Complex Numbers in Engineering Mathematics: A Comprehensive Introduction, Lecture notes of Engineering Mathematics

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Eski¸sehir Technical University, Turkey

October 22, 2018

Today

Complex Numbers

What is a complex number

What is a complex number?

A simple interpretation

A complex number is a point on a two-dimensional real vector space Complex numbers z can be defined as ordered pairs

z = (x, y),

where x is called real part and y is called imaginary part and they are often denoted as

x = Re(z) y = Im(z),

What is a complex number

What is a complex number?

A simple interpretation

A complex number is a point on a two-dimensional real vector space Complex numbers z can be defined as ordered pairs

z = (x, y),

where x is called real part and y is called imaginary part and they are often denoted as

x = Re(z) y = Im(z),

Two complex numbers (x 1 , y 1 ) and (x 2 , y 2 ) are said to be equal whenever x 1 = x 2 y 1 = y 2 ,

Properties of complex numbers

Let z 1 = (x 1 , y 1 ) and z 2 = (x 2 , y 2 ). Then,

z 1 + z 2 = (x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ),

z 1 z 2 = (x 1 x 2 − y 1 y 2 , x 1 y 2 + x 2 y 1 ).

Properties of complex numbers

Let z 1 = (x 1 , y 1 ) and z 2 = (x 2 , y 2 ). Then,

z 1 + z 2 = (x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ),

z 1 z 2 = (x 1 x 2 − y 1 y 2 , x 1 y 2 + x 2 y 1 ).

Then, (x, 0) + (0, y) = (x, y),

Properties of complex numbers

Let z 1 = (x 1 , y 1 ) and z 2 = (x 2 , y 2 ). Then,

z 1 + z 2 = (x 1 , y 1 ) + (x 2 , y 2 ) = (x 1 + x 2 , y 1 + y 2 ),

z 1 z 2 = (x 1 x 2 − y 1 y 2 , x 1 y 2 + x 2 y 1 ).

Then, (x, 0) + (0, y) = (x, y),

(0, 1)(y, 0) = (0, y)

(x, y) = (x, 0) + (0, 1)(y, 0)

Let z 1 = (x 1 , 0) and z 2 = (x 2 , 0). Then,

z 1 + z 2 = ( x ︸ 1 ︷︷+ x (^2) ︸

Real space^ sum in

, 0) ; z 1 z 2 = ( x ︸ ︷︷ ︸ 1 x 2 multip. in Real space

Let z 1 = (x 1 , 0) and z 2 = (x 2 , 0). Then,

z 1 + z 2 = ( x ︸ 1 ︷︷+ x (^2) ︸

Real space^ sum in

, 0) ; z 1 z 2 = ( x ︸ ︷︷ ︸ 1 x 2 multip. in Real space

By sum and multiplication definitions of a complex number

(x, y) = (x, 0) + (0, 1)(y, 0)

Let z 1 = (x 1 , 0) and z 2 = (x 2 , 0). Then,

z 1 + z 2 = ( x ︸ 1 ︷︷+ x (^2) ︸

Real space^ sum in

, 0) ; z 1 z 2 = ( x ︸ ︷︷ ︸ 1 x 2 multip. in Real space

By sum and multiplication definitions of a complex number

(x, y) = (x, 0) + (0, 1)(y, 0)

Let z 1 = (x 1 , 0) and z 2 = (x 2 , 0). Then,

z 1 + z 2 = ( x ︸ 1 ︷︷+ x (^2) ︸

Real space^ sum in

, 0) ; z 1 z 2 = ( x ︸ ︷︷ ︸ 1 x 2 multip. in Real space

By sum and multiplication definitions of a complex number

(x, y) = (x, 0) + (0, 1)(y, 0)

Then, if we let i = (0, 1),

z = (x, y) = (x, 0) + (0, 1)(y, 0) =: x + iy,

Let z 1 = (x 1 , 0) and z 2 = (x 2 , 0). Then,

z 1 + z 2 = ( x ︸ 1 ︷︷+ x (^2) ︸

Real space^ sum in

, 0) ; z 1 z 2 = ( x ︸ ︷︷ ︸ 1 x 2 multip. in Real space

By sum and multiplication definitions of a complex number

(x, y) = (x, 0) + (0, 1)(y, 0)

Then, if we let i = (0, 1),

z = (x, y) = (x, 0) + (0, 1)(y, 0) =: x + iy,

Show that i^2 = − 1

Basic Algebraic Properties

I (^) The commutative and associate laws

z 1 + z 2 = z 2 + z 1 , z 1 z 2 = z 2 z 1 ,

(z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) , (z 1 z 2 )z 3 = z 1 (z 2 z 3 ), I (^) 0 = (0, 0) and 1 = (1, 0). Then, for any z ∈ C,

z + 0 = z z · 1 = z

Basic Algebraic Properties

I (^) The commutative and associate laws

z 1 + z 2 = z 2 + z 1 , z 1 z 2 = z 2 z 1 ,

(z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) , (z 1 z 2 )z 3 = z 1 (z 2 z 3 ), I (^) 0 = (0, 0) and 1 = (1, 0). Then, for any z ∈ C,

z + 0 = z z · 1 = z

I (^) For each z ∈ C, there exists a unique z∗^ ∈ C satisfying z + z∗^ = 0, where z∗^ is called additive inverse of z.