COMPLEX NUMBERS

 

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The Algebra of Complex Numbers

 

·        Objectives:  Define complex numbers and present basic algebraic operations: addition, subtraction, multiplication, complex conjugate, and division.

 *This section uses rectangular form. We will introduce simplified operations for multiplication and division with polar form and phasor notation in later sections.

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* Please note that occasionally we use i in place of j.

 

1.      Definition: Complex Numbers

A complex number is a number with a real and an imaginary part, usually expressed in Cartesian form

 

a is called the real part and b is called the imaginary part.


Complex numbers can also be expressed in polar form

The polar form can also be expressed in terms of trigonometric functions using the Euler relationship

 

         * We will discuss the polar form in more detail in Section 4

 

Examples:

 

Given a.        and b.

 

·        a.,  is the real part and  is the imaginary part

 

·        b.,  we note that the real part = 0;

we call the number purely imaginary.

 

 

 

 

 

2.    Addition:

We note that complex numbers must be in rectangular form to add them.

Rule: Add the real parts of each complex number to get

 the real part of the sum. Then add the j parts of each complex

 number to get the j part of the sum.

 

Example: Add 8+j5 and 2+j1   

Solution: (8+j5) + (2+j1) = (8+2) + j (5+1) = 10 + j6

                   

 

                                     

                                     

3.    Subtraction:

As in addition, the numbers must in rectangular form to be subtracted.

Rule: Subtract the real parts of the numbers to get the real part

of the difference, and subtract the j parts to get the j part of the

difference.

 

Example: Subtract 1 +j2 from 3+j4

Solution: (3+j4) – (1+j2) = (3 – 1) + j (4 – 2) = 2+j2

 

 

 

4.    Multiplication:

Multiplication of two complex numbers in rectangular form is accomplished by multiplying, in turn, each term in one number by both terms in the other number and then combining the resulting real terms and the resulting j terms.

(Recall that j x j = -1).

 

Example:  (5 + j3)(2 – j4) = 10 –j20 +j6 +12 = 22 – j14.

 

 

 

5.    Complex Conjugate:

The complex conjugate can be found by simply

changing the sign of the imaginary part in the

rectangular form.

 

Example: The complex conjugate of 2 +j3 is 2 –j3.


 

 

6.    Division:

 

Division of two complex numbers in rectangular form is accomplished by multiplying both the numerator and the denominator by the complex conjugate of the denominator and then combining terms and simplifying.

 

Example:

 

 

 

EXERCISES

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