Extending the Number System

Transcription

1 Analytical Geometry Extending the Number System Extending the Number System Remember how you learned numbers? You probably started counting objects in your house as a toddler. You learned to count to ten on your fingers. You may have even learned some fractions such as half a cookie or a quarter of a cake. When you got a little older you learned about negative numbers. If you live where there is cold weather, you might have learned this sooner. Later on you learned about squaring numbers and square roots and many other operations on numbers. Have you ever thought there was something more? What happens in between all of those numbers? Is that all there really is? Or could there be numbers out there that we haven't even dreamed of yet... Essential Questions How can I use the properties of exponents with rational numbers? What are the properties of rational and irrational numbers? How can I perform arithmetic operations with complex numbers? How can I perform arithmetic operations on polynomials? Module Minute Rational 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 of one half. This allows us to use different forms of notation and therefore open up our number system to many other types of numbers. Irrational numbers are numbers with no end and no repeating pattern. Operations performed with irrational numbers result in another irrational number. Rational and irrational numbers are part of the real number system. When we try to use 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 real numbers. Key Words Complex 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 and operation 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 non zero denominator. Rational Number A number that can be expressed as a fraction written with integers. Whole Numbers The numbers 0, 1, 2, 3, /10

2 A handout of these key words and definitions is also available in the sidebar. What To Expect Polynomial Patterns Assignment Complex Number Quiz Rational Exponents Discussion Extending the Number System Quiz Extending the Number System Test Amusement Park Project To view the standards from this unit, please download the handout from the sidebar. Polynomials Polynomials are expressions in math that can be manipulated with basic mathematical operations. In the same 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 video below to learn more. Monomial A monomial is a type of polynomial. A monomial is made up of a number, a variable, or the product of a number and one or more variables with whole number exponents. Examples of monomials: 9 1 4x 4 Notice that there are no terms being added or subtracted. Polynomial A polynomial is a monomial or a sum of monomials: 8x 9 2x 8 + 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. 10x 6 + 3x /10

3 Even though there are these special names, they are all polynomials. Each polynomial can then be broken into special parts. The largest exponent of a polynomial is called the degree of the polynomial. 8x 9 2x 8 + 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 of the first term is called the leading coefficient. Adding and Subtracting Polynomials Now that we know about the parts of a polynomial we can think about adding and subtracting them. Recall that like terms are terms with the same variables and the variables have the same exponents. Terms can only be added or subtracted when they are like terms. x 6, 9x 6, 8x 6 are all like terms 5xy, 9 xy, 2xy are all like terms 13x 7 y 5, 16 x 7 y 5, 3 x 7 y 5 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 Polynomials When multiplying polynomials we will often need to use the distributive property. 3/10

4 Notice in the above to examples, a is multiplied by both b and c and then it is added or subtracted. This follows the order of operations. When multiplying binomials we get to use a special form of the distributive property called the FOIL method. 4/10

5 Example (6x + 3)(11x + 12) 6x(11x) + 6x(12) + 3(11x) + 3(12) 66x x + 33x x x + 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 a separate word document. Please save your work and submit it. Complex Numbers Writing Complex Numbers Up until this point, all numbers that you have ever used have been part of the Real Number System. There are some situations where the Real Number System is not enough. For example, what is the square root of 4? We know that square root means to find a number that can be squared to get the given number. What number can we square that will give us a negative number? Is it possible? In the Real Number System, when we square a negative number it becomes positive. However, square root of 4 does exist. It exists as an Imaginary Number. There is an imaginary number i, such that the square root of a negative number can be found. Imaginary numbers can be added and subtracted with real numbers. If an imaginary number is not added 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 Pure Imaginary 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 root symbol. This can be accomplished by using imaginary numbers. The square root of 6 is not a rational number so we will leave that alone. However, the square root of a negative will be a number in the imaginary number system. Therefore, standard form for this number is: The i will always be written before the square root when it is in standard form. 5/10

6 Example 3: Write the following complex number in standard form. To write a square root in standard form it must meet 3 conditions. 1. The radicand (number under the square root symbol) must not have any perfect 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 a negative 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 Numbers When adding and subtracting complex numbers, combine the like terms, that is, combine the real number together 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) 6/10

7 3 + 1 are the real numbers and i + i are the imaginary numbers. In this case, the imaginary numbers will cancel each other out. So our answer is: 4 Try these on your own: Multiplying and Dividing Complex Numbers When multiplying complex numbers, we follow the same mathematical rules that are used to multiply polynomials. 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 6i 2 30i ( 6) 30i i 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) i 3i 2i i i Example 3: Multiply complex numbers. ( 6 2i)(5 + 2i) 30 12i 10i 4i i i 7/10

8 Example 4: Multiply complex numbers. (1 + 3i)(1 6i) 1 6i + 3i 18i 2 1 3i i Example 5: Multiply complex numbers. 9i(3 + 6i) 27i 54i 2 27i i To divide complex numbers, you must always multiply the numerator and denominator by the complex conjugate. The complex conjugate is the complex number in the denominator but with the opposite sign on the imaginary term. Try some on your own. Complex Number Quiz It is now time to complete the "Complex Numbers" Quiz. complete your quiz; please plan accordingly. You will have a limited amount of time to Rational Exponents In years past, you have learned properties of exponents. You have learned how to add, subtract, multiply and divide terms that had the same base with differing exponents. Up to this point, exponents have always been whole numbers. Now, we will examine exponents that are made of rational numbers. Recall that rational numbers 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. For example, This works the same with cubed roots and cubed numbers: 8/10

9 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 Discussion Come up with two expressions that contain roots. Then respond to a classmate's post by rewriting their two expressions with rational exponents instead of roots. After you have rewritten your classmate'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 Quiz It is now time to complete the "Extending the Number System" Quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Module Wrap Up Assignment Checklist In this module you were responsible for completing the following assignments. Polynomial Patterns Assignment Complex Number Quiz Rational Exponents Discussion Extending the Number System Quiz Extending the Number System Test Amusement Park Project 9/10

10 Review Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Then, continue to the next page for your final assessment instructions. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may 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 Assessments Extending the Number System Test It is now time to complete the "Extending the Number System" Test. Once you have completed all self assessments, assignments, and the review items and feel confident in your understanding of this 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 Project Download the Amusement Park Project from the sidebar and complete your answers on a separate word document. Please save your work and submit your completed assignment. 10/10