Engineering Mathematics 13 1 Complex Number Complex Plane

Engineering MathematicsⅡ 제 13 -1 : Complex Number Complex Plane 제 13 -2 : Polar form of Complex Number Powers and Roots

Complex Number Why do we use complex number? History of complex number What is the complex number?

Why do we use complex number?

Complex Number?

Four arithmetical Operations 1. Addition 2. Multiplication

Four Arithematical Operations 3. Substraction 4. Division

Complex Plane Geometrical representation of complex numbers

Geometrical representation of complex numbers

Complex Conjugate Numbers

Polar Form of Complex Numbers (1) (2) (3) (“r” is called the absolute value or modulus of z) (Geometrically |z| is the distance of point z from the origin)

is called the argument of z

Example 1

Polar Form of Complex Numbers Triangle Inequality

We obtain from 6 the generalized triangle inequality ☞ Proof

Reversed triangle Inequality

Polar Form of Complex Numbers Multiplication in polar form

Polar Form of Complex Numbers Division in polar form

De Moivre’s Formula

Roots

Roots

Taking z=1 |z| = r = 1

Example