In one line
A complex number can be written as \(x+iy\) (Cartesian), \(r(\cos\theta+i\sin\theta)\) (polar) or \(re^{i\theta}\) (exponential); all three describe the same point in the Argand plane.
Explained simply
Think of giving directions to a friend's house. You can say "go 3 km east and 4 km north" or "go 5 km in the direction 53° north of east". Same house, two descriptions. Cartesian form is the first style, polar form is the second.
Technically, a complex number \(z=x+iy\) (with \(i^2=-1\)) is the point \((x,y)\) in the Argand plane. Its distance from the origin is the modulus \(r=|z|\) and the angle it makes with the positive real axis is the argument \(\theta=\arg z\). Using Euler's formula \(e^{i\theta}=\cos\theta+i\sin\theta\), the polar form becomes the compact exponential form \(z=re^{i\theta}\).
How it works
- Find \(r=\sqrt{x^2+y^2}\).
- Find the acute angle \(\alpha=\tan^{-1}\left|\dfrac{y}{x}\right|\).
- Fix the quadrant of \((x,y)\) and get the principal argument \(\theta\) in \((-\pi,\pi]\):
Quadrant Signs of x, y \(\theta\) I +, + \(\alpha\) II −, + \(\pi-\alpha\) III −, − \(-(\pi-\alpha)\) IV +, − \(-\alpha\) - Write \(z=r(\cos\theta+i\sin\theta)=re^{i\theta}\).
- To go back, use \(x=r\cos\theta,\ y=r\sin\theta\).
Key formula
\(z=x+iy=r(\cos\theta+i\sin\theta)=re^{i\theta}\), where \(x\) = real part, \(y\) = imaginary part, \(r=\sqrt{x^2+y^2}\) = modulus (a non-negative real number), \(\theta\) = argument in radians. Useful results: \(z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}\) and \(\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}e^{i(\theta_1-\theta_2)}\).
Quick example
Convert \(z=-1-i\). Here \(r=\sqrt{1+1}=\sqrt2\), \(\alpha=\tan^{-1}1=\pi/4\). The point is in quadrant III, so \(\theta=-(\pi-\pi/4)=-3\pi/4\). Hence \(z=\sqrt2\left[\cos\left(-\tfrac{3\pi}{4}\right)+i\sin\left(-\tfrac{3\pi}{4}\right)\right]=\sqrt2\,e^{-3i\pi/4}\).
Check: \(\sqrt2\cos(-3\pi/4)=\sqrt2(-1/\sqrt2)=-1\) and \(\sqrt2\sin(-3\pi/4)=-1\). Correct.
Common mistakes
- Writing \(\theta=\tan^{-1}(y/x)\) blindly: for \(-1-i\) the calculator gives \(\pi/4\), which is the argument of \(1+i\), not \(-1-i\).
- Using degrees inside \(e^{i\theta}\); the exponential form needs \(\theta\) in radians.
- Taking \(r\) as \(x^2+y^2\) instead of its square root.
How MU asks it
This is a pre-requisite, so it rarely comes as a full question by itself, but every Module 1 and 2 problem starts with it. You may get a 3-mark part such as "Express \(1+i\sqrt3\) in polar and exponential form". For full marks show \(r\), \(\alpha\), the quadrant, the final \(\theta\) and both forms. In longer 5/6-mark problems on De Moivre's theorem or roots, a wrong argument at this step makes the whole answer wrong, so always show the quadrant check.