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Algebra of Complex Variables

Complex variable A is made of two real numbers:

A = ar + jai ar and aj are both real and j = 1Since a complex number A is constructed of two real numbers (ar and ai), it can be viewed

as a point in a two-dimensional plane, called the complex plane as is shown. ar and aidenote the Cartesian coordinates of the point. In a 2-D plane, points can also be represented

by polar coordinates (r and ) or |A| and A for the complex number A.

A

Im

A Re

|A|

ai

ar

Magnitude of A: |A|Phase ofA: A

Real part ofA: ar = Re{A}Imaginary part of A ai = Im{A}

Rectangular form of A A = ar + jai

Polar form of A A = |A|

From the diagram, we get:

|A

|= a

2r + a

2i

A = tan1

aiar

or

ar = |A

|cos(A)

ai = |A| sin(A)

Note that A angle ranges from -180 to 180 (or from zero to 360) while tan1 values range

from -90 to 90. Value of A depends on the signs of both ar and ai and not the sign of

ai/ar only. For example, if ar = ai = +1 as well as ar = ai = 1, tan1(ai/ar) = 45, whilethe correct values are A = 45

for the first case and A = 135 for the second case. Thefollowing table helps find the correct value of A (It assumes that the calculator gives tan

1

values from 90 to 90.)

ar > 0 ai > 0 0 < A < 90 tan1 value is correctar > 0 ai < 0 90 < A < 0 tan1 value is correctar < 0 ai > 0 90 < A < 180 Add 180 to tan1 valuear < 0 ai < 0 180 < A < 90 Add 180 to tan1 value

ECE60L Notes (Math Summaries), Spring 2001 1

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Eulers Formula relates the rectangular and polar form of complex numbers:

Eulers Formula: ej = cos + j sin

A = ar + jai = |A| cos + j|A| sin = |A|ej = |A|

Algebra of Complex Variables:

Consider two complex variables, A = ar + jai = |A|A and B = br + jbi = |B|B . Then,

A + B = (ar + jai) + (br + jbi) = (ar + br) + j(ai + bi)

Note: |A + B| = |A| + |B| A+B = A + B

A B = (ar + jai) (br + jbi) = (arbr aibi) + j(arbi + aibr)

A B = |A|ej

A |B| ej

B = |A||B|ej(A+

B)

Note: |A.B| = |A|.|B| A.B = A + B

A

B=

|A| ejA|B| ejB =

|A||B| e

j(AB)

Note:AB = |A||B| A/B = A B

Complex Conjugate:

A = ar + jai = |A|AA = ar jai = |A|A Complex Conjugate of AA + A = (ar + jai) + (ar jai) = 2arA A = (ar + jai) (ar jai) = j2aiA.A = (|A|A) (|A|A) = |A|2 A real number

To find the ratio of two complex number, we multiply the both nominator and denominator

with the complex conjugate of the denominator:A

B=

A.B

B.B=

1

|B|2 (A.B)

A

B=

ar + jaibr + jbi

=(ar + jai)(br jbi)(br + jbi)(br jbi) =

(arbr + aibi) + j(arbi + aibr)b2r + b

2i

ECE60L Notes (Math Summaries), Spring 2001 2