Complex Numbers and Quadratic Equations revision notes

The real number system falls short when attempting to solve quadratic equations of the form $ax^2 + bx + c = 0$ where the discriminant $D = b^2 - 4ac < 0$, specifically equations like $x^2 + 1 = 0$. To resolve this, the real number system is extended to a larger system called Complex Numbers.

• Historical Note: The inability to find square roots of negative numbers was recognized by Indian mathematicians Mahavira (850 AD) and Bhaskara (1150 AD). Cardan (1545) solved $x+y=10, xy=40$ obtaining $5 \pm \sqrt{-15}$ but deemed them "useless".
• Euler was the first to introduce the symbol $i$ for $\sqrt{-1}$.
• W.R. Hamilton established the purely mathematical definition of a complex number as an ordered pair of real numbers $(a, b)$, avoiding the misleading term "imaginary".

Operations on complex numbers $z_1 = a + ib$ and $z_2 = c + id$:

• Difference: $z_1 - z_2 = z_1 + (-z_2) = (a-c) + i(b-d)$.

• Division: $\frac{z_1}{z_2} = z_1 z_2^{-1}$ provided $z_2 \neq 0$.

• General Formula: For any integer $k$: $i^{4k} = 1$, $i^{4k+1} = i$, $i^{4k+2} = -1$, $i^{4k+3} = -i$.
• Inverse of $i$: $\frac{1}{i} = -i$. JEE Tip Always remember $\frac{1}{i} = -i$ to quickly simplify denominators.
• Square Roots of Negative Reals: If $a$ is a positive real number, $\sqrt{-a} = i\sqrt{a}$.

• Modulus ($|z|$): For $z = a + ib$, the modulus is the non-negative real number $|z| = \sqrt{a^2 + b^2}$. Geometrically, it is the distance between the point $P(a, b)$ and the origin $(0, 0)$ in the Argand plane.
• Conjugate ($\bar{z}$): For $z = a + ib$, the conjugate is $\bar{z} = a - ib$. Geometrically, the point $(a, -b)$ is the mirror image of the point $(a, b)$ on the real axis.
• Multiplicative Inverse ($z^{-1}$): For a non-zero complex number $z = a + ib$, $z^{-1} = \frac{1}{a+ib} = \frac{a - ib}{a^2 + b^2} = \frac{\bar{z}}{|z|^2}$.
• Relation between Modulus and Conjugate: $z\bar{z} = |z|^2$. JEE Tip This is the most powerful algebraic tool in complex numbers. Whenever you see a denominator with a complex number, multiply numerator and denominator by its conjugate.

For any complex numbers $z_1, z_2$:

• $\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}$
• $\overline{z_1 z_2} = \overline{z_1} \overline{z_2}$
• $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}$ (provided $z_2 \neq 0$)

Standard algebraic identities for real numbers hold true for complex numbers due to commutativity and distributivity:

• $(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2$
• $(z_1 - z_2)^2 = z_1^2 - 2z_1z_2 + z_2^2$
• $(z_1 + z_2)^3 = z_1^3 + 3z_1^2z_2 + 3z_1z_2^2 + z_2^3$
• $(z_1 - z_2)^3 = z_1^3 - 3z_1^2z_2 + 3z_1z_2^2 - z_2^3$
• $z_1^2 - z_2^2 = (z_1 - z_2)(z_1 + z_2)$

• The Argand Plane: An ordered pair $(x, y)$ plotted on a 2D Cartesian plane where X is the Real Axis and Y is the Imaginary Axis.
• Mirror Image: A complex number $z = x + iy$ plotted in Q1 has its conjugate $\bar{z} = x - iy$ plotted in Q4. The geometric interpretation of the conjugate is a pure reflection across the x-axis (Real Axis).

Polar and Euler Representation

• Argument ($\theta$): The angle made by the vector joining origin to $P(x,y)$ with the positive x-axis. Principal argument: $-\pi < \theta \le \pi$.
• Polar Form: $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$.
• Euler Form: $z = r e^{i\theta}$. JEE Tip Always convert to Euler form when multiplying, dividing, or finding powers/roots of complex numbers.

De Moivre's Theorem (DMT)

• For any integer $n$, $(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$.

Cube Roots of Unity

• Solutions to $z^3 = 1$ are $1, \omega, \omega^2$, where $\omega = \frac{-1 + i\sqrt{3}}{2} = e^{i 2\pi/3}$.
• Properties: $1 + \omega + \omega^2 = 0$ and $\omega^3 = 1$.

$n$th Roots of Unity

• Solutions to $z^n = 1$ are $1, \alpha, \alpha^2, \dots, \alpha^{n-1}$, where $\alpha = e^{i 2\pi/n}$.
• Sum of roots = 0. Product of roots = $(-1)^{n-1}$.

Geometry of Complex Numbers (Coni Method / Rotation Theorem)

• Distance between $z_1$ and $z_2$ is $|z_1 - z_2|$.
• Equation of circle with center $z_0$ and radius $R$: $|z - z_0| = R$.
• Rotation of $z_1$ about $z_0$ by angle $\alpha$: $\frac{z_2 - z_0}{z_1 - z_0} = |\frac{z_2 - z_0}{z_1 - z_0}| e^{i\alpha}$.

• General form: $z = a + ib$
• Modulus: $|z| = \sqrt{a^2 + b^2}$
• Conjugate: $\bar{z} = a - ib$
• Multiplicative Inverse: $z^{-1} = \frac{a - ib}{a^2 + b^2} = \frac{\bar{z}}{|z|^2}$
• Powers of i relation: $i^{4k} + i^{4k+1} + i^{4k+2} + i^{4k+3} = 0$ (Sum of four consecutive integer powers of $i$ is zero).
• Identity of squares: $|x+iy|^2 = x^2 + y^2$.

• Equality of Complex Numbers limit: You can equate real to real and imaginary to imaginary ONLY if both sides are strictly cast into $a+ib$ format where $a, b \in \mathbb{R}$.
• Division validity: $\frac{z_1}{z_2}$ is only defined when the denominator modulus $|z_2| \neq 0$.

• Conjugate Sign Convention: The conjugate ONLY flips the sign of the imaginary part, not the real part. E.g., Conjugate of $-3 + 4i$ is $-3 - 4i$, NOT $3 - 4i$.

1. Algebraic manipulation: Expressing complex fractions into $a+ib$ form using rationalization (multiplying by conjugate).
2. Locus Problems: Finding the geometric trajectory of a complex number given a modulus/argument condition (e.g., $|z-z_1| = |z-z_2|$ represents the perpendicular bisector).
3. Triangle Inequalities: Used heavily to find minimum and maximum distances in the Argand plane.
4. Roots of Unity: Problems mixing quadratic roots and $\omega, \omega^2$.
5. Euler Form Exponentiation: Raising complex numbers to very high powers (e.g., $(\frac{1+i}{1-i})^{100}$).