# 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.

## JEE Conceptual Theory

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 \(\frac{z_2-z_0}{z_1-z_0}=\frac{|z_2-z_0|}{|z_1-z_0|} e^{i \theta}\)

## JEE Mains Problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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## JEE Advanced Problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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

**Cite as:**JEE Complex Numbers - Advanced Conceptual Understanding.

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