\cos(-\pi/3) = 4 \times \frac{1}{2} = 2$ and $b = 4\sin(-\pi/3) = 4 \times (-\frac{\sqrt{3}}{2}) = -2\sqrt{3}$. So $z = 2 - 2\sqrt{3}\,i$.

Common Pitfalls

1. Misapplying Algebraic Rules: Forgetting that $i^2 = -1$ during multiplication is a common error. Always simplify $i^2$ to $-1$ as your final step in any expansion.

Incorrect: $(2i)(3i) = 6i$. Correct: $(2i)(3i) = 6i^2 = 6(-1) = -6$.

1. Argument Calculation Errors: Using $\theta = \arctan(b/a)$ without quadrant checking will give the wrong argument about 50% of the time. Always plot the point $(a, b)$ mentally or on a quick sketch to determine the correct quadrant for $\theta$.

1. Confusing Modulus with Absolute Value: While the notation $|z|$ is similar, the modulus is a distance in the 2D plane. Remember the formula: $|a+bi| = \sqrt{a^2 + b^2}$, not just $|a| + |b|$ or $\sqrt{a^2}$.

1. Sign Errors with Conjugates: The conjugate of $a + bi$ is $a - bi$. A frequent mistake is to only change the sign of the real part or to misapply it during division.

Incorrect: The conjugate of $3 - 7i$ is $-3 + 7i$. Correct: The conjugate of $3 - 7i$ is $3 + 7i$.

Summary

• The imaginary unit $i$ is defined by $i^2 = -1$, enabling the complex number system $\mathbb{C}$.
• The standard Cartesian form is $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.
• Arithmetic follows algebraic rules, with division requiring multiplication by the complex conjugate $\overline{z} = a - bi$.
• Geometrically, a complex number is a point on an Argand diagram. Its modulus $|z| = \sqrt{a^2+b^2}$ is its distance from the origin, and its argument $\arg(z)$ is the angle it makes with the positive real axis.
• The polar form $z = r(\cos\theta + i\sin\theta)$ expresses the number using its modulus $r$ and argument $\theta$, providing a crucial foundation for advanced operations like multiplication, division, and powers via de Moivre's theorem.