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Unit 1 Complex Numbers

Section 12.1 Basic Definitions

Definition 1:

For any real number x (x ≥ 0), √𝑥 = 𝑦 𝑖𝑓 𝑦2 = 𝑥.

Example:

√9 = 3 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 32 = 9.

Note:

If 𝑥 < 0, √𝑥 does not exist as a real number.

Example:

√−9 is undefined (as a real number).

Such numbers, however, do exist and have applications in certain fields (electrical engineering for

example). We can work with such radicals with the following definition.

Definition 2:

The imaginary unit is denoted as j. In particular:

𝒋 = √−𝟏 and 𝒋𝟐 = −𝟏

This definition, used in conjunction with the property below, enables us to evaluate and simplify negative

square roots.

Property of Radicals:

√𝑎 × 𝑏 = √𝑎 × √𝑏 [provided at least one of a or b is positive]

Example:

√−9 = √9 × −1 =

=

Problem 1:

Express each radical in terms of j.

𝑎) √−25 𝑏) √−7 𝑐) 3√−20

𝑑) − √−32 𝑒) √−3

16

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Property of negative radicals:

√−𝑎 = 𝑗 √𝑎

Note:

The property √𝒂 × √𝒃 = √𝒂 × 𝒃 can be used to combine a product of two radicals in many cases.

As noted on the previous page, however, this property may not be applied if both a and b are negative. In

such cases, first simplify each radical individually.

Problem 2:

Find and simplify each product.

𝑎) √4 × √9 𝑏) √−4 × √9

𝑐) √−4 × √−9 𝑑) √−3 × √−4

𝑒) − √−2 × √−10 𝑓) − (√(−2)(−5) ) (√−10 )

𝑔) √−5

3 × √

4

15 ℎ) √

−9

2 × √

−16

25

Do #’s 5,7,9,11,13,17,19,21, p. 360 text

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Simplifying an expression of the form 𝑗𝑛 (where n is a whole number)

Example:

Simplify:

𝑎) 𝑗8 𝑏) 𝑗11

Useful strategy:

If the power n is even, rewrite 𝑗𝑛 as (𝑗2)𝑚. Use the definition of 𝑗2 = −1 to complete.

If the power n is odd, rewrite 𝑗𝑛 as (𝑗𝑛−1) 𝑗. Proceed as before.

Problem 3:

Simplify the following.

𝑎) 𝑗26 𝑏) 𝑗31 𝑐) 𝑗9 − 𝑗7

𝑑) − 𝑗4 𝑒) (−𝑗)4

Do #’s 25,27,29,31, p. 360 text

Complex Numbers

Definition 3:

A number of the form 𝐚 + 𝐛𝐣, where a and b are real numbers, is called a complex number.

Notes:

If a = 0, the resulting number bj is a pure imaginary number.

If b = 0, the resulting number a is a real number.

Conclusion:

Complex numbers include the set of real numbers as well as the set of pure imaginary

numbers.

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For example:

The pure imaginary number 2𝑗 could be rewritten as 0 + 2𝑗.

The real number -5 could be rewritten as −5 + 0𝑗.

Note:

The form 𝐚 + 𝐛𝐣 is known as the rectangular form of a complex number.

In this form:

“a” is the real part.

“b” is the imaginary part.

Problem 4:

Express each of the following complex numbers in rectangular form.

𝑎) − √9 − √−9 𝑏)√54 − √−24

𝑐) 𝑗5 − 4 𝑑) √−25𝑗2 + √−16

Do #’s 33,35,37,39,41,43, p. 360 text

Note:

Each complex number is unique. Therefore, two complex numbers (𝑥 + 𝑦𝑗) 𝑎𝑛𝑑 (𝑎 + 𝑏𝑗) are equal

only if 𝑥 = 𝑎 𝑎𝑛𝑑 𝑦 = 𝑏.

Example:

If 𝑥 + 𝑦𝑗 = 3 − 2𝑗, then 𝑥 = 3 𝑎𝑛𝑑 𝑦 = −2.

Problem 5:

Find the values of x and y which satisfy the following equations.

