Prereq 13: Complex Numbers

Summary

In this final lesson of the AP pre-calculus prerequisites review, Judy introduces complex numbers, emphasizing their importance for pre-calculus. She explains the imaginary unit 'i' and its powers, then discusses complex numbers as the sum of a real and an imaginary part. Judy covers operations with complex numbers, their properties, and the concept of complex conjugates. She provides examples of addition, subtraction, and multiplication of complex numbers, as well as simplifying expressions involving complex numbers in denominators. The lesson aims to build procedural and symbolic fluency, preparing students for AP pre-calculus.

Takeaways

• 📚 This is the final lesson in a series of 13, focusing on complex numbers, a topic important for pre-calculus students.
• 🧠 Complex numbers consist of a real part and an imaginary part, written in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part.
• 🔢 The imaginary unit 'i' is defined as the square root of negative one. Powers of 'i' follow a repeating pattern: i, -1, -i, and 1.
• 🔄 To simplify powers of 'i' greater than 4, divide the exponent by 4 and use the remainder to find the corresponding value of 'i'.
• ✏️ Addition and subtraction of complex numbers involve combining the real parts and the imaginary parts separately.
• ✖️ Multiplication of complex numbers is done using the distributive property (FOIL method), and the result is simplified by substituting i² with -1.
• 📈 Complex numbers can be plotted on a coordinate plane with the horizontal axis representing real numbers and the vertical axis representing imaginary numbers.
• 🔄 A complex conjugate is formed by changing the sign of the imaginary part of a complex number. It's used to eliminate imaginary numbers from denominators.
• ➗ Simplifying expressions involving complex numbers in the denominator requires multiplying by the complex conjugate of the denominator.
• 🎯 The lesson emphasizes the importance of understanding complex numbers as a foundation for success in AP pre-calculus.

Q & A

What is the main topic of the 13th lesson in the AP pre-calculus prerequisites review?

The main topic of the 13th lesson is complex numbers, which are a prerequisite for AP pre-calculus.

What is an imaginary number?

An imaginary number is a number that can be written as the square root of a negative number, with 'i' defined as the square root of negative one.

What happens when you square the imaginary unit 'i'?

When you square 'i', the result is negative one, since i^2 = -1.

How can you simplify higher powers of 'i'?

You can simplify higher powers of 'i' by dividing the exponent by 4 and using the remainder as the new exponent, following the pattern of i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1.

What is a complex number and how is it represented?

A complex number is the sum of a real number and an imaginary number, represented in the form a + bi, where 'a' is the real part and 'bi' is the imaginary part.

Do the properties of real numbers also apply to complex numbers?

Yes, all the properties that apply to real numbers, such as commutative, associative, and distributive properties, also apply to complex numbers.

How can complex numbers be plotted?

Complex numbers can be plotted on a real and imaginary coordinate plane, with the horizontal axis for real numbers and the vertical axis for imaginary numbers.

What is a complex conjugate and why is it important?

A complex conjugate is the result of changing the sign of the imaginary part of a complex number, e.g., the conjugate of 3 + 4i is 3 - 4i. It is important for simplifying expressions and eliminating complex numbers from the denominator.

How do you simplify the expression (4 + 3i) + (7 - 5i)?

You simplify by combining like terms: (4 + 7) + (3i - 5i), which results in 11 - 2i.

What is the result of multiplying complex numbers (4 + 3i) * (7 - 5i)?

You distribute the multiplication as with polynomials, resulting in 28 - 20i + 21i - 15, which simplifies to 43 + i after combining like terms and considering i^2 = -1.

How do you simplify the expression (2 + 5i) / (4 - 2i)?

You multiply the numerator and denominator by the complex conjugate of the denominator (4 + 2i), simplify the resulting expressions, and then reduce the fraction to get (-1 + 12i) / 10.

