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.

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 \(r=|z|, \theta=\arg(z)\)

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

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

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Once you are confident of Complex Numbers, move on to JEE Roots of Unity.