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Analytical GeometryExtending the Number System

Extending the Number SystemRemember how you learned numbers? Youprobably started counting objects in yourhouse as a toddler. You learned to count toten on your fingers. You may have evenlearned some fractions such as half a cookieor a quarter of a cake. When you got a littleolder you learned about negative numbers. Ifyou live where there is cold weather, youmight have learned this sooner. Later on youlearned about squaring numbers and squareroots and many other operations on numbers.Have you ever thought there was somethingmore? What happens in between all of thosenumbers? Is that all there really is? Or couldthere be numbers out there that we haven'teven dreamed of yet...

Essential QuestionsHow can I use the properties ofexponents with rational numbers?What are the properties of rational andirrational numbers?How can I perform arithmeticoperations with complex numbers?How can I perform arithmetic operations on polynomials?

Module MinuteRational numbers can be used as exponents to represent radical notation. For example,taking the square root of a number is the same as having that number with an exponent ofone half. This allows us to use different forms of notation and therefore open up our numbersystem to many other types of numbers. Irrational numbers are numbers with no end and norepeating pattern. Operations performed with irrational numbers result in another irrationalnumber. Rational and irrational numbers are part of the real number system. When we try touse some operations on real numbers, we end up with imaginary numbers. Operations on

imaginary numbers can result with more imaginary numbers or lead us back to the familiar territory of realnumbers.

Key WordsComplex Number­ A complex number is the sum of a real number and an imaginary number.Expression­ A mathematical phrase involving at least one variable and sometimes numbers andoperation symbols.Nth Roots­ The number that must be multiplied by itself n times to equal a given value.Polynomial Function­ A function that is a monomial or the sum of monomials.Rational Exponents­ An exponent that can be written as a ratio of two integers.Rational Expression­ A quotient of two polynomials with a non­zero denominator.Rational Number­ A number that can be expressed as a fraction written with integers.Whole Numbers­ The numbers 0, 1, 2, 3, ...

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A handout of these key words and definitions is also available in the sidebar.

What To Expect

Polynomial Patterns AssignmentComplex Number QuizRational Exponents DiscussionExtending the Number System QuizExtending the Number System TestAmusement Park Project

To view the standards from this unit, please download the handout from the sidebar.

PolynomialsPolynomials are expressions in math that can be manipulated with basic mathematical operations. In thesame way that we add, subtract, and multiply real numbers. We can do the same with polynomials.

Before we complete operations on polynomials, let's discuss what a polynomial is. Watch the videobelow to learn more.

MonomialA monomial is a type of polynomial. A monomial is made up of a number, a variable, or the product of anumber and one or more variables with whole number exponents.

Examples of monomials:

914x

4

Notice that there are no terms being added or subtracted.

PolynomialA polynomial is a monomial or a sum of monomials:

8x9 ­ 2x8 + 6x + 3

Some polynomials have special names, such as the monomial.

A polynomial with only two terms is called a binomial.

6x + 9

A polynomial with only three terms is called a trinomial.

10x6 + 3x + 9

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Even though there are these special names, they are all polynomials. Each polynomial can then be brokeninto special parts. The largest exponent of a polynomial is called the degree of the polynomial.

8x9 ­ 2x8 + 6x + 3 This polynomial has a degree of 9.

It is standard form to write polynomials so that the exponents decrease from left to right.

When a polynomial is written so that the exponents of a variable decrease from left to right, the coefficient ofthe first term is called the leading coefficient.

Adding and Subtracting PolynomialsNow that we know about the parts of a polynomial we can think about adding and subtracting them. Recallthat like terms are terms with the same variables and the variables have the same exponents. Terms canonly be added or subtracted when they are like terms.

x6, 9x6, ­8x6 are all like terms5xy, 9 xy, ­2xy are all like terms13x7y5, 16 x7y5, 3 x7y5 are all like terms

Watch the video below to learn how to add and subtract polynomials.

To add or subtract polynomials we will follow two steps:

1. If subtracting, distribute the negative sign to each term in the following polynomial.2. Combine like terms by adding or subtracting the coefficients.

Multiplying PolynomialsWhen multiplying polynomials we will often need to use the distributive property.

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Notice in the above to examples, a is multiplied by both b and c and then it is added or subtracted. Thisfollows the order of operations.

When multiplying binomials we get to use a special form of the distributive property called the FOIL method.

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Example

(6x + 3)(11x + 12)

6x(11x) + 6x(12) + 3(11x) + 3(12)

66x2 + 72x + 33x + 36

66x2 + 94x + 36

Watch the video below to learn more about multiplying and simplifying.

Polynomial Patterns Assignment Download the Polynomial Patterns Assignment from the sidebar and complete your answers on aseparate word document. Please save your work and submit it.

Complex NumbersWriting Complex NumbersUp until this point, all numbers that you have ever used have been part of the Real Number System. Thereare some situations where the Real Number System is not enough. For example, what is the square root of­4? We know that square root means to find a number that can be squared to get the given number. Whatnumber can we square that will give us a negative number? Is it possible? In the Real Number System, whenwe square a negative number it becomes positive. However, square root of ­4 does exist. It exists as anImaginary Number. There is an imaginary number i, such that the square root of a negative number can befound. Imaginary numbers can be added and subtracted with real numbers. If an imaginary number is notadded or subtracted with a real number, it is called a pure imaginary number.

Example 1:

Take the following numbers and decide which ones are Real Numbers, Imaginary Numbers, and PureImaginary Numbers.

