DREAM IDEA PLAN IMPLEMENTATION Introductory to Complex Numbers

DREAM IDEA PLAN IMPLEMENTATION

Introductory to Complex Numbers Present to: Amirkabir University of Technology (Tehran Polytechnic) & Semnan University Dr. Kourosh Kiani Email: kkiani 2004@yahoo. com Email: Kourosh. kiani@aut. ac. ir Web: aut. ac. com 2

Introductory to Complex Numbers

In algebra we encounter the problem of finding the roots of the polynomial: If , there are two real roots. If , we obtain the single root (of multiplicity 2) To deal with the case number: Then for we introduce the imaginary

Example:

Definition: A complex number is a number of the form Where x and y are real numbers. x is called the real part of z and is denoted Re(z). y is called the imaginary part of z and is denoted Im(z). Representation (1) is sometimes called the Cartesian form of complex number z. Complex number Re(z) Im(z) z = 1+2 i 1 2 z = -3 +5 i -3 5 z=4 4 0 z = -8 i 0 -8 Notation: We shall denote the set of complex numbers by symbol C. Thus

Equality of Complex Numbers When are two complex numbers equal? Real part must be equal AND Imaginary part must be equal

Zero Complex Numbers A complex number is zero iff Real part equals zero AND Imaginary part equals zero

Relational operations The following relational operations are NOT defined for complex numbers

Basic operations Given two complex numbers Addition of complex numbers

Examples:

Addition Imaginary Axis A+B B A Real Axis

Subtraction of complex numbers

Subtraction Imaginary Axis B A A-B Real Axis

Scalar Multiplication of a complex number If a and then

Multiplication of complex numbers

Multiplication A B Imaginary Axis B A Real Axis

Examples:

Examples:

Conjugate of complex Number Let . Then the conjugate of z, denoted y x is given by

Examples: y x

Multiplication of two conjugate complex numbers

complex Conjugate Similarly:

Inverse of complex Number Let

Division of complex numbers

Division Imaginary Axis B A A/B Real Axis

Examples:

Examples:

Complex Plane We can plot complex numbers in the xy-plane by plotting Re(z) along the x-axis and Im(z) along the y-axis. Thus, each complex number can be thought of as a point or vector in the xy-plane. The complex plane is also known as Argand Digram. y y -3+2 i 1+2 i 2+3/2 i x x -1 -i 2 -3/2 i

Absolute value of a complex number For. We define the magnitude or length or absolute value of z, denoted by

Examples: y= Im (z) x= Re (z)

Questions? Discussion? Suggestions ? 34

35