Outlines

00:00

📚 Introduction to Complex Numbers

The first paragraph introduces the topic of complex numbers in the context of AP pre-calculus prerequisites. Judy, the presenter, explains that while students are expected to be familiar with complex numbers, they do not need to be proficient. The lesson includes a review of imaginary numbers, starting with the definition of 'i' as the square root of -1. It proceeds to explain the powers of 'i', demonstrating the cyclical nature of these powers and how to simplify expressions involving 'i' raised to higher powers. The paragraph also covers the definition of a complex number as a sum of a real and an imaginary part, the properties that apply to complex numbers, and the concept of the complex conjugate, which is essential for simplifying expressions with complex numbers in the denominator.

05:01

🔍 Complex Number Operations and Simplification

This paragraph delves into the practical application of complex numbers, starting with addition and subtraction. Judy illustrates how to simplify expressions by combining like terms, using the commutative property. She then moves on to multiplication, showing the process of distributing and simplifying the product of two complex numbers, including the substitution of 'i' squared with -1. The paragraph also addresses division involving complex numbers, explaining the use of the complex conjugate to eliminate complex numbers from the denominator and simplify the expression. Through examples, Judy demonstrates the process of simplifying complex fractions by multiplying the numerator and denominator by the conjugate of the denominator, resulting in a simplified form without complex numbers in the denominator.

10:03

🎓 Conclusion and Encouragement for AP Pre-Calculus

In the final paragraph, Judy wraps up the lesson by emphasizing the importance of understanding imaginary and complex numbers for success in AP pre-calculus. She reminds students of the mathematical practices and skills they've been building throughout the series, which will aid in their pre-calculus journey. Judy expresses excitement for the students' progress and encourages them to feel confident as they begin their AP pre-calculus course, wishing them good luck and reinforcing the idea that they are well-prepared for the challenges ahead.

Keywords

💡Imaginary Number

An imaginary number is a number that, when squared, results in a negative value. It is the fundamental concept for understanding complex numbers. In the script, imaginary numbers are introduced with the symbol 'i', defined as the square root of negative one. The script demonstrates how powers of 'i' cycle through a pattern, which is crucial for simplifying expressions involving imaginary numbers.

💡Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, typically written in the form a + bi, where 'a' is the real number and 'b' is the coefficient of the imaginary part. The video script uses complex numbers to illustrate various operations such as addition, subtraction, and multiplication, emphasizing their importance in pre-calculus.

💡Square Root of Negative One

This concept is integral to the definition of imaginary numbers. In the script, it is explained that the square root of negative one is represented by 'i'. This concept is foundational as it sets the stage for understanding how imaginary numbers behave under different mathematical operations.

💡Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. In the script, the complex conjugate is used to eliminate complex numbers from the denominator of a fraction, which is a key technique in simplifying complex expressions and is crucial for understanding the properties of complex numbers.

💡Commutative Property

The commutative property states that the order in which two numbers are added or multiplied does not change the result. In the context of the video, this property is applied when adding complex numbers, as it allows for the reordering of terms to facilitate the combination of like terms.

💡Associative Property

The associative property allows for the grouping of numbers differently without changing the result, such as (a + b) + c being equal to a + (b + c). The script mentions this property in the context of addition of complex numbers, where it is used to group real and imaginary parts for easier addition.

💡Distributive Property

The distributive property is used to multiply a sum by a number, which results in the multiplication of each addend by the number, and then summing the products. In the script, this property is applied when multiplying complex numbers, breaking down the process into manageable parts.

💡Real and Imaginary Coordinate Plane

This is a graphical representation where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. The script mentions this plane as a way to visualize complex numbers, which is helpful for understanding their geometric interpretation.

💡I Squared

In the context of imaginary numbers, 'i squared' refers to the square of the imaginary unit 'i', which equals -1. The script uses this fact to simplify expressions involving powers of 'i', showing how it cycles through a pattern of values.

💡Simplification

Simplification in mathematics refers to the process of making an expression as simple as possible while maintaining its value. The script provides examples of simplifying complex numbers by combining like terms and using the properties of imaginary numbers, such as 'i squared equals -1'.

💡Pre-Calculus Mathematical Practices

These are the skills and habits of mind that are necessary for success in pre-calculus and beyond. The script reminds viewers of these practices as they relate to the manipulation and understanding of complex numbers, emphasizing the importance of procedural and symbolic fluency in mastering pre-calculus concepts.

