Summary & JEE Exam Tips: Complex Numbers

Basics: A complex number is of the form $z=a+ib$ , where $i=\sqrt{-1}$ . The algebra of complex numbers follows standard algebraic rules with the additional property $i^2=-1$ .
Modulus & Conjugate:
The conjugate of $z=a+ib$ is $\bar{z}=a-ib$ .
The modulus is $|z| = \sqrt{a^2+b^2}$ .
A crucial property is $z\bar{z} = |z|^2$ .
Argand Plane & Polar Form:
A complex number can be visualized as a point $(a,b)$ on the Argand plane.
Polar Form: $z = r(\cos\theta + i\sin\theta)$ , where $r=|z|$ and $\theta$ is the argument.
Euler's Form: $z = re^{i\theta}$ , where $e^{i\theta}=\cos\theta+i\sin\theta$ .
De Moivre's Theorem: For an integer n, $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$ . This is used to find powers and roots.
n-th Roots of Unity: The solutions to $z^n=1$ . Their sum is 0. The cube roots of unity ( $1, \omega, \omega^2$ ) are particularly important, with properties $1+\omega+\omega^2=0$ and $\omega^3=1$ .
Quadratic Equations: An equation $ax^2+bx+c=0$ with real coefficients and a negative discriminant ( $b^2-4ac < 0$ ) has two complex conjugate roots.

Think Geometrically: Always try to visualize the problem on the Argand plane. Locus problems are very common.
$|z-z_1| = k$ is a circle with center $z_1$ and radius $k$ .
$|z-z_1| = |z-z_2|$ is the perpendicular bisector of the line segment joining $z_1$ and $z_2$ .
$\arg(\frac{z-z_1}{z-z_2}) = \alpha$ represents an arc of a circle.
Master Cube Roots of Unity: Problems involving $\omega$ are frequent and often test your ability to simplify expressions quickly using $1+\omega+\omega^2=0$ and $\omega^3=1$ . This is a high-yield topic.
Rotation is Multiplication: Remember that multiplying a complex number $z$ by $e^{i\alpha}$ rotates the vector for $z$ by an angle $\alpha$ counter-clockwise. This is a powerful concept for solving geometry problems in JEE Advanced.
Memorize Standard Forms: Know these results by heart to save time:
$\frac{1+i}{1-i} = i$
$(1+i)^2 = 2i$ and $(1-i)^2 = -2i$
$|1+i| = \sqrt{2}$
Combine Chapters: Be prepared for problems that link Complex Numbers with other topics like Coordinate Geometry, Binomial Theorem, Sequences & Series, and Trigonometry.

Complex Numbers and Quadratic Equations