Example 2:

Write the following complex number in standard form.

In order for a number to be written in standard form, there must not be a negative under the square rootsymbol. This can be accomplished by using imaginary numbers. The square root of 6 is not a rational numberso we will leave that alone. However, the square root of a negative will be a number in the imaginary numbersystem. Therefore, standard form for this number is:

The i will always be written before the square root when it is in standard form.

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Example 3:

Write the following complex number instandard form.

To write a square root in standard form itmust meet 3 conditions.

1. The radicand (number under thesquare root symbol) must not have anyperfect square factors.

2. There are no fractions in the radicand.3. There are no square roots in the

denominator of a fraction.

In this problem, 48 does have perfect square

factors. We can break into

which can then be simplified into .

Example 4:

Write the following complex number in standard form.

In our previous example, we already found the standard from of the square root of 48. Now since, it has anegative under the radical, we know that this is an imaginary number.

Example 5:

Write the following complex number in standard form.

This can be factored into . Therefore, the standard form is .

Adding and Subtracting Complex NumbersWhen adding and subtracting complex numbers, combine the like terms, that is, combine the real numbertogether and combine the imaginary numbers together.

Watch the video below on adding complex numbers to get a better understanding.

Example 1:

Write the following expression as a complex number in standard form.

(3 + i) + (1 ­ i)

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3 + 1 are the real numbers and i + ­i are the imaginary numbers. In this case, the imaginary numbers willcancel each other out.

So our answer is: 4

Try these on your own:

Multiplying and Dividing Complex NumbersWhen multiplying complex numbers, we follow the same mathematical rules that are used to multiplypolynomials.

Example 1:

Multiply complex numbers.

3i(­10 ­ 2i)

To simplify, we need to distribute 3i to both terms in the binomial:

3i(­10) + 3i(­2i)

­30i ­ 6i2

­30i ­ (­6)

­30i + 6

6 ­ 30i

Example 2:

Multiply complex numbers.

(5 ­ i)(3 + 2i)

For this problem we will have to FOIL, or use the distributive property.

5(3) + 5(2i) + (­i)(3) + (­i)(2i)

15 + 10i ­ 3i ­2i2

15 + 7i + 2

17 + 7i

Example 3:

Multiply complex numbers.

(­6 ­2i)(5 + 2i)

­30 ­ 12i ­ 10i ­ 4i2

­30 ­ 22i + 2

­26 ­ 22i

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Example 4:

Multiply complex numbers.

(1 + 3i)(1 ­ 6i)

1 ­ 6i + 3i ­ 18i2

1 ­ 3i +18

19 ­ 3i

Example 5:

Multiply complex numbers.

­9i(3 + 6i)

­27i ­ 54i2

­27i + 54

54 ­ 27i

To divide complex numbers, you must always multiply the numerator and denominator by the complexconjugate. The complex conjugate is the complex number in the denominator but with the opposite sign onthe imaginary term.

Try some on your own.

Complex Number QuizIt is now time to complete the "Complex Numbers" Quiz. You will have a limited amount of time tocomplete your quiz; please plan accordingly.

Rational ExponentsIn years past, you have learned properties of exponents. You have learned how to add, subtract, multiply anddivide terms that had the same base with differing exponents. Up to this point, exponents have always beenwhole numbers. Now, we will examine exponents that are made of rational numbers. Recall that rationalnumbers are numbers that can be expressed as fractions.

We already know that taking the square root of a squared number will result in the base number. Forexample,

This works the same with cubed roots and cubed numbers:

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There is another way to write this which will allow the math to make a lot more sense. When we take the

square root of a number, we are actually raising the number to the power. Cubed roots are the same

raising the base to power.

You can check this on your calculator. (Be sure to put the in parenthesis.)

Writing our exponents in this manner, as rational numbers, we can apply all of the properties of exponents.This makes the math much easier to do.

Watch the videos below to learn more.

Rational Exponents DiscussionCome up with two expressions that contain roots. Then respond to a classmate's post by rewritingtheir two expressions with rational exponents instead of roots. After you have rewritten yourclassmate's expressions, multiply them together. Finally, respond to the classmate that completed

your problem telling them if they got the correct answer or not.

Extending the Number System QuizIt is now time to complete the "Extending the Number System" Quiz. You will have a limitedamount of time to complete your quiz; please plan accordingly.

Module Wrap UpAssignment Checklist

In this module you were responsible for completing the following assignments.

Polynomial Patterns AssignmentComplex Number QuizRational Exponents DiscussionExtending the Number System QuizExtending the Number System TestAmusement Park Project

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ReviewNow that you have completed the initial assessments for this module, review the lesson material with thepractice activities and extra resources. Then, continue to the next page for your final assessmentinstructions.

Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you maysee questions asked on a standardized test. Please follow the directions closely. Remember that youmay have to use prior knowledge from previous units in order to answer the question correctly. If you

have any questions or concerns, please contact your instructor.

Final AssessmentsExtending the Number System Test

It is now time to complete the "Extending the Number System" Test. Once you have completedall self­assessments, assignments, and the review items and feel confident in your understanding ofthis material, you may begin. You will have a limited amount of time to complete your test and once

you begin, you will not be allowed to restart your test. Please plan accordingly.

Amusement Park ProjectDownload the Amusement Park Project from the sidebar and complete your answers on aseparate word document. Please save your work and submit your completed assignment.