Complex NumbersJEE Main

Concept

A complex number $z = a + bi$ is a point (or arrow) on the Argand plane. Its length is the modulus $|z|$ and the angle it makes with the positive real axis is the argument $\arg z$. This geometric view turns algebra into rotations and scalings.

Key formula

Derivation

Writing $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ (Euler) makes multiplication elegant: moduli multiply and arguments add. So $z^2$ has modulus $r^2$ and argument $2\theta$ — squaring rotates by $\theta$ and scales by $r$.

The conjugate $\bar z = a - bi$ is the reflection in the real axis, and $z\bar z = |z|^2$ is always real.

Scenarios to explore

• Complex Numbers — The Argand plane — modulus and argument.
• Purely imaginary — On the imaginary axis the argument is 90°.
• On the negative real axis — The argument is π and the modulus is just |Re z|.

Real-world applications

• AC circuit analysis (phasors are complex numbers).
• Rotations in 2D geometry via multiplication by $e^{i\theta}$.
• Roots of unity and polynomial factorisation.

JEE exam tips

• De Moivre: $z^n = r^n(\cos n\theta + i\sin n\theta)$ — powers are easy in polar form.
• Multiplying by $i$ rotates a point $90°$ anticlockwise about the origin.

Common mistakes

• Reading the argument from the wrong quadrant — use the signs of $a$ and $b$.
• Forgetting $i^2 = -1$ when expanding $z^2$.
• Confusing $|z|^2$ (real) with $z^2$ (generally complex).

Exam traps to avoid

• $\arg z$ is usually taken in $(-\pi, \pi]$ — watch the principal-value convention.
• $|z_1 z_2| = |z_1||z_2|$ but $|z_1 + z_2| \le |z_1| + |z_2|$ (triangle inequality).