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
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