Just as we were able to define the sum of two complex numbers, we can also define their product. The multiplication of complex numbers is based on the multiplication of binomials with real coefficients. This operation is performed without regard for the fact that has a special meaning. However, before performing the multiplication, all the complex numbers must be expressed in terms of . The multiplication of two binomials that have real coefficients is completed by applying the distributive property. In general, . Since these same operations are valid for complex numbers, multiplication can be defined as:
for all real numbers and .
Example 1: Determine the product of the following complex numbers:
Solution:
Example 2: Determine the product of the following complex numbers:
Solution:
These numbers must first be expressed in terms of .
The operation of division of complex numbers involves the same process that is used for rationalizing the denominator of a fraction that has a radical in the denominator. Therefore, to divide a complex number, the numerator and the denominator must be multiplied by the conjugate of the denominator. This procedure makes it possible to write the solution in the standard form of a complex number. As a result, the operation of division of complex numbers can be defined as:
for all real numbers
Example 3: Determine the quotient of the following complex numbers:
Solution:
The conjugate of is .
Therefore the numerator and denominator of the fraction will be multiplied by this conjugate.
Another way to express the answer is . However, most results that are in the form of a fraction are usually written as a single fraction.
Let us return to the problem of representing the current of the alternating circuit in the form of a complex number. We were given the formula but to solve for the current the formula must be used.
Solution:
Lesson Summary
In this lesson you learned how to perform the basic operations of multiplication and division on complex numbers. The processes involved in both are very similar to performing the operations on binomials with real coefficients.
Points to Consider
Will these operations be performed the same way for numbers in a complex number plane?
Are there other forms of complex numbers that may facilitate these operations on complex numbers in a complex number plane?
Review Questions
Perform the indicated operations and express all answers in the form .
Review Answers
Applications, Trigonometric Tools
Operations on Complex Numbers
Learning Objectives
A student will be able to:
Understand real-world applications of complex numbers.
Introduction
In this lesson we will explore examples of real-world problems that use complex numbers in the solutions of these problems.
Example 1: The voltage in a particular circuit is the product of the current and the impedance (the resistance) . Calculate the voltage in a circuit that has a current of and an impedance of .
Solution:
Example 2: An airplane heads north of west with a velocity that can be represented by . The wind is blowing from south of west with a velocity that be represented by . Determine the resultant velocity of the plane graphically and algebraically.
Solution:
Using the TI Calculator
Learning Objectives
A student will be able to:
Use the TI calculator to perform basic operations on complex numbers.
Introduction
The TI calculator is programmed to perform operations with complex numbers
Turn on the calculator and press [MODE] Cursor down to Real and over to
Press [ENTER] Now press [ MODE](quit) to return to home screen.
To express a complex number in standard form , simply enter the number into the calculator and press [ENTER] The result will be the complex number in standard form.
Example 1: Express in standard form.
Press and press [ENTER] appears on the screen.
To multiply complex numbers that are in standard form requires you to access by pressing decimal.
Example 2:
Press ( minus decimal)( plus decimal)
[ENTER] appears on the screen.
The other basic operations can all be done in the same manner on the calculator.
Trigonometric Form of Complex Numbers: Relationships among x, y, r, and θ
Learning Objectives
A student will be able to:
Understand the relationship between the rectangular form of complex numbers and their corresponding polar form.
Introduction
Despite their names, complex numbers and imaginary numbers have very real and significant applications in both mathematics and in the real world. The fields of physics and electronics and use these numbers to model phenomena all the time. In particular, the fields of mechanics, circuit analysis, and acoustics also use complex and imaginary numbers extensively. “say. The abstract mathematical formalism of trigonometry and complex notation carry important physical meanings in these disciplines. Complex numbers are also useful for pure mathematics, providing a more consistent and flexible number system that helps solve algebra and calculus problems. We will see some of these applications in the examples throughout this lesson, though our focus will be on understanding of the notation and manipulation, not engineering or science. It is remarkable that an abstract mathematical theory invented over three centuries ago could find important applications in modern electronics. Mathematics is like that. It surprises us.
We have just seen the relationship between vectors and complex numbers by representing the addition of two complex numbers on the complex plane. The resulting vector was the sum of the two complex numbers. Since we can use one to represent the other, we will apply this fact to write complex numbers in another form. This new form will prove to be advantageous when performing the basic operations of multiplication and division on complex numbers.
The following diagram will help you understand the relationship between complex numbers and the new form of complex numbers.
In the figure above, the point that represents the number was plotted and a vector was drawn from the origin to this point. The relation between vectors and complex numbers can be seen. As a result, an angle in standard position, , has been formed. In addition to this, the point that represents is units from the origin. Therefore, any point in the complex plane can be found if the angle and the value are known. The following equations relate and .
The Trigonometric or Polar Form of a Complex Number (r cis )
The Trigonometric or Polar Form of a Complex Number (r cis )
Learning Objectives
A student will be able to:
Recognize the equations for converting complex numbers from standard form to polar form and vice versa.
Introduction
This short lesson will expose you to the equations used to convert complex numbers written in standard form to their polar form. In the previous section, you were introduced to the equations that showed the relationship between and .
Recall the equations that you learned in the previous lesson.
These demonstrated the relationship between rectangular coordinates and polar coordinates
If we now apply the first two equations to the point the result would be:
The right side of this equation is called the polar or trigonometric form of a complex number. A shortened version of this polar form is written as . The length is called the absolute value or the modulus, and the angle is called the argument of the complex number. Therefore, the following equations define the polar form of a complex number:
Trigonometric Form of Complex Numbers: Steps for Conversion
Learning Objectives
A student will be able to:
>
Convert complex numbers from standard form to polar form and vice versa.
