# 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 \(z\) is represented by \(|z|=\sqrt{x^2+y^2}.\)
- Properties: triangle inequality \(\Big| |z_1|-|z_2| \Big| \leq |z_1+z_2| \leq |z_1|+|z_2|\)

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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