UNIT 1 • EXTENDING THE NUMBER SYSTEM Lesson … · • The polar form of a complex number is an...

UNIT 1 • EXTENDING THE NUMBER SYSTEMLesson 3: Operating with Complex Numbers

Instruction

© Walch EducationU1-151

UCSS Secondary Mathematics II Teacher Resource 1.3.5 HONORS

IntroductionSince complex numbers consist of both real and imaginary parts, they cannot be graphed on a traditional coordinate plane. Instead, complex numbers must be graphed on a similar but somewhat different plane called a complex plane. This plane allows complex numbers to be graphed and it enables them to be rewritten in different formats.

Key Concepts

• Complex numbers can be graphed on a complex plane.

• The complex plane is similar to the Cartesian coordinate plane, but the y-axis represents imaginary numbers and is often called the imaginary axis. The x-axis still represents real numbers and is referred to as the real axis. On a complex plane, the positive y-axis is typically labeled 1i, 2i, 3i, etc., and the negative y-axis is labeled –1i, –2i, –3i, and so on. However, the x-axis is simply labeled with numbers (i.e., 1, 2, 3), as shown.

Prerequisite Skills

This lesson requires the use of the following skills:

• understanding complex numbers

• recognizing that i2 = –1

• adding and subtracting complex numbers

• finding the moduli of complex numbers

• multiplying radical expressions

• simplifying radicals


Instruction

U1-152© Walch EducationUCSS Secondary Mathematics II Teacher Resource

1.3.5 HONORS

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

• A complex number in the form a + bi can be graphed on the complex plane by identifying the location of the real part of the number (a) on the real axis (the x-axis) and the imaginary part of the number (bi) on the imaginary axis (the y-axis). Consider the following diagram.

Real axis0

Imaginary axis (a, bi)

br

aθ

• When complex numbers are graphed on a complex plane, a new way of writing them is revealed. Notice that if a line is drawn from the origin to the point, the line segment has both a direction and a magnitude, or length. The length of the line segment, r, is equal to the modulus of the complex number. The angle formed (θ) is called the argument of the complex number. Drawing another line from the point to the real axis forms a right triangle with angle θ and sides a and b. In previous lessons, you may have seen the variable r used for the remainder. In this sub-lesson, it has a different meaning.

• The argument of a complex number, which is the angle that describes the direction of the complex number on the complex plane, gives the measurement of the angle between the positive real axis (the x-axis) and the line segment.


Instruction



• Using this information, complex numbers can be written in polar form.

• The polar form of a complex number is an expression representing the polar coordinates of the number on the complex plane. In this form, a complex number is represented as r(cos θ + i sin θ), in which r is the modulus and θ is the argument of the complex number. (The Greek letter theta, θ, is commonly used to refer to unknown angle measures.)

• To write a number in polar form, calculate r and θ. r can be found by calculating the modulus of the complex number or by using the Pythagorean Theorem. One convenient way to find the modulus of a complex number a + bi is to square both the a and b values of the number, and

then take the square root of the sum. In other words, the modulus equals 2 2+a b . This is the same as multiplying the complex number by its conjugate and then taking the square root of the product.

• The argument, θ, can be found by using basic trigonometry. In the form r(cos θ + i sin θ), cos θ stands for “the cosine of θ” and sin θ stands for “the sine of θ.” Cosine and sine are trigonometric functions of an acute angle in a right triangle.

• The cosine is the ratio of the length of the side adjacent to the length of the hypotenuse; the

cosine of θ = cos θ = length of adjacent side

length of hypotenuse.

• The sine is the ratio of the length of the opposite side to the length of the hypotenuse; the sine

of θ = sin θ = length of opposite side

length of hypotenuse.

• The tangent of an acute angle in a right triangle is the ratio of the length of the opposite

side to the length of the adjacent side; the tangent of θ = tan θ = length of opposite side

length of adjacent side.

Therefore, for a complex number, b

atanθ = . To find the value of θ, take the arctangent of

b

a.

The arctangent is the inverse of the tangent function, written tan–1 θ or arctan θ.

• The arctangent can be calculated on a scientific or graphing calculator. Follow the directions appropriate to your calculator model.

On a TI-83/84:

Step 1: Make sure your calculator is in degree mode.

Step 2: Press [2ND][TAN].

Step 3: Enter the value of b

a, then press [)].

Step 4: Press [ENTER].


Instruction


1.3.5 HONORS

On a TI-Nspire:


Step 2: From the home screen, arrow down to the Calculate icon and press [enter].

Step 3: Press [ctrl][tan] and then enter the value of b

a. Note: On some

models of the TI-Nspire, you can press [trig] to bring up the menu

of trigonometric functions, and then use the keypad to select “tan–1.”

Step 4: Press [enter].

• Note that this process only works if the complex number is in Quadrant I.

• Addition and subtraction of complex numbers can easily be analyzed on the complex plane by comparing the locations of the numbers.

