Chapter 1: Complex Numbers — Class 11 Maths Notes

What is a complex number?

A complex number is an ordered pair \((a, b)\) of real numbers, written as \(z = a + ib\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i = \sqrt{-1}\) with \(i^2 = -1\). Har woh equation jiska real numbers mein koi hal na ho, jaise \(x^2 + 1 = 0\), complex numbers mein hal ho jati hai.

Key point: \(\text{Im}(z)\) is the real number \(b\), not \(ib\). Is ghalti se bachein — board paper mein yeh aksar pucha jata hai.

Equality of two complex numbers

Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal.

Key point: Yahi rule Exercise 1.2 ke "find x and y" wale sawalon ki bunyad hai: pehle dono sides ko \(x + iy\) form mein lao, phir real aur imaginary parts compare karo.

Conjugate and modulus

The conjugate of \(z = a + ib\) is \(\bar{z} = a - ib\) (sign of the imaginary part flips). The modulus \(|z|\) is the distance of \(z\) from the origin in the Argand plane.

Useful properties: \(\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}\), \(\overline{z_1 z_2} = \bar{z_1}\,\bar{z_2}\), \(\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}\), and \(|z_1 z_2| = |z_1|\,|z_2|\).

Key point: Division mein denominator ko real banane ke liye uske conjugate se multiply karte hain: \(\frac{1}{a+ib} = \frac{a-ib}{a^2+b^2}\).

Powers of \(i\)

Powers of \(i\) repeat every four steps, so any \(i^n\) reduces to one of \(i, -1, -i, 1\) by taking the remainder of \(n\) divided by 4.

Here \(r\) is the remainder when \(n\) is divided by 4 (division algorithm: \(n = 4q + r\)).

Key point: Four consecutive powers always add to zero: \(i^n + i^{n+1} + i^{n+2} + i^{n+3} = 0\).

Square root of a complex number

To find \(\sqrt{a + ib}\), set \(\sqrt{a+ib} = x + iy\) with real \(x, y\). Squaring gives the system \(x^2 - y^2 = a\) and \(2xy = b\). A shortcut uses \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 = a^2 + b^2\), so \(x^2 + y^2 = \sqrt{a^2 + b^2}\).

Key point: \(x\) aur \(y\) ke signs \(2xy = b\) se decide hote hain: agar \(b > 0\) to dono same sign, agar \(b < 0\) to opposite signs.

Complex polynomials as a product of linear factors

Over complex numbers, every polynomial of degree \(n\) splits completely into \(n\) linear factors (Fundamental Theorem of Algebra). A sum of squares, which never factors over real numbers, factors over complex numbers using \(i\).

Quadratic equations with negative discriminant give conjugate-pair roots: for \(az^2 + bz + c = 0\), \(z = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), and \(\sqrt{\text{negative}} = i\sqrt{|\text{value}|}\).

Key point: Pehle rational root ya grouping se real factors nikalo, phir bache hue quadratics ko quadratic formula se todo.

Cube roots of unity

The three cube roots of unity are \(1, \omega, \omega^2\), where \(\omega = \frac{-1 + \sqrt{3}\,i}{2}\). They satisfy two golden relations used in almost every proof of this section.

The three cube roots of any number \(a\) are \(a^{1/3}, a^{1/3}\omega, a^{1/3}\omega^2\). The four fourth roots of a positive real number \(a\) are \(\pm a^{1/4}, \pm i\,a^{1/4}\), and their sum is always zero.

Key point: \(1 + \omega + \omega^2 = 0\) se \(\omega + \omega^2 = -1\) milta hai — yeh substitution proofs mein bar bar kaam aata hai.

Polar form and the Argand plane

Every complex number \(z = x + iy\) can be written as \(z = r(\cos\theta + i\sin\theta)\), where \(r = |z| = \sqrt{x^2+y^2}\) is the modulus and \(\theta = \arg(z)\) is the argument — the angle from the positive real axis. The point \((x, y)\) plotted in the plane is the Argand diagram of \(z\).

Quadrant ka khayal rakhein: \(\tan^{-1}(y/x)\) hamesha sahi quadrant nahi deta. First quadrant mein \(\theta = \tan^{-1}(y/x)\), second mein \(\pi - \tan^{-1}|y/x|\), third mein \(\pi + \tan^{-1}|y/x|\), fourth mein \(-\tan^{-1}|y/x|\).

Key point: Polar form mein multiplication aur division asaan ho jate hain: moduli multiply/divide hote hain, arguments add/subtract hote hain.

Operations in polar form and argument properties

If \(z_1 = r_1(\cos\theta_1 + i\sin\theta_1)\) and \(z_2 = r_2(\cos\theta_2 + i\sin\theta_2)\), then:

Hence \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\) and \(\arg(z_1/z_2) = \arg(z_1) - \arg(z_2)\) (up to multiples of \(2\pi\)).

Key point: \(|z - a| = r\) is a circle (centre \(a\), radius \(r\)), and \(|z - a| = |z - b|\) is the perpendicular bisector of the segment joining \(a\) and \(b\) — a straight line.

Complex numbers in the real world

In AC circuits, voltage, current and impedance are written as complex numbers (phasors). Ohm's law becomes \(V = IZ\), so current is \(I = V/Z\), computed exactly like complex division. The same multiply-by-a-complex-key idea is used in simple cryptographic ciphers: encrypt by multiplying with a key, decrypt by dividing (multiplying by the multiplicative inverse).

Key point: Real-world sawalon mein pehle quantities ko \(a + ib\) ya polar form mein likho, phir normal complex operations apply karo.