JEE Complex Numbers - Advanced Conceptual Understanding

This page will teach you how to master JEE Complex Numbers up to JEE Advanced level. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

To have mastery over Complex Numbers for JEE Advanced, the main concepts you should be confident of are square root of a complex number, logarithm of a complex number, modular inequalities, De Moivre's theorem and rotation of a complex number.

Square root of a complex number

• $\sqrt{a+ib}=x+iy \Rightarrow x+iy=\pm \left[ \left( \frac{\sqrt{a^2+b^2}+a}{2} \right)^\frac{1}{2}+i \left( \frac{\sqrt{a^2+b^2}-a}{2} \right)^\frac{1}{2} \right]$

Logarithm of a complex number

• $\log_ez=\log_e|z|+i(2k\pi+\theta)$ where $\theta=arg(z)$, $k \in \mathbb Z$

Modular inequalities

• Minimum value of $|z-z_1|+|z-z_2|$ is $|z_1-z_2|$

• $|z_1-z_2| \geq \left| |z_1|-|z_2| \right|$

• $|z_1+z_2|=|z_1-z_2| \Leftrightarrow arg(z_1)-arg(z_2)=\pm \frac{\pi}{2}$

• $|z_1+z_2|^2=|z_1|^2+|z_2|^2 \Leftrightarrow \frac{z_1}{z_2}$ is purely imaginary number

De Moivre's theorem

• $(\cos \theta +i \sin \theta)^n=\cos n\theta+i \sin n\theta$, when $n$ is an integer

• $\cos n\theta+i \sin n\theta$ is one of the values of $(\cos \theta+i\sin \theta)^n$, when $n$ is a fraction

Rotation of a complex number

• When we rotate $z_1$ through an angle $\theta$ in anticlockwise direction about origin $(0,0)$, say we get $z_2$, then $\frac{z_2-0}{z_1-0}=\frac{|z_2-0|}{|z_1-0|} e^{i \theta}$

• When we rotate $z_1$ through an angle $\theta$ in anticlockwise direction about another complex number $z_0$, say we get $z_2$, then $\frac{z_2-z_0}{z_1-z_0}=\frac{|z_2-z_0|}{|z_1-z_0|} e^{i \theta}$

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Once you are confident of Complex Numbers, move on to JEE Binomial Theorem.