IAS Buddha - UPSC Preparation

The imaginary number is a fine and wonderful resource of the human spirit. — Gottfried Wilhelm Leibniz

4.1 The Imaginary Unit

No real $x$ satisfies $x^2=-1$ . We introduce $i$ with

$i^2=-1$

A complex number is $z=a+ib$ with $a,b\in\mathbb{R}$ . Here $a=\operatorname{Re}(z)$ (real part) and $b=\operatorname{Im}(z)$ (imaginary part).

$z_1=z_2$ iff real and imaginary parts match.

Addition, subtraction, multiplication: treat $i$ like a variable with $i^2=-1$ .

Conjugate: $\bar{z}=a-ib$ . Then $z\bar{z}=a^2+b^2$ .

Modulus: $|z|=\sqrt{a^2+b^2}$ .

$|z_1 z_2|=|z_1|\,|z_2|,\qquad \overline{z_1 z_2}=\bar{z}_1\bar{z}_2$

Division: $\dfrac{z_1}{z_2}=\dfrac{z_1\bar{z}_2}{|z_2|^2}$ if $z_2\mathrel{\htmlClass{course-neq}{\char"2260}} 0$ .

Argand plane: $z=a+ib$ is the point $(a,b)$ .

$(z_1+z_2)^2=z_1^2+z_2^2+2z_1 z_2$

and the usual binomial expansions still hold.