Highlights

Introduction to imaginary numbers and their basic properties.

Definition of the imaginary number i as the square root of -1.

Explanation of the powers of i and how they repeat in a cycle of four.

Simplification of higher powers of i by dividing the exponent by 4.

Introduction to complex numbers as the sum of a real number and an imaginary number.

Properties of operations with complex numbers follow the same rules as real numbers.

Plotting complex numbers on a real and imaginary coordinate plane.

Definition and importance of complex conjugates in simplifying expressions.

Example of adding complex numbers: (4 + 3i) + (7 - 5i).

Example of subtracting complex numbers: (4 + 3i) - (7 - 5i).

Example of multiplying complex numbers: (4 + 3i) * (7 - 5i).

Explanation of distributing multiplication in complex numbers similarly to polynomials.

Using complex conjugates to simplify complex fractions.

Example of simplifying a complex fraction: (2 + 5i) / (4 - 2i).

Final review and encouragement for students preparing for AP pre-calculus.

Transcripts

hi everyone I'm Judy from the AP program

welcome back to the AP pre-calculus

prerequisites review and thanks for

tuning in lesson number 13 is the final

lesson and complex numbers is a topic

that you should be familiar with but

you're not expected to be proficient in

this topic so this should cover the

basics of what you need to know as you

enter pre-calculus

as with the previous lessons after this

instructional video you have the option

to complete a practice problem set check

your answers using our answer key and

annotated Solutions and watch a video

demonstrating the solution for the

problem or problems that you found the

most difficult that way you can decide

if you need extra support from a teacher

or a tutor or one of your peers and

you'll be able to be specific about what

you need help with

reminders for imaginary numbers are what

we're going to start out with today

and the imaginary number is a square

root of a negative number

in particular the imaginary number I is

defined as I equals the square root of

negative one

when we look at powers of I if I is the

square root of negative 1 then I squared

is the square root of negative 1 squared

the squaring and square rooting cancel

each other out and so we end up with a

value of negative one

so I is the square root of negative one

I squared is negative 1.

