Complex Numbers and Applications- Advanced Engineering Mathematics
Complex Numbers and Applications
ME50 ADVANCED ENGINEERING MATHEMATICS
1 Complex Numbers √ A complex number is an ordered pair (x, y) of real numbers x and y. For example, (−2.1, 3.5), (π, 2), (0, 0) are complex numbers. Let z = (x, y) be a complex number. The real part of z, denoted by Re z, is the real number x. The imaginary part of z, denoted by Im z, is the real number y. Re z = x Im z = y Two complex numbers z1 = (a1, b1) and z2 = (a2, b2) are equal, written z1 = z2 or (a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2. For example, if (x, 2) = (3, c) then x = 3 and c = 2. Since a complex number is denoted by an ordered pair (x, y) of real numbers x and y, then we may view the complex number (x, y) as the point with abscissa x and ordinate y. The complex plane consists of all the points that represent the complex numbers. For example, let us indicate the following complex numbers in the complex plane: z1 = (−3, −2), z2 = (0, 1), z3 = (4, 2), z4 = (5, −1)
. . . . . ....... . . . . . . . . . ...... . . . . . . . . . . . . ...... . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . ... . . . . . . . . . ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ... . . . . . . . . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . .
z3
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z2
•
z1
•
•
z4
The complex plane showing four complex numbers z1, z2, z3, and z4
1.1 Operations on Complex Numbers
Some binary operations on complex numbers are addition, multiplication, and division. They are defined as follows:
ME50 ADVANCED ENGINEERING MATHEMATICS
1 Complex Numbers √ A complex number is an ordered pair (x, y) of real numbers x and y. For example, (−2.1, 3.5), (π, 2), (0, 0) are complex numbers. Let z = (x, y) be a complex number. The real part of z, denoted by Re z, is the real number x. The imaginary part of z, denoted by Im z, is the real number y. Re z = x Im z = y Two complex numbers z1 = (a1, b1) and z2 = (a2, b2) are equal, written z1 = z2 or (a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2. For example, if (x, 2) = (3, c) then x = 3 and c = 2. Since a complex number is denoted by an ordered pair (x, y) of real numbers x and y, then we may view the complex number (x, y) as the point with abscissa x and ordinate y. The complex plane consists of all the points that represent the complex numbers. For example, let us indicate the following complex numbers in the complex plane: z1 = (−3, −2), z2 = (0, 1), z3 = (4, 2), z4 = (5, −1)
. . . . . ....... . . . . . . . . . ...... . . . . . . . . . . . . ...... . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . ... . . . . . . . . . ................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ... . . . . . . . . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . .
z3
•
z2
•
z1
•
•
z4
The complex plane showing four complex numbers z1, z2, z3, and z4
1.1 Operations on Complex Numbers
Some binary operations on complex numbers are addition, multiplication, and division. They are defined as follows: