CK-12 Trigonometry
Page 30
Using the same trend, two complex numbers are equal if the real parts are equal and the imaginary parts are equal. In other words only if and . This definition for equal complex numbers can be applied to equations. Remember that the solution for an equation is the value that makes both sides equal.
Example 1: Perform the indicated operations and simplify each complex number to its standard form.
a)
b)
c)
Solution:
a)
b)
c)
Example 2: What values of and satisfy the equation ?
Solution:
Complex numbers also have conjugates. The conjugate of is and vice versa. To obtain the conjugate of a complex number, the sign of the imaginary part is changed.
Example 3: Find the conjugate of each complex number.
a)
b)
c)
Solution:
a)
b)
c)
Lesson Summary
In this lesson you learned that a complex number was the sum of a real part and an imaginary part. Using this definition, you were able to express a complex number in standard form. You also explored the equality of complex numbers and applied this definition to solving equations. The final topic you learned about was the conjugate of a complex number that is obtained by changing the sign of the imaginary part.
Points to Consider
What operations can be performed using complex numbers?
Are there specific rules or laws for performing these operations?
Will the results of these operations also be complex numbers?
Review Questions
Perform the indicated operations and simplify each complex number to its standard form. Write the conjugate for each solution.
What values of and satisfy the equation .
Review Answers
The Set of Complex Numbers (complex, real, irrational, rational, etc)
Learning Objectives
A student will be able to:
Recognize the Complex Number System.
Position numbers in the correct category within the system.
Introduction
Every number that you can imagine belongs to the complex number system. The set of complex numbers is made up of all real and imaginary numbers and all possible combinations. They take the form of where and are real numbers and . We will explore the subsets of this number system and present the results in a flow chart representation.
The complex number system includes the real numbers and the imaginary numbers. Real numbers include all decimals…rational and irrational numbers. Every real number can be found on a line. Rational numbers consist of the quotient of two integers and yield decimals that repeating patterns. Some examples of rational numbers are and . Included in the rational numbers are the integers. Integers are rational numbers that consist of positive and negative whole numbers including zero. Another subset of the rational numbers is the whole numbers. These include zero and the counting numbers The irrational numbers are also part of the real numbers. Irrational numbers produce decimals that have no repeating patterns. Some examples of irrational numbers are , and . The imaginary numbers that are included in the complex number system, are those that cannot be expressed as decimals. Examples of imaginary numbers are and all of these use . The following flow chart demonstrates the structure of the complex number system
This lesson is meant as a conveyor of information to familiarize you with the complex number system. Therefore there are no exercises that need to be completed for the lesson. However, you should concentrate on learning the members of each subset of the complex number system.
Lesson Summary
In this lesson you explored the subsets that make up the complex number system. You also learned of the types of numbers that belong to each one.
Points to Consider
If all of these numbers are included in the complex number system, can complex numbers be represented on a graph?
If complex numbers can be graphed, which coordinate will be represented by the real part? By the imaginary part?
While the Complex plane looks like the Cartesian plan, the horizontal axis the real part of a complex number and the vertical axis represents the imaginary part of a complex number. A single complex number is plotted on this plane with determining its coordinate and determining its coordinate.
Complex Number Plane
Learning Objectives
A student will be able to:
Graph complex numbers in the complex plane.
Assign coordinates to points plotted in the complex plane.
In the same way that ordered pairs of real numbers are assigned to points in a plane, so are complex numbers. Beginning with two perpendicular number lines that intersect at the origin, like the axis of a Cartesian graph, place real numbers on the horizontal line and numbers on the vertical line. To plot a point on a Cartesian coordinate system, the value was located on the horizontal axis and from here the point was moved upward or downward the value of . The point was plotted here. A complex number in standard form has as the real part and as the imaginary part. Therefore, the is the value and the is the value in a complex plane. A big distinction between the real Cartesian plane and the complex plane is that in the former, pairs of real numbers are plotted as points, and in the latter single complex numbers are plotted as points.
This is a model of the complex plane. The horizontal number line is called the real axis. Every real number is the coordinate of a point on this axis. The vertical line is called the imaginary axis. Each pure imaginary number or number is the coordinate of a point on the axis.
Every point in the complex plane has a complex number as its coordinate to define the position of the point with respect to the axes. There is a correlation between the Cartesian coordinate system and the complex number plane. This can be seen by letting the real axis be the axis and the imaginary axis be the axis. Thus, the point with coordinates in the complex number plane has coordinates in the Cartesian coordinate system.
The distance from the origin to the point with coordinate is called the absolute value of the complex number . In the complex number plane the coordinate of is often referred to as . This distance, according to the Pythagorean Theorem, is . Therefore,
OR
Now that the complex number plane has been explored, it is time to plot some points.
Example 1: Plot each number on the complex number plane and determine the distance from the origin of points and .
a)
b)
c)
d)
Solution:
Distance from the origin:
It is time to return to the two students who are walking home to determine who walked the greater distance. If the distance walked by each student is represented by and , respectively, the following system of equations could represent the problem.
Solving this system of equations:
Substituting into the first equation:
The solutions we obtain for are and . These solutions are confusing because if we look at them on a number line, we would see:
In the first solution, Jacob walked to home while Kyle walked .
The second solution indicates that both Jacob and Kyle each walked .
If we take another look at the problem, it does not specify which distance is one-half the square of the other. As a result, the equations and could have been used to represent the problem.
Solving this system of equations:
Using the quadratic formula
and
and
and
Substituting into the first equation
The solutions we obtain for are and The distance walked by Jacob and Kyle can be represented on a complex number plane.
Kyle walks to point B with coordinate and Jacob walks to point A with coordinate .
According to our definition of absolute value
The distances from the origin (school) to the points on the complex plane (home) are not confusing. Kyle walked the greater distance.
Lesson Summary
In this lesson you learned how to plot complex numbers on a complex number plane. You also learned of the similarities between this plane and the Cartesian number plane. The absolute value of a complex number was shown to be an asset when solving a problem.
Points to Consider
Are there other times when solutions to problems are best determined by using complex numbers?
If we could perform basic operations on complex numbers, would the results be useful?
What are the applications of complex numbers in the real world?
Review Questions
Give the coordinates of each point plotted on the complex number plane and calculate the absolute value of any two of the points.
Review Answers
Vocabulary
Complex Number
Any number that can be written in the form , where and are real numbers and is the imaginary part.
Complex Number Plane
A coordinate plane used to represent complex numbers. This plane looks like the Cartesian plan, except that instead of both axes representing real numbers, the horizontal axis the real part of a complex number and the vertical axis represents the imaginary part of a complex number. A single complex number is plotted on this plane with determining its x-coordinate and determining its coordinate.
Conjugate of a Complex Number
The conjugate of the complex number is .
Imaginary Number
A complex number of the form where .
Quadratic Formula
Learning Objectives
A student will be able to:
Find complex zeros of quadratic equations.
Understand the concept of the conjugate with respect to the roots of a quadratic equation and complex numbers.
Introduction
Consider the graph of . You can see that the graph does not intersect the axis. Does this mean that there are no roots for the quadratic function We will explore this later in this lesson.
The quadratic formula is used to determine the roots of a quadratic equation where and are real numbers and .The radicand of the formula is known as the discriminant and is very useful in determining the nature of the roots of the equation. The following table summarizes the results:
Value of the discriminant Nature of the roots
Two different real roots
One repeated real root
A complex conjugate pair of roots
Note that in the function graphed in the figure above, the value of is negative, corresponding to the fact that the function has no roots. Unless the parabola depicting touches the axis exactly at its vertex, it will cross the axis twice and have exactly two roots.
Complex roots do not appear in the graph of a quadratic function, as they do not lie in the real numbers. Any quadratic equation that has a root of the form also has a root of the form . These two roots are called conjugates.
Example 2: For the following equations, evaluate the discriminant and describe the roots of the equation.
a)
b)
Solution:
a)
A complex conjugate pair of roots
b)
Two different real roots
Example 3: Solve the equation
Solution:
Let us return to the graph of . As we saw, the parabola did not intersect the axis. We can learn about the roots if we evaluate the discriminant.