• To multiply complex numbers in polar form, first multiply the moduli (the values for r), and then add the angles (the values for θ).

• To find the distance between two complex numbers, first subtract the complex numbers and then find the modulus of the difference.

• To find the midpoint of a segment, the point on a line segment that divides it into two equal parts, average the values at the segment’s endpoints.

• Use the midpoint formula, 2

,2

1 2 1 2x x y y+ +

, to find the midpoint of a line segment.

Common Errors/Misconceptions

• confusing the real and imaginary axes

• dividing a by b instead of b by a when finding the argument

• multiplying the angles rather than the moduli when multiplying complex numbers in polar form

• forgetting to divide by 2 when averaging the values of the endpoints to find the midpoint of a segment


Instruction



Example 1

Plot 5 – 2i on a complex plane.

1. Sketch a complex plane.

The complex plane is very similar to the Cartesian coordinate plane, except that on the complex plane the y-axis represents imaginary numbers and is called the imaginary axis. The x-axis still represents real numbers and is called the real axis. The axis labeling on your sketch should resemble that of the graph shown.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

Guided Practice 1.3.5 HONORS


Instruction


1.3.5 HONORS

2. Plot the point 5 – 2i.

The real part of the complex number 5 – 2i is 5. Locate 5 on the real axis or x-axis, 5 units to the right of the origin.

The imaginary part of the complex number 5 – 2i is –2i. Locate –2i on the imaginary axis or y-axis, 2 units down from the origin. In other words, count two spaces down from the 5 on the real axis and plot the point 5 – 2i.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

5 – 2i


Instruction



Example 2

Rewrite 3 + 5i in polar form.

1. Sketch the point on the complex plane.

While sketching the point is not necessary, it helps to clarify the problem and solution.

Plot 3 + 5i on a complex plane. 3 corresponds to 3 on the real axis, or x-axis, and 5i corresponds to 5 on the imaginary axis, or y-axis. Count over to the right 3 spaces and then count up 5 spaces in order to plot 3 + 5i.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

3 + 5i


Instruction


1.3.5 HONORS

2. Draw r and θ, and complete the triangle.

Draw a line segment from the origin to the point 3 + 5i. Label the line r. Then label the angle the line segment makes with the real axis as θ.

Next, complete the triangle by drawing a vertical line segment from the point to the real axis. The vertical leg of the triangle is b since it is the length of the imaginary part of the complex number a + bi. The horizontal leg of the triangle is a since it is the length of the real part of the complex number.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

3 + 5i

br

aθ

3. Determine the length of r.

The length of r is equal to the modulus of the complex number, 3 + 5i.

To find the modulus, square the real and imaginary parts of the number, and then take the square root of the sum.

Notice that this is the same as using the Pythagorean Theorem to find the length of the hypotenuse of the triangle drawn.

2 2= +r a b Formula to find the modulus, r

(3) (5)2 2= +r Substitute 3 for a and 5 for b.

9 25= +r Simplify.

34=r Combine like terms.

The length of r is 34 , which is in its simplest form.


Instruction



4. Determine the value of θ.

The value of θ, the argument, can be found using basic trigonometry.

Since the tangent is the ratio of the length of the opposite side to the

length of the adjacent side, for a complex number, b

atanθ = .

We have already determined that a = 3 and b = 5, so substitute these

values into the formula for tangent.

b

atanθ = Formula for the tangent of θ

tan(5)

(3)θ = Substitute 5 for b and 3 for a.

arctan5

3θ = Take the arctangent of

5

3.

Compute the arctangent to solve for the angle measure. Follow the directions appropriate to your calculator model.

On a TI-83/84:


Step 2: Press [2ND][TAN].

Step 3: Enter the value of b

a: [5][÷][3]. Then press [)].

Step 4: Press [ENTER].(continued)


Instruction


1.3.5 HONORS

On a TI-Nspire:

Step 1: Make sure your calculator is in degree mode.Step 2: From the home screen, arrow down to the Calculate icon

and press [enter].

Step 3: Press [ctrl][tan] and then enter the value of b

a: [5][÷][3].

Note: On some models of the TI-Nspire, you can press [trig]

to bring up the menu of trigonometric functions, and then

use the keypad to select “tan–1.”Step 4: Press [enter].

The result of either calculator is θ ≈ 59°.

5. Write the complex number in polar form.

The polar form of a complex number is r(cos θ + i sin θ), in which r is the modulus and θ is the argument of the complex number. Substitute the known values for r and θ into this form.

r(cos θ + i sin θ) Polar form of a complex number

34 cos(59 ) sin(59 )( )[ ]= ° + °i Substitute 34 for r and 59° for θ.

Written in polar form, the complex number 3 + 5i is

34 cos59 sin59( )°+ °i .


