# JEE Complex Numbers

This page will teach you how to master JEE Complex Numbers. 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

As per JEE syllabus, the main concepts under Complex Numbers are introduction to complex numbers, argument of a complex number, modulus of a complex number, conjugate of a complex number and different forms of a complex number.

### Introduction to complex numbers

Properties of \(i\)

Real and imaginary parts: \(z=x+iy\) where \(x,y \in \mathbb R\) and \(i=\sqrt{-1}\)

Complex numbers - arithmetic: \(z_1+z_2\), \(z_1-z_2\), \(z_1 \cdot z_2\) and \(\frac{z_1}{z_2} \ ( z_2 \neq 0)\)

### Modulus of a complex number

If \(z=x+iy\), then modulus of \(z\), represented by \(|z|=\sqrt{x^2+y^2}\)

Properties: Triangle inequality \(\left| |z_1|-|z_2| \right| \leq |z_1+z_2| \leq |z_1|+|z_2|\)

### Argument of a complex number

If \(z=x+iy\), then argument of \(z\), say \(\theta\) satisfies \(|z| \cos \theta=x\) and \(|z| \sin \theta=y\)

Properties: \(arg(z) \in (-\pi,\pi]\)

### Conjugate of a complex number

Properties: \(\overline{ (\bar z)}=z\) ; \(\overline{z_1 \pm z_2}=\overline{z_1} \pm \overline{z_2}\) ; \(\overline{z_1 \cdot z_2}=\bar{z_1}\cdot \bar{z_2}\) ; \(\overline{\left( \frac{z_1}{z_2} \right)}=\frac{\bar{z_1}}{\bar{z_2}}\)

Geometrical significance

### Different forms of a complex number

Cartesian form: \(z=x+iy\) where \(x,y \in \mathbb R\)

Polar form: \(z=r(\cos \theta+i \sin \theta\) where \(r=|z|, \theta=arg(z)\)

Euler form: \(z=re^{i\theta}\) where \(r=|z|, \theta=arg(z)\)

## 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 Roots of Unity.

**Cite as:**JEE Complex Numbers.

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