Revise Complex numbers: Cartesian and polar form for applications for IB Mathematics: Applications and Interpretation (1.8)

The imaginary unit $i$ is defined by the single rule $i^2 = -1$. From it we build a complex number in Cartesian form, $z = a + bi$, where $a$ and $b$ are ordinary real numbers. We call $a = \operatorname{Re}(z)$ the real part and $b = \operatorname{Im}(z)$ the imaginary part; note that the imaginary part is the real number $b$, not $bi$. When $b = 0$ the number is purely real, and when $a = 0$ it is purely imaginary.

Because $i^2 = -1$, the powers of $i$ cycle in fours: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then the pattern repeats. Addition and subtraction of complex numbers work componentwise, exactly like vectors: $(3 + 2i) + (1 - 5i) = 4 - 3i$. Multiplication expands like any bracket, using $i^2 = -1$ to simplify: $(2 + i)(1 + 3i) = 2 + 6i + i + 3i^2 = 2 + 7i - 3 = -1 + 7i$.

The complex conjugate of $z = a + bi$ is $\bar{z} = a - bi$: same real part, opposite imaginary part. Conjugates matter because their product is real: $z\bar{z} = (a + bi)(a - bi) = a^2 + b^2$, which is $|z|^2$. That is the trick for dividing in Cartesian form: multiply top and bottom by the conjugate of the denominator to make the denominator real.

The Argand diagram plots $z = a + bi$ as the point $(a, b)$, with the real part on the horizontal axis and the imaginary part on the vertical axis. This turns a complex number into a point, and turns algebra into geometry. From the picture, two natural measurements appear: how far the point is from the origin, and in what direction it lies.

The distance from the origin to $(a, b)$ is the modulus, $|z| = r = \sqrt{a^2 + b^2}$; it is always non-negative. The direction, the angle the line from the origin makes with the positive real axis measured anticlockwise, is the argument, $\theta = \arg z$. Together $r$ and $\theta$ are the polar coordinates of the point, and they are the ingredients of the polar form.

Modulus: $|z| = r = \sqrt{a^2 + b^2}$. \n Argument: $\theta = \arg z$, found from $\arctan\!\left(\dfrac{b}{a}\right)$ adjusted for the quadrant of $(a, b)$.

Substituting $a = r\cos\theta$ and $b = r\sin\theta$ into $z = a + bi$ gives the polar, or modulus-argument, form $z = r(\cos\theta + i\sin\theta)$, often abbreviated $r\,\text{cis}\,\theta$. Euler's relation $e^{i\theta} = \cos\theta + i\sin\theta$ then lets us write the same number even more compactly as the Euler form $z = re^{i\theta}$. These are not three different numbers; they are three notations for one point on the Argand diagram, and you convert freely between them.

To go from Cartesian to polar or Euler, find $r$ and $\theta$ as above. To go the other way, expand: $a = r\cos\theta$ and $b = r\sin\theta$. Keep $\theta$ in radians whenever Euler form is involved, since $e^{i\theta}$ is only meaningful in radians.

The reason polar and Euler forms are worth the conversion effort is how simply they multiply and divide. Because exponents add, $r_1 e^{i\theta_1} \times r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$: the moduli multiply and the arguments add. Dividing reverses this, $\dfrac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \dfrac{r_1}{r_2}\, e^{i(\theta_1 - \theta_2)}$: the moduli divide and the arguments subtract. Geometrically, multiplying by a complex number scales the Argand diagram by its modulus and rotates it by its argument.

Multiplication: $r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$. \n Division: $\dfrac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \dfrac{r_1}{r_2}\, e^{i(\theta_1 - \theta_2)}$. \n Moduli multiply/divide; arguments add/subtract.

This is why complex numbers live in an applied course. A sinusoidal quantity such as a voltage $V\cos(\omega t + \phi)$ has two features that matter: its amplitude $V$ and its phase $\phi$. Those are precisely the modulus and argument of a complex number $Ve^{i\phi}$, called a phasor. Every quantity in an AC circuit oscillates at the same frequency $\omega$, so the shared time factor $e^{i\omega t}$ cancels out of any ratio, and questions about amplitudes and phases become questions about moduli and arguments.

The payoff is Ohm's law for AC circuits, $I = \dfrac{V}{Z}$, where the voltage $V$, the impedance $Z$ and the current $I$ are all complex. In polar form the division is a single line: divide the moduli to get the current's amplitude, subtract the arguments to get its phase. What would be an awkward trigonometric identity becomes one modulus division and one argument subtraction.

The two-part idea at the heart of this lesson, that a complex number is a modulus and an argument, is the foundation for everything the topic goes on to do. Repeated multiplication in polar form, where the argument keeps adding, is exactly De Moivre's theorem, the engine for powers and roots of complex numbers. The phasor picture you met for a single AC division extends to whole networks of impedances, and the same modulus-and-argument reasoning underlies the way sinusoids of the same frequency are added in signal processing. Master converting between Cartesian, polar and Euler form, and multiplying and dividing by combining moduli and arguments, and the rest of the complex-number course becomes variations on a move you already own.