For z 0, let r=|z|>0 and let theta be an argument in radians. The same number has three useful representations. **Representation** | **Form** | **Conversion** Cartesian | z=x+yi | x=rcostheta, y=rsintheta Polar | z=r(costheta+isintheta) | r=sqrt(x2+y2); choose the correct quadrant. Euler | z=r eitheta | eitheta=costheta+isintheta cistheta abbreviates costheta+isintheta. Arguments differing by 2kpi, k in Z, describe the same point. Zero has modulus zero and no argument. **Operation** | **Rule** | **Geometry** Product | z_1z_2=r_1r_2 ei(_1+ _2) | Multiply lengths; add angles. Quotient | z_1z_2= r_1r_2 ei(_1- _2) | Divide lengths; subtract angles. Sum or difference | Convert to Cartesian form first. | Add or subtract the corresponding vectors. Example - 3+i=2(cos 5pi6+isin 5pi6)=2 e(5pi i)/(6). For z_1=2 e(5pi i)/(6) and z_2=3 e(pi i)/(3), z_1z_2=6 e(7pi i)/(6). Its principal argument is - 5pi6, not 7pi6.

For \(z\ne0\), let \(r=|z|>0\) and let \(\theta\) be an argument in radians. The same number has three useful representations.

\(r=\sqrt{x^2+y^2}\); choose the correct quadrant.

\(\mathrm e^{i\theta}=\cos\theta+i\sin\theta\)

\(\operatorname{cis}\theta\) abbreviates \(\cos\theta+i\sin\theta\). Arguments differing by \(2k\pi\), \(k\in\mathbb Z\), describe the same point. Zero has modulus zero and no argument.

\(z_1z_2=r_1r_2\mathrm e^{i(\theta_1+\theta_2)}\)

\(\dfrac{z_1}{z_2}=\dfrac{r_1}{r_2}\mathrm e^{i(\theta_1-\theta_2)}\)

Add or subtract the corresponding vectors.

\(-\sqrt3+i=2\left(\cos\dfrac{5\pi}{6}+i\sin\dfrac{5\pi}{6}\right)=2\mathrm e^{\frac{5\pi i}{6}}\).

For \(z_1=2\mathrm e^{\frac{5\pi i}{6}}\) and \(z_2=3\mathrm e^{\frac{\pi i}{3}}\), \(z_1z_2=6\mathrm e^{\frac{7\pi i}{6}}\). Its principal argument is \(-\dfrac{5\pi}{6}\), not \(\dfrac{7\pi}{6}\).

Complex Numbers

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Polar and Euler forms

Polar and Euler forms