A complex conjugate pair of roots
If the roots are a complex pair of roots, the parabola will NOT intersect the axis.
Lesson Summary
If the radicand of the quadratic formula produced a negative value, you carefully checked your calculations for an error because the square root of a negative number did not exist. In this lesson you learned that you no longer have to check your calculations, if you are certain that they are correct, because the square root of a negative number does exist and it is in the form of a complex number. We applied this fact to determining the roots of a quadratic equation by using the quadratic formula. You also learned that if you calculated the value of the discriminant, you could predict the nature of the roots of the equation.
Allowing complex roots enables a much more robust theory. The Theorem of Algebra proved by Gauss states that in the complex system, a polynomial of degree has roots. Finding the algebraic expression for these roots leads to much more difficult problems, but the extension of the real numbers to the complex plane guarantees a number of roots equal to the degree of the equation.
Points to Consider
What does the complex conjugate pair of roots tell us about the graph of the quadratic function?
What does the graph of a quadratic equation of the form tell us about the roots of the function?
Review Questions
For the following quadratic equation, describe the nature of the roots and solve the equation to determine the exact roots.
What does the following graph tell you about its quadratic function?
Review Answers
A complex conjugate pair of roots
The graph does not intersect the axis. The value of the discriminant, will be less than zero. This means that the roots of the quadratic function will be a complex conjugate pair.
Sums and Differences of Complex Numbers
Learning Objectives
A student will be able to:
Add and subtract complex numbers.
The sum or difference of two pure imaginary numbers is consistent with the rules of arithmetic. If is considered to be and the distributive property is applied to the operation of addition then can be expressed as or . The same is true for subtraction. can be written as or A complex number consists of a real part and an imaginary part. The real parts are added or subtracted and the imaginary parts are added or subtracted as shown above. Therefore these basic operations of complex numbers can be defined as:
for all real numbers and
and
for all real numbers and .
Many of the properties of real numbers are also applicable to complex numbers. The commutative property is one that applies to both real and complex numbers for addition.
If then
In a similar way, we can show that the addition of complex numbers is associative.
From the above, we can conclude that zero is the additive identity element for the complex number system.
The negative of a complex number in standard form is . Therefore
From this we can conclude that the additive inverse of a complex number is .
Example 1: Perform the indicated operations in each the following:
a)
b)
c)
Solution:
a)
b)
c)
Two complex numbers and their sum can be represented graphically in a complex plane. If two complex numbers are graphed in a plane and lines are drawn from the origin to each point, we can consider these complex numbers as being vectors. Therefore the sum of the two numbers can be called the vector sum. To represent this graphically, plot one of the complex numbers and draw a line from the origin to the point. Repeat this process for the second complex number. Complete a parallelogram with the lines drawn as adjacent sides. The resulting fourth vertex is the point that represents the sum.
Lesson Summary
In this lesson you learned that the properties of real numbers apply to complex numbers. You also learned the method for adding and subtracting complex numbers. By representing two complex numbers graphically, you saw one way in which these numbers can be applied to real-world problems.
Points to Consider
If complex numbers in a complex plane are related to real numbers of a Cartesian coordinate system, are they related to polar numbers in a polar plane?
Is there a way
to convert complex numbers to a polar form?
Review Questions
Perform the indicated operations graphically and check the results algebraically.
Review Answers
(a) Subtracting from is equivalent to adding . Therefore we graph the solution by adding and the result is .
Check:
(b) Adding these two complex numbers Graphically produced the result
Check:
Products and Quotients of Complex Numbers (conjugates)
Learning Objectives
A student will be able to:
Multiply and divide complex numbers.
Introduction
The impedance of an electric circuit is the total effective resistance to the flow of current by a combination of the elements in the circuit. In an alternating-current circuit, the voltage is given by where is the current in amperes and is the impedance in ohms. If and , what is the complex number representation for We will determine this value later in this lesson.