𝑎) 𝑥 − 𝑦𝑗 = −5 + 𝑗 𝑏) 4𝑥 + 3𝑗𝑦 = 20 − 6𝑗

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𝑐) 4 − 3𝑗 + 𝑥 = 6𝑗 + 𝑗𝑦

𝑑) 𝑥 + 3𝑥𝑗 + 3𝑦 = 5 − 𝑗 − 𝑗𝑦

Do #’s 49,51,53, p. 360 text

One final point:

The conjugate of the complex number (𝐚 + 𝐛𝐣) is (𝐚 − 𝐛𝐣) and vice versa.

Example:

Write the conjugate of the following.

𝑎) 3 − 5𝑗 𝑏) 6j 𝑐) √−49 + 4𝑗2

Do #’s 45,47,55,57, pp. 360-361 text

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Section 12.2 Basic Operations with Complex Numbers (See 12.3 – 12.6, p. 361 text)

Case 1: The given numbers are in the form 𝐚 + 𝐛𝐣.

Problem 1:

Perform the indicated operation and simplify.

𝑎) (3 + 2𝑗) + (−5 − 𝑗) 𝑏) (−1 − 3𝑗) − (5 − 2𝑗)

𝑐) 1.5𝑗 (2𝑗 − 1.4) 𝑑) (2 − 𝑗)(5 + 3𝑗)

𝑒) (−1 − 𝑗)2

Do #’s 5,7,9,13,15,23,25, p. 363 text

Problem 2:

Find the product of 2 + 𝑗 and its conjugate.

Do # 45, p. 363 text

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Division of Complex Numbers

To divide two complex numbers, use the same process as rationalizing the denominator of a rational expression:

multiply the numerator and the denominator by the conjugate of the denominator.

Example:

7−2𝑗

3+4𝑗

=7−2𝑗

3+4𝑗×

3−4𝑗

3−4𝑗

=21−28𝑗−6𝑗+8𝑗2

9−16𝑗2

=21−34𝑗−8

9+16

=13−34𝑗

25

=𝟏𝟑

𝟐𝟓−

𝟑𝟒

𝟐𝟓𝒋

Problem 3:

Simplify the following quotients. Express your final answer in the form 𝐚 + 𝐛𝐣.

a) 2−j

1+2j 𝑏)

4+3𝑗

2𝑗

𝑐) 2𝑗

3−𝑗+

1

2 𝑗 (𝑗 + 4)

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Extra Practice:

1. 12+10𝑗

6−8𝑗

2. 1

𝑗+

2

3+𝑗

3. (6𝑗+5)(2−4𝑗)

(5−𝑗)(4𝑗+1)

Do #’s 27,29,31,35,47, p. 363 text

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Case 2: The given numbers require some simplification before proceeding.

Problem 4:

Simplify.

𝑎) (3√−16 + √9 ) − (7 − 2√−4) 𝑏) (5 − √−36)(−√−1)

𝑐) 𝑗7−𝑗

2𝑗−𝑗2

Do #’s 33,39, p. 363 text

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Miscellaneous Problems

1. Express 𝑗−4 + 2𝑗−1 in rectangular form.

2. Write the reciprocal of 2 − 3𝑗 in rectangular form.

3. Solve for x:

(𝑥 + 5𝑗)2 = 11 − 60𝑗

Solutions:

Do #’s 49,51, p. 363 text

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Section 12.3 Graphical representation of a complex number

Recall: Rectangular form of a complex number: 𝑥 + 𝑦𝑗

The complex plane:

A complex number 𝐱 + 𝐲𝐣 may be represented by the point (𝐱, 𝐲) in the complex plane

where:

𝑥 = the real part 𝑦 = the imaginary part

Example:

Represent each number as a point in the complex plane.

𝑎) 2 + 𝑗 𝑏) − 3 − 2𝑗

Extending the concept: Drawing a line segment from the origin to the point (𝐱, 𝐲)in the plane gives an

alternate way to represent a complex number – as a vector.

Example: Represent each number as a vector in the complex plane.

𝑎) 3 + 𝑗 𝑏) − 1 − 4𝑗

Do #’s 3,5,7, p. 363 text

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Adding complex numbers graphically:

Step 1: Find the point in the plane corresponding to the first number. Draw a vector from the origin to this

point.

