Complex Numbers – Formula Sheet (Class 12 Maths, Chapter 2)
Imaginary unit
\[ i^{2}=-1,\quad i^{3}=-i,\quad i^{4}=1,\qquad i^{n}+i^{n+1}+i^{n+2}+i^{n+3}=0. \]
Rectangular form
\( z=x+iy \), with \(\operatorname{Re}(z)=x\), \(\operatorname{Im}(z)=y\). Equality: \(x_{1}=x_{2}\) and \(y_{1}=y_{2}\).
Operations
\[ (a+ib)\pm(c+id)=(a\pm c)+i(b\pm d) \]
\[ (a+ib)(c+id)=(ac-bd)+i(ad+bc) \]
\[ \dfrac{a+ib}{c+id}=\dfrac{(a+ib)(c-id)}{c^{2}+d^{2}} \]
Conjugate
\( \bar z=x-iy,\qquad z\bar z=x^{2}+y^{2}=|z|^{2} \)
\[ \overline{z_{1}\pm z_{2}}=\bar z_{1}\pm\bar z_{2},\quad \overline{z_{1}z_{2}}=\bar z_{1}\bar z_{2},\quad \overline{\left(\tfrac{z_{1}}{z_{2}}\right)}=\dfrac{\bar z_{1}}{\bar z_{2}} \]
\[ \operatorname{Re}(z)=\dfrac{z+\bar z}{2},\qquad \operatorname{Im}(z)=\dfrac{z-\bar z}{2i} \]
\(z\) is real \(\iff z=\bar z\); \(z\) is purely imaginary \(\iff z=-\bar z\).
Modulus
\[ |z|=\sqrt{x^{2}+y^{2}},\quad |z_{1}z_{2}|=|z_{1}||z_{2}|,\quad \left|\dfrac{z_{1}}{z_{2}}\right|=\dfrac{|z_{1}|}{|z_{2}|},\quad |z^{n}|=|z|^{n} \]
\[ |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|,\qquad |z_{1}-z_{2}|\ge\big||z_{1}|-|z_{2}|\big| \]
Square root
\[ \sqrt{a+ib}=\pm\left(\sqrt{\dfrac{|z|+a}{2}}+i\,\dfrac{b}{|b|}\sqrt{\dfrac{|z|-a}{2}}\right),\qquad b\neq 0 \]
Locus
\( |z-z_{0}|=r \): circle, centre \(z_{0}\), radius \(r\). \( |z-a|=|z-b| \): perpendicular bisector of the segment \([a,b]\).
Polar and Euler form
\[ z=r(\cos\theta+i\sin\theta)=re^{i\theta},\quad r=|z|,\quad \theta=\arg(z),\quad -\pi<\operatorname{Arg}(z)\le\pi \]
\[ z_{1}z_{2}=r_{1}r_{2}\big[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\big] \]
\[ \dfrac{z_{1}}{z_{2}}=\dfrac{r_{1}}{r_{2}}\big[\cos(\theta_{1}-\theta_{2})+i\sin(\theta_{1}-\theta_{2})\big] \]
De Moivre's theorem
\[ (\cos\theta+i\sin\theta)^{n}=\cos n\theta+i\sin n\theta \qquad (n\in\mathbb{Z}) \]
\(n\)th roots
\[ z^{1/n}=r^{1/n}\left(\cos\dfrac{\theta+2k\pi}{n}+i\sin\dfrac{\theta+2k\pi}{n}\right),\qquad k=0,1,\dots,n-1 \]
\(n\)th roots of unity: sum \(=0\); product \(=(-1)^{\,n+1}\). Cube roots: \(1,\omega,\omega^{2}\) with \(\omega^{3}=1\) and \(1+\omega+\omega^{2}=0\).