I to the third power is the same as I

squared negative 1 times I so that comes

out to negative I

so I is the square root of negative 1 I

squared is negative 1 I to the third

power is negative I

and then I to the fourth power is the

same as I squared which is negative 1

times I squared which is negative one

and when you multiply those you get one

this whole process and this whole

pattern continues to repeat over and

over and over again so I to the fifth

power would be the square root of

negative one I to the sixth power would

be negative one I to the seventh power

would be negative i i to the eighth

power would be one

I to the ninth power would be the square

root of would be I the square root of

negative 1 and it continues on and on

and on over and over in other words

any time that I is raised to a power

greater than 4 if you want to simplify

it you simply divide the exponent by 4

and the remainder becomes your

simplified expression exponent so for

example I to the 15th power 15 divided

by 4 would be 12 remainder 3 and so that

would be the same thing as I to the

third power which is negative one

and now that we know a little bit more

about imaginary numbers let's look at a

few reminders for complex numbers

first a complex number is the sum of a

real number and an imaginary number

written in the form a plus bi with the

real number is the first term and the

imaginary number is the last term so for

example 5 plus 3i 5 is a real number 3i

is an imaginary number and it's the sum

of the two

in terms of properties for operations

with real number real numbers and

complex numbers all of the properties

that apply to real numbers also apply to

complex numbers so things like the

commutative property the associative

property the distributive property all

of the properties you've been working

with for a long long time with real

numbers apply to complex numbers as well

next complex numbers can be plotted on a

real and imaginary coordinate plane

where the horizontal axis it represents

real numbers and the vertical axis

represents the imaginary numbers

and the last reminder is a complex

conjugate that's probably a term you

haven't heard in a while and maybe

you've only heard once along the way a

complex conjugate is the expression that

results from changing the sign of the

imaginary part of a complex number so

for example 3 plus 4i is the complex

conjugate of three minus 4i

this is important because simplified

Expressions don't have any imaginary

numbers in the denominator so if you see

a complex number in the denominator you

need to use the complex conjugate in

order to eliminate that complex number

in the denominator

and I'll show you an example like that

so that you can see what I'm talking

about

now that we've been reminded what

imaginary numbers are and complex

numbers are we're going to go ahead and

work through a few examples in example

one the first expression that we're

going to simplify is the quantity 4 plus

3i plus the quantity 7 minus 5i and if

you recall whenever you've got addition

between sets of parentheses you can

simply drop the parentheses and when we

do that because of the commutative

property we can rearrange the order of

things as well so we're going to Cluster

together the real numbers and we're

going to Cluster together the com the

imaginary numbers so that we can go

ahead and add like terms so 4 plus 7 is

11 and 3i minus 5i is minus 2i

and there's our simplified answer

for example B 4 plus 3i minus the

quantity 7 minus 5i let's not forget

that we need to take that minus sign and

distribute it to everything in the

second set of parentheses so it's going

to be a minus 7 and a plus 5i that we

use to go ahead and do our

simplification so 4 minus 7 is negative

3 and 3i plus 5i is positive 8i so our

simplified answer is negative 3 plus 8i

and then when we're multiplying complex

numbers for part C to this example 4

plus 3i times 7 minus 5i we're going to

go ahead and distribute multiplication

and in the same way that we did when we

were multiplying polynomials so we're

going to start by multiplying the first

terms of each set of parentheses 4 times

7 then the outside terms so 4 times

negative 5i then the inside terms so 3

times 7 and then the last terms in each

set of parentheses so 3i times negative

5i

and when we do that we get 28

minus 20 I

plus 21

minus 15 I squared

we're going to further simplify that we

know that I squared is the same thing as

negative 1.

so that's negative 15 times negative 1.

and we can also combine the negative 20i

and the 21 I to get 1 I

so 28 plus I plus 15

gives us a final simplified answer of 43

plus I

for example two we're going to simplify

the expression 2 plus 5i over 4 minus 2i

remember we can't have any complex

numbers in the denominator in a

simplified response right so we're going

to start by multiplying the top and the

bottom we're going to multiply the

numerator and denominator by the same

value which is the complex conjugate of

the denominator so we're going to

multiply top and bottom by 4 plus 2i

we're going to foil together the

Expressions on the top of the fractions

and we're going to foil together the

expressions in the denominator of the

fractions

so for the top when we foil those

together we're going to get 8 plus 4i

plus 20i plus 10i squared

and on the bottom when we foil those

expressions together we're going to end

up with 16 plus 8i minus 8i minus 4i

squared

let's not forget that I squared is

negative 1 so when we combine our like

terms on the top and put together the 4i

and the 24 and the 20i to get 24i we can

also change the I squared to a negative

1.

and then we'll do something similar on

the bottom when we combine the 8i and

the negative 8i they'll cancel each

other out and we'll go ahead and replace

that I squared with negative 1.

so the numerator becomes 8 plus 24i

minus 10 and the denominator becomes 16

plus 4. so when we combine our like

terms in the numerator we end up with

negative 2 plus 24i and when we do the

same in the denominator we end up with

20.

we're almost done

the reason that we're not done is

because all of the numbers in this can

be reduced by a value of 2. so we're

going to go ahead and reduce this

fraction by 2.

and when we do we're going to get

negative 1 plus 12i over 10.

there's our final simplified answer

so those are all of the new things that

you needed to hear about

imaginary numbers and complex numbers

and I just wanted to remind you once

again of the pre-calculus mathematical

practices and skills

so that you can understand and make ties

to what you're learning and how it's

going to play into AP pre-calculus and

once again what we've done today has

been largely in the area of procedural

and symbolic fluency so you're building

this skill deeply and that's going to

help you be more successful when you

arrive in your AP pre-calculus soon

thank you so much for joining me I'm so

excited that you've stayed with me all

through 13 lessons and I know that you

must be feeling really good about

starting AP pre-calculus knowing that

these are the topics that you should

know before you go in and good luck with

your AP pre-calculus you've got this