You already know that multiplying complex numbers in rectangular form works a lot like multiplying binomials. We just use FOIL (First, Outer, Inner, Last) and remember that $i^2 = -1$.

Division is a bit trickier. We can't just divide term by term. The goal is to remove the imaginary unit, $i$, from the denominator. To do this, we use a clever tool called the complex conjugate., a powerful concept first explored by the Irish mathematician William Rowan Hamilton.

The complex conjugate of a number $a + bi$ is simply $a - bi$. You just flip the sign of the imaginary part. The magic happens when you multiply a complex number by its conjugate. The imaginary parts cancel out, leaving a real number.

To divide two complex numbers, you multiply both the numerator and the denominator by the conjugate of the denominator. This clears the $i$ from the bottom, leaving a standard rectangular form.

Let's try dividing $(4 + i)$ by $(2 - 3i)$.

This method always works, but it can be tedious. There is a more elegant way to think about multiplication and division, which starts with visualising the numbers.

Instead of just treating complex numbers as algebraic expressions, we can plot them on a two-dimensional plane. This visual representation is called an Argand diagram.. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. The number $a + bi$ corresponds to the point $(a, b)$.

This geometric view reveals two key properties of any complex number (other than 0): its distance from the origin and its angle relative to the positive real axis. The distance is called the modulus, written as $r$ or $|z|$. The angle is the argument, written as $\theta$ or $arg(z)$.

Together, the modulus and argument define a complex number's polar form. Instead of using coordinates $(a, b)$, we use a distance and an angle $(r, \theta)$.

Knowing $a$ and $b$ lets us find $r$ and $\theta$. Conversely, knowing $r$ and $\theta$ lets us find $a$ and $b$ using basic trigonometry.

$a = r \cos(\theta)$ $b = r \sin(\theta)$

Substituting these into $z = a + bi$ gives us the polar form:

Let's convert $z = 1 + \sqrt{3}i$ to polar form.

First, find the modulus $r$:

Next, find the argument $\theta$:

So, the polar form of $1 + \sqrt{3}i$ is $2(\cos(\pi/3) + i\sin(\pi/3))$.

This might seem like more work, but it makes multiplication and division incredibly simple.

When we multiply two complex numbers in polar form, we multiply their moduli and add their arguments.

If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then:

For division, we divide the moduli and subtract the arguments.

Geometrically, this is powerful. Multiplying a complex number by $z_2$ scales its length by a factor of $r_2$ and rotates it counter-clockwise by an angle of $\theta_2$. Division is the reverse: it scales by $1/r_2$ and rotates clockwise by $\theta_2$.

This approach turns complex calculations into simple arithmetic and geometric transformations, which is essential for more advanced topics like De Moivre's theorem and Euler's formula.