Step 2: Repeat this process for the second number.

Step 3: Complete a parallelogram with the lines drawn as adjacent sides. The resulting fourth vertex is the

point representing the sum.

Problem:

Perform the indicated operations graphically.

𝑎) (1 + 3𝑗) + (3 + 2𝑗) 𝑏)(4 − 𝑗) + (−2 + 3𝑗)

𝑐) (3 − 𝑗) − (2 + 2𝑗)

Do #’s 9,11,13,15,17,21,23, p. 363 text

− − −

−

−

−

Imaginary

Real

− − −

−

−

−

Imaginary

Real

− − −

−

−

−

Imaginary

Real

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Section 12.4 Polar form of a complex number

Example:

Represent the number 3 + 4𝑗 as a vector in the complex plane.

Notes:

As a vector possesses both magnitude and direction, so too does a complex number.

The magnitude is simply the length of the vector.

The direction is the angle 𝜃 formed between the positive real axis and the vector itself.

The form:

Consider the complex number 𝒙 + 𝒚𝒋 represented as a vector in the complex plane.

Expressing x and y in terms of r using basic trigonometry we get:

cos 𝜃 = 𝑥

𝑟 →

sin 𝜃 = 𝑦

𝑟 →

Substituting for x and y, the number 𝒙 + 𝒚𝒋 =

− − −

−

−

−

Imaginary

Real

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To convert the number 𝒙 + 𝒚𝒋 to polar form, we need to find the values of “r” and “θ” .

How to solve for “r” and “θ”:

𝑟2 = 𝑥2 + 𝑦2 (r > 0)

tan 𝜃 = 𝑦

𝑥 (0 ≤ 𝜃 < 360°)

Problem 1:

Represent each of the following complex numbers graphically and determine its polar form.

𝑎)3 + 4𝑗 𝑏) − 2 − 𝑗√5

Do #’s 3,5,7,9,11,15,17, p. 368 text

− − −

−

−

−

Imaginary

Real

− − −

−

−

−

Imaginary

Real

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Alternate notation for polar form:

𝑟∠𝜃 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) Examples:

𝑎) 5(𝑐𝑜𝑠 53.1° + 𝑗𝑠𝑖𝑛 53.1°) =

𝑏) 1.82 ∠ 320° =

Problem 2:

Express each complex number in rectangular form.

𝑎) 2.57(𝑐𝑜𝑠 112.3° + 𝑗 𝑠𝑖𝑛 112.3°) 𝑏) 1.82 ∠ 320°

Problem 3:

Represent the number 4(𝑐𝑜𝑠270° + 𝑗𝑠𝑖𝑛270°) graphically and express it in rectangular

form.

Do #’s 19,21,25,27,29,33,35, p. 363 text

Multiplying and Dividing complex numbers in polar form

Basic Results:

𝑖) (𝑟1∠𝜃1)(𝑟2∠𝜃2) = 𝑟1𝑟2 ∠ (𝜃1 + 𝜃2)

𝑖𝑖) (𝑟1∠𝜃1)

(𝑟2∠𝜃2)=

𝑟1

𝑟2 ∠(𝜃1 − 𝜃2)

− − −

−

−

−

Imaginary

Real

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Problem 4:

Perform the indicated operation.

𝑎) (1.5 ∠ 230°)(2.4 ∠ 150°) 𝑏) 3.2∠67.3°

1.3∠154.1°

Do #’s 5,7,9,11,17,19,25,27,29, pp. 375-376 text

Consider the problem below:

Find the sum: 2 ∠ 30° + ∠ 60°

Notes:

• Unlike multiplication and division of numbers in polar form, addition cannot be done.

• The above sum can only be found by first converting each number to rectangular form,

adding the results and then converting this number to polar form.

Solution:

Do #’s 21,23, p. 375 text

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Applications:

a.) Given that the current in a circuit is 3.90 − 6.04𝑗𝑚𝐴 and the impedance is 5.16 + 1.14𝑗𝑘Ω, find the

magnitude of the voltage.

b.) Given that the voltage in a given circuit is 8.3 − 3.1𝑗𝑉 and the impedance is 2.1 − 1.1𝑗Ω, find the

magnitude of the current.