Introduction
Now that the various equations have been explored for converting complex numbers from standard form to polar form, we will now put these equations into action. The polar form of complex numbers is used extensively in the field of optics and electricity. We will discover their use in solving electrical problems later in the lesson.
It is now time to implement the equations explored earlier to perform the operation of converting complex numbers in standard form to complex numbers in polar form. The following equations will be used to complete the conversions:
Example 1: Represent the complex number graphically and express it in its polar form.
Solution:
From the rectangular form and
The polar form is
Another widely used notation for the polar form of a complex number is . This is not a new form – merely a shorthand way of writing . Now there are three ways to write the polar form of a complex number.
Example 2: Express the following polar form of each complex number using the shorthand representations.
a)
b)
Solution:
a)
b)
Example 3: Represent the complex number graphically and give two notations of its polar form.
Solution:
From the rectangular form of and
This is the reference angle so now we must determine the measure of the angle in the third quadrant.
One polar notation of the point is
Another polar notation of the point is So far we have expressed all values of theta in degrees. Polar form of a complex number can also have theta expressed in radian measure. This would be beneficial when plotting the polar form of complex numbers in the polar plane.
The answer to the above example with theta expressed in radian measure would be:
Now that we have explored the polar form of complex numbers and the steps for performing these conversions, we will look at an example in circuit analysis that requires a complex number given in polar form to be expressed in standard form. The field of circuit analysis was one that was mentioned at the beginning of the lesson as using complex and imaginary numbers frequently.
Example 4: The impedance , in ohms, in an alternating circuit is given by . Express the value for in standard form. (In electricity, negative angles are often used. The physical rationale for representing quantities in circuits as vectors rather than simple scalers is beyond the scope of the study of trigonometry. Electrical quantities in alternating circuits are vectors with magnitude and direction.)
Solution:
The value for is given in polar form. From this notation, we know that and Using these values, we can write:
Therefore the standard form is .
Lesson Summary
In this lesson you learned how to convert complex numbers expressed in standard form to their corresponding polar form and vice versa. You were also introduced to a shorthand notation for the polar form of a complex number. The relation between the two forms was readily seen when both were related to graphical representations.
Points to Consider
A polar form of a complex number exists. Is it possible to perform basic operations on complex numbers in this form?
If operations can be performed, do the processes change for polar form or remain the same as for standard form?
Review Questions
Express the complex number graphically and write it in its polar form.
Graph the complex number and express it in standard form.
Review Answers
and
Since is in the fourth quadrant then Expressed in polar form is or
The standard form of the polar complex number is
Vocabulary
Argument
In the complex number , the argument is the angle .
Modulus
In the complex number , the modulus is . It is the distance from the origin to the point in the complex plane.
Polar Form
Also called trigonometric form is the complex number written as where and .
Product Theorem
Learning Objectives
A student will be able to:
Determine the product theorem of complex numbers in polar form.
Introduction
In previous lessons, we have implemented the formula to determine the voltage or the current of an alternating current. To determine involved calculating the product of and . This calculation was done with all quantities expressed as complex numbers in standard form. In lesson 7.3, the calculations will be done by using the polar form of the complex numbers.
Multiplication of complex numbers in polar form is similar to the multiplication of complex numbers in standard form. However, to determine a general rule for multiplication, the trigonometric functions will be simplified by applying the sum/difference identities for cosine and sine. To obtain a general rule for the multiplication of complex numbers in polar from, let the first number be and the second number . Now that the numbers have been designated, proceed with the multiplication of these binomials.
To arrive at the general rule, and the sum identity and were applied. Therefore:
OR
Quotient Theorem
Learning Objectives
A student will be able to:
Determine the quotient theorem of complex numbers in polar form.
Introduction
In previous lessons, we have implemented the formula to determine the voltage or the current of an alternating current. To determine involved calculating the quotient of and . This calculation was done with all quantities expressed as complex numbers in standard form. In lesson 7.3, the calculations will be done by using the polar form of the complex numbers.
Division of complex numbers in polar form is similar to the division of complex numbers in standard form. However, to determine a general rule for division, the denominator must be rationalized by multiplying the fraction by the conjugate. In addition, the trigonometric functions must be simplified by applying the sum/difference identities for cosine and sine as well as one of the Pythagorean identities. To obtain a general rule for the division of complex numbers in polar from, let the first number be and the second number The conjugate of is . Now that the numbers have been designated, proceed with the division of these binomials.
To arrive at the general rule, the difference identity and and the Phythagorean identity were applied. Therefore:
OR
Using the Quotient and Product Theorem
Learning Objectives
A student will be able to:
Determine the product and the quotient of complex numbers in polar form.
Introduction
In previous lessons, we have implemented the formula to determine the voltage or the current of an alternating current. To determine involved calculating the product of and . To determine involved calculating the quotient of and . These calculations were done with all quantities expressed as complex numbers in standard form. The calculations can be done now by using the product theorem and the quotient theorem for the polar form of complex numbers.
Now that general rules have been obtained for the multiplication and division of complex numbers in polar form, they can now be implemented. Recall that these rules are:
AND
Example 1: Find the product of the complex numbers and
Solution:
Note: Angles are expressed unless otherwise stated.
Example 2: Find the product of
Solution:
Doing these calculations prior to substituting into the rule simplifies the process.
Example 3: Find the quotient of
Solution:
Express each number in polar form.
Example 4: Find the quotient of the two complex numbers and
Solution:
Lesson Summary
In this lesson you learned how to apply the general rules for the multiplication and the division of complex numbers in polar form. If the numbers are given in polar form and the basic operations of multiplication and division are performed, the product or quotient can then be converted to standard form, if required.
Points to Consider
We have performed the basic operations of arithmetic on complex numbers, but we have not dealt with any exponents other than 2 or any roots other than .
CK-12 Trigonometry Page 31
