TN 12th Maths — Learn, Visualize, Practice

1. Why complex numbers?

Some equations simply have no real solution. The equation $x^2 + 1 = 0$ asks for a number whose square is $-1$ — and no real number squares to a negative value. To solve such equations we extend the number system beyond the real line.

DefinitionImaginary unit

The imaginary unit $i$ is defined by $i = \sqrt{-1}$ , so that $i^2 = -1$ .

Powers of $i$ cycle with period 4, which is worth memorising because it makes simplifying large powers instant:

i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1,\quad i^5 = i,\ \dots

Example

Simplify

i^{50}

.

2. What is a complex number?

DefinitionComplex number

A complex number is an expression of the form $z = a + bi$ , where $a$ and $b$ are real numbers. Here $a = \operatorname{Re}(z)$ is the real part and $b = \operatorname{Im}(z)$ is the imaginary part.

Two complex numbers are equal exactly when both their real parts and their imaginary parts are equal. A number with $b = 0$ is purely real; one with $a = 0$ (and $b \ne 0$ ) is purely imaginary.

Addition and multiplication follow ordinary algebra, treating $i$ as a symbol with the single rule $i^2 = -1$ :

(a+bi) + (c+di) = (a+c) + (b+d)i

(a+bi)(c+di) = (ac - bd) + (ad + bc)i

Example

Multiply

(2 + 3i)(1 - 4i)

.

3. The Argand plane

Every complex number $z = a + bi$ can be pictured as the point $(a, b)$ in a plane: the horizontal axis carries the real part, the vertical axis the imaginary part. This picture is called the Argand plane (or complex plane), and it turns algebra about complex numbers into geometry.

Key idea: Addition of complex numbers is exactly vector addition of their points; multiplication scales and rotates. Open the Visualize tab and drag the point to feel this.

4. Conjugate of a complex number

DefinitionConjugate

The conjugate of $z = a + bi$ is $\bar z = a - bi$ — the reflection of $z$ across the real axis.

The conjugate is the key tool for dividing complex numbers and for extracting real/imaginary parts:

z\bar z = a^2 + b^2 = |z|^2,\qquad \operatorname{Re}(z) = \dfrac{z + \bar z}{2},\qquad \operatorname{Im}(z) = \dfrac{z - \bar z}{2i}

Properties of the conjugate

$\overline{z_1 + z_2} = \bar z_1 + \bar z_2$
$\overline{z_1 z_2} = \bar z_1\,\bar z_2$
$z$ is real $\iff z = \bar z$ ; $\ z$ is purely imaginary $\iff z = -\bar z$

Example

Write

\dfrac{1}{2 + i}

in the form

a + bi

.

5. Modulus of a complex number

Just as the absolute value of a real number measures its distance from $0$ on the number line, the modulus of a complex number measures its distance from the origin in the Argand plane — the hypotenuse of the right triangle with legs $x$ and $y$ .

DefinitionModulus (Definition 2.4)

If $z = x + iy$ , then the modulus of $z$ , denoted $|z|$ , is $|z| = \sqrt{x^2 + y^2}$ .

Properties of the modulus

$|z| = |\bar z|$ and $\ z\bar z = |z|^2$
$|z_1 z_2| = |z_1||z_2|$
$\left|\dfrac{z_1}{z_2}\right| = \dfrac{|z_1|}{|z_2|},\ z_2 \ne 0$
$|z_1 + z_2| \le |z_1| + |z_2|$ (triangle inequality)
$|z^n| = |z|^n$ for integer $n$

Example

Find

|3 + 4i|

.

6. Polar (trigonometric) form and argument

A non-zero complex number is fixed by two pieces of data: how far it is from the origin (its modulus $r$ ) and the direction it points (its angle $\theta$ ). Writing $x = r\cos\theta$ and $y = r\sin\theta$ gives the polar form.

DefinitionPolar form (Definition 2.6)

For a non-zero $z = x + iy$ with modulus $r = |z|$ and angle $\theta$ , the polar form is $z = r(\cos\theta + i\sin\theta)$ , sometimes written $r\,\text{cis}\,\theta$ . The angle $\theta = \arg(z)$ is the argument, found from $\tan\theta = \dfrac{y}{x}$ (adjusted for the correct quadrant).

Note: The argument is only determined up to multiples of $2\pi$. The unique value in $(-\pi, \pi]$ is the principal argument, written $\operatorname{Arg}(z)$; then $\arg(z) = \operatorname{Arg}(z) + 2n\pi$.

Properties of the argument

$\arg(z_1 z_2) = \arg z_1 + \arg z_2$
$\arg\!\left(\dfrac{z_1}{z_2}\right) = \arg z_1 - \arg z_2$

Example

Express

z = 1 + i

in polar form.

7. Euler's form

Euler's formula packages the polar form into a compact exponential, which makes multiplication, powers, and roots especially clean:

e^{i\theta} = \cos\theta + i\sin\theta \qquad\Longrightarrow\qquad z = r\,e^{i\theta}

In this form, multiplying two numbers multiplies their moduli and adds their angles: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$ .

8. De Moivre's theorem and roots

DefinitionDe Moivre's theorem

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

Applied in reverse, it produces the $n$ distinct $n$ -th roots of a complex number, which sit at equal angular spacing of $\tfrac{360^\circ}{n}$ around a circle. The $n$ -th roots of unity are:

z_k = \cos\dfrac{2\pi k}{n} + i\sin\dfrac{2\pi k}{n},\quad k = 0, 1, \dots, n-1

Key idea: Explore the roots wheel in the Visualize tab — change $n$ and watch the roots redistribute evenly around the circle.

Example

Use De Moivre to compute

(\cos 20^\circ + i\sin 20^\circ)^3

.