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c.) Two resistors have resistances that can be expressed as 𝑅1 = 5∠30𝑜 and 𝑅2 = 8∠120𝑜. What is

the total resistance if the resistors are in series? Parallel?

Do # 55, p. 363 text

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Section 12.5 Exponential form of a complex number

Exponential form of a complex number is written as:

𝒓𝒆𝒋𝜽

As with polar form:

r is the magnitude of the representative vector

𝜃 is the angle formed by this vector (in radians)

Recall:

An angle is converted from degrees to radians by multiplying by 𝜋

180.

Example:

127.1° =127𝜋

180 = 2.22

Probkems:

1. Express the complex number 5.24 ∠ 118.2° in:

a) exponential form b) rectangular form

2. Express 4.15 𝑒5.60𝑗 in polar form.

Summary of the three forms:

Rectangular: 𝒙 + 𝒚𝒋

Polar: 𝒓(𝐜𝐨𝐬 𝜽 + 𝒋 𝐬𝐢𝐧 𝜽) = 𝒓∠𝜽 (𝜽 𝒊𝒏 𝒅𝒆𝒈𝒓𝒆𝒆𝒔)

Exponential: 𝒓𝒆𝒋𝜽 (𝜽 𝒊𝒏 𝒓𝒂𝒅𝒊𝒂𝒏𝒔)

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3. Express −3 − 2𝑗 in exponential form.

4. Simplify the following product and express the result in rectangular form.

(12.3 𝑒1.54𝑗)(5.9 𝑒3.17𝑗)

Do #’s 3,5,7,9,11,15,17,19,21,23,25,27,29,31,33 pp. 370-371 text

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Section 12.6 Powers and Roots of complex numbers

Topic 1:

Raising a complex number (in polar form) to a power

Demoivre’s Theorem:

(𝒓∠𝜽)𝒏 = 𝒓𝒏 ∠𝒏𝜽

Problems:

1. Simplify the following. Leave your answer in polar form.

𝑎) (0.5 ∠ 100°) 6 𝑏) [ 2(𝑐𝑜𝑠 15° + 𝑗 𝑠𝑖𝑛 15°)]4

2. Change 2 − 𝑗 to polar form. Using this result, simplify (2 − 𝑗)5.

Do #’s 13,15,33,35, pp. 375-376 text

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Topic 2:

Using DeMoivre’s Theorem, find the “n” 𝒏𝒕𝒉 roots of a complex number (in polar form).

Basic Steps:

• Express the given number in polar form.

• Apply the appropriate fractional exponent (as per the root of interest) and determine

the first root…using DeMoivre’s Theorem.

• Add 360° 𝑡𝑜 𝜃 and repeat step 2 to find the second root.

• Continue adding 360° and repeating the process of step 2 until all required roots

have been found.

Example: Find the 3 cube roots of 3 − 4𝑗

Step 1: 3 − 4𝑗 = 5∠306.8699°

Step 2: cube root = exponent of 1

3

Step 3: (5∠306.8699°)1

3 = 51

3∠1

3(306.8699°) = 1.71∠102.3° (this is the first root)

Step 4: (5∠666.8699°)1

3 = 51

3∠1

3(666.8699°) = 1.71∠222.3° (second root)

(5∠1026.8699°)1

3 = 51

3∠1

3(1026.8699°) = 1.71∠342.3° (third root)

Example:

Find the cube roots of −1:

−1 = −1 + 0𝑗 = 1∠180°

First root: (1∠180°)1

3 = 11

3∠1

3(180°) = 1∠60° = 0.5 + 0.87𝑗

Second root: (1∠540°)1

3 = 11

3∠1

3(540°) = 1∠180° = −1

Third root: (1∠900°)1

3 = 11

3∠1

3(900°) = 1∠300° = 0.5 − 0.87𝑗

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Example:

Find the square roots of 2𝑗. Give your answers in rectangular form.

Problem:

Find the cube roots of each of the following:

𝑎) 2 − 𝑗

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b) 27

Do #’s 37,39,41,43,49, p. 376 text