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 — memorise this and large powers become 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}

.

Example

Evaluate

i^{15} + i^{16} + i^{17} + i^{18}

.

2. What is a complex number?

DefinitionComplex number

A complex number has the form $z = a + bi$ with $a, b$ real. Here $a = \operatorname{Re}(z)$ is the real part and $b = \operatorname{Im}(z)$ is the imaginary part.

Two complex numbers are equal only when their real parts match AND their imaginary parts match. Addition and multiplication follow ordinary algebra 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)

.

Example

Find real

x, y

if

(x + yi) + (2 - 3i) = 5 + i

.

3. The Argand plane

Every complex number $z = a + bi$ is the point $(a, b)$ : the horizontal axis carries the real part, the vertical axis the imaginary part. This picture — the Argand plane — turns algebra into geometry.

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

4. Conjugate and division

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 division and for extracting parts:

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

Example

Write

\dfrac{1}{2 + i}

in the form

a + bi

.

Example

Express

\dfrac{3+i}{1-i}

in the form

a+bi

.

5. Modulus of a complex number

The modulus measures the distance of $z$ from the origin — the hypotenuse of the right triangle with legs $x$ and $y$ .

DefinitionModulus (Definition 2.4)

If $z = x + iy$ , then $|z| = \sqrt{x^2 + y^2}$ .

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)(1+i)|

.

6. Square root of a complex number

To find $\sqrt{a+bi}$ , set it equal to $x+yi$ , square, and match parts. A clean shortcut uses the modulus.

\sqrt{a+bi} = \pm\left(\sqrt{\tfrac{|z|+a}{2}} + i\,\operatorname{sgn}(b)\sqrt{\tfrac{|z|-a}{2}}\right)

Example

Find

\sqrt{6 - 8i}

.

Note: Always check by squaring your answer — a $30$-second verification that catches sign slips: $(2\sqrt2 - \sqrt2 i)^2 = 8 - 8i + 2i^2 = 6 - 8i$. ✓

7. Polar (trigonometric) form and argument

A non-zero complex number is fixed by how far it is from the origin (modulus $r$ ) and the direction it points (angle $\theta$ ). With $x = r\cos\theta$ and $y = r\sin\theta$ :

DefinitionPolar form (Definition 2.6)

$z = r(\cos\theta + i\sin\theta)$ , where $r = |z|$ and $\theta = \arg(z)$ satisfies $\tan\theta = \dfrac{y}{x}$ (chosen for the correct quadrant).

Note: The argument is only fixed up to multiples of $2\pi$. The unique value in $(-\pi,\pi]$ is the principal argument $\operatorname{Arg}(z)$.

In polar form, multiplication and division become rotations — angles add or subtract:

z_1 z_2 = r_1 r_2\big[\cos(\theta_1+\theta_2) + i\sin(\theta_1+\theta_2)\big]

Example

Express

z = 1 + i

in polar form.

Example

Where is

z = -1 + i

? Give its polar form.

8. Geometry and locus

Because $|z_1 - z_2|$ is the distance between two points, equations in $z$ describe familiar shapes. This is a favourite exam theme.

$|z - z_0| = r$ is a circle of radius $r$ centred at $z_0$ .
$|z - a| = |z - b|$ is the perpendicular bisector of the segment joining $a$ and $b$ .
$|z - z_1| + |z - z_2| = k$ is an ellipse with foci $z_1, z_2$ .

Example

Describe the locus of

|z - (2 + 3i)| = 4

.

9. Euler's form

Euler's formula packs the polar form into an exponential, which makes powers and roots especially clean:

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

Multiplying multiplies the moduli and adds the angles: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2} = r_1 r_2\, e^{i(\theta_1 + \theta_2)}$ . Setting $\theta=\pi$ gives the famous $e^{i\pi} = -1$ .

10. De Moivre's theorem and roots

DefinitionDe Moivre's theorem

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

Example

Compute

\left(\cos\tfrac{\pi}{6} + i\sin\tfrac{\pi}{6}\right)^{6}

.

Run in reverse, De Moivre gives the $n$ distinct $n$ -th roots, equally spaced by $\tfrac{360^\circ}{n}$ around a circle:

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

Example

Find the cube roots of unity and their sum.

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