Revise Complex numbers: Cartesian, polar and Euler form for IB Mathematics: Analysis and Approaches (1.6)

Real numbers cannot solve $x^2=-1$, because no real number squares to a negative. We therefore define a new number, the imaginary unit $i$, by the single rule $i^2=-1$. Every complex number is then built from a real part and an imaginary part in Cartesian form $z=a+bi$, where $a$ and $b$ are real. 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 $bi$.

For $z=a+bi$: \ Conjugate: $z^*=a-bi$ \ Modulus: $|z|=\sqrt{a^2+b^2}$, so that $z\,z^*=|z|^2=a^2+b^2$

Addition and subtraction work component by component: add the real parts and add the imaginary parts. Multiplication uses ordinary expansion, replacing $i^2$ with $-1$ wherever it appears. Division is the one operation that needs a trick: multiply numerator and denominator by the conjugate of the denominator, which makes the denominator real because $(c+di)(c-di)=c^2+d^2$.

We picture a complex number $z=a+bi$ as the point $(a,b)$ on the Argand diagram: the horizontal axis is the real axis and the vertical axis is the imaginary axis. This geometric view is the key to everything that follows. The distance from the origin to the point is the modulus $r=|z|$, and the angle the line to the point makes with the positive real axis (measured anticlockwise, in radians) is the argument $\theta=\arg(z)$. Conjugation reflects a point in the real axis; addition of two complex numbers corresponds to adding their position vectors.

Instead of the coordinates $(a,b)$ we can locate the same point using its distance $r$ and direction $\theta$. Reading off the right-angled triangle in the Argand diagram gives $a=r\cos\theta$ and $b=r\sin\theta$, so $z=a+bi=r\cos\theta+ir\sin\theta=r(\cos\theta+i\sin\theta)$. This is the modulus-argument, or polar, form.

Cartesian $\to$ polar: \ $r=\sqrt{a^2+b^2}$ \ $\theta=\arctan\!\left(\dfrac{b}{a}\right)$, adjusted for the quadrant of $(a,b)$ \ Polar form: $z=r(\cos\theta+i\sin\theta)$

Euler's formula states that $e^{i\theta}=\cos\theta+i\sin\theta$. Substituting this into the polar form collapses it to the compact Euler form $z=re^{i\theta}$. The two forms are the same object written differently, but Euler form is usually the most convenient for calculation because it makes multiplication, division and powers obey the ordinary rules of indices. Remember that $\theta$ must be in radians.

Euler's formula: $e^{i\theta}=\cos\theta+i\sin\theta$ \ Euler form: $z=re^{i\theta}$, where $r=|z|$ and $\theta=\arg(z)$ in radians

This is where the polar and Euler forms earn their keep. Multiplying in Cartesian form is fiddly; in Euler form it is just index laws. Because $e^{i\theta_1}\cdot e^{i\theta_2}=e^{i(\theta_1+\theta_2)}$, multiplying two complex numbers multiplies their moduli and adds their arguments. Geometrically, multiplying by $z_2$ scales by $r_2$ and rotates by $\theta_2$. Division reverses this: divide the moduli and subtract the arguments.

Let $z_1=r_1e^{i\theta_1}$ and $z_2=r_2e^{i\theta_2}$. Then \ Multiplication: $z_1 z_2=r_1 r_2\,e^{i(\theta_1+\theta_2)}$ \ Division: $\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\,e^{i(\theta_1-\theta_2)}$ \ Power: $z^n=r^n e^{in\theta}$