Instruction



Example 3

Find the sum of the two points shown on the following complex plane.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

1. Determine the complex number represented by each point.

The point in Quadrant I is located at 2 on the real axis and 4i on the imaginary axis. Thus, this point represents 2 + 4i.

The point in Quadrant II is located at –1 on the real axis and 3 (or 3i) on the imaginary axis. Therefore, this point represents –1 + 3i.


Instruction


1.3.5 HONORS

2. Add the complex numbers.

The complex numbers can be added by combining like terms.

(2 + 4i) + (–1 + 3i) Add to find the sum.

= 1 + 7i Combine like terms.

Looking at the problem graphically results in the same answer:

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

1 + 7i

• For the real part, 2 means 2 units to the right and –1 means 1 unit to the left. Starting at the origin, move to the right 2 units and then back to the left 1 unit to end at 1 on the real axis.

• For the imaginary part, 4i means 4 units up and 3i means 3 units up. Starting at the origin, move up 4 units and then move up 3 units more to end at 7 on the imaginary axis, or 7i.

The sum is 1 + 7i, the same answer that was found by combining like terms; thus, the sum of the two points is 1 + 7i.


Instruction



Example 4

Multiply 5(cos 37° + i sin 37°) by 5 cos 27 sin 27( )°+ °i .

1. Multiply the r-values.

To multiply complex numbers in polar form, multiply the r-values and add the angle measures.

Since complex numbers in polar form follow the pattern r(cos θ + i sin θ), r is the value in front of the parentheses.

The r-value of 5(cos 37° + i sin 37°) is 5.

The r-value of 5 cos 27 sin 27( )°+ °i is 5 .

Therefore, the product of the r-values is 5• 5 or 5 5 .

2. Add the angle measures.

The angle measure of 5(cos 37° + i sin 37°) is 37°.

The angle measure of 5 cos 27 sin 27( )°+ °i is 27°.

Therefore, the sum of the angle measures is 37° + 27° or 64°.

3. Write the product in polar form.

Complex numbers in polar form follow the pattern r(cos θ + i sin θ).

As determined in steps 1 and 2, 5 5=r and θ = 64°.

Substitute these values into the pattern.

r(cos θ + i sin θ) Polar form of a complex number

5 5 cos 64 sin 64i( ) ( ) ( )= ° + ° Substitute 5 5 for r and 64° for θ.

The product of 5(cos 37° + i sin 37°) and 5 cos 27 sin 27( )°+ °i

is 5 5 cos 64 sin 64( )°+ °i .


Instruction


1.3.5 HONORS

Example 5

For the two points shown, find the distance between the points and the midpoint of the line segment connecting them.

–10 –8 –6 –4 –2 2 4 6 8 10

Real axis

10i

8i

6i

4i

2i

0

–2i

–4i

–6i

–8i

–10i

Imaginary axis

1. Identify the complex number each endpoint represents.

The endpoint in Quadrant I is located at 5 on the real axis and 2i on the imaginary axis. Thus, this point represents 5 + 2i.

The endpoint in Quadrant IV is located at 1 on the real axis and–4i on the imaginary axis. Therefore, this point represents 1 – 4i.

2. Find the distance between the two endpoints.

To find the distance between two complex numbers, first subtract the complex numbers and then find the modulus of the difference. It does not matter which number is subtracted from the other.

(1 – 4i) – (5 + 2i) Subtract the complex numbers.

= 1 – 4i – 5 – 2i Distribute the negative sign across 5 + 2i.

= –4 – 6i Combine like terms.

Now find the modulus of –4 – 6i. (continued)


Instruction



To find the modulus, first square both the a and b values of the number.

(–4)2 + (–6)2 Square each part of the complex number.

= 16 + 36 Simplify.

= 52

Next, take the square root of the sum.

52 Take the square root of the sum.

2•2•13= Determine the prime factors of the radicand.

2 13= Extract any prime factor that occurs twice.

The distance between the two endpoints is 2 13 .

3. Find the midpoint of the line segment connecting the two endpoints.

To find the midpoint of a segment, or the point on a line segment that divides it into two equal parts, average the values at its endpoints.

Recall that the coordinates of a point on a complex plane are of the form (a, b).

The a and b values for the coordinates of the endpoint 5 + 2i are (5, 2); for the endpoint 1 – 4i, they are (1, –4).

Average the real values, or the x-values, 5 and 1. To average two numbers, add them and divide by 2.

5 1

2

6

23

+= =

The real value, or the x-value, of the midpoint is 3.

Average the imaginary values, or the y-values, 2 and –4.2 ( 4)

2

2

21

+ −=−

=−

The imaginary value, or the y-value, of the midpoint is –1.

The coordinates of the midpoint are (3, –1), which represents the complex number 3 – i.

This answer makes sense because on the complex plane, the point (3, –i) appears to be midway between the two endpoints of the line segment.

UNIT 1 • EXTENDING THE NUMBER SYSTEM Lesson … · • The polar form of a complex number is an...

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