Intermediate Algebra – Student Workbook – Second Edition 2013

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$Lesson 2a – Assessment - Scottsdale Community College - Math Blog$

Lesson 5b - Solving Quadratic Equations Mini-Lesson Complex Numbers – a HIGH LEVEL overview Suppose we are asked to solve the quadratic equation x 2 = -1. Well, right away you should think that this looks a little weird. If I take any real number times itself, the result is always positive. Therefore, there is no REAL number x such that x 2 = -1. [Note: See explanation of Number Systems on the next page] Hmmm…well, let’s approach this using the Quadratic Formula and see what happens. To solve x 2 = -1, I need to write in standard form as x 2 + 1 = 0. Thus, a = 1 and b = 0 and c = 1. Plugging these in to the quadratic formula, I get the following: 0 x 2 0 4(1)(1) 2(1) 4 2 4( 1) 2 4 2 1 2 1 2 1 Well, again, the number 1 does not live in the real number system nor does the number 1 yet these are the two solutions to our equation x 2 + 1 = 0. The way mathematicians have handled this problem is to define a number system that is an extension of the real number system. This system is called the Complex Number System and it has, as its base defining characteristic, that equations such as x 2 + 1 = 0 can be solved in this system. To do so, a special definition is used and that is the definition that: i = 1 With this definition, then, the solutions to x 2 + 1 = 0 are just x = i and x = -i which is a lot simpler than the notation with negative under the radical. When Will We See These Kinds of Solutions? We will see solutions that involve the complex number “i” when we solve quadratic equations that never cross the x-axis. You will see several examples to follow that will help you get a feel for these kinds of problems. Complex Numbers Complex numbers are an extension of the real number system. Standard form for a complex number is a + bi 2 where a & b are real numbers, i 1 and i 1 To work with i on your calculator, press MODE then change REAL to a+bi by using your arrow keys. The i button is on the bottom row. You can use this button for the problems on the next page when simplifying and performing operations with i. Scottsdale Community College Page 222 Intermediate Algebra
Lesson 5b - Solving Quadratic Equations Mini-Lesson COMPLEX NUMBER SYSTEM Complex Numbers – all numbers of the form a + bi where a, b are real numbers and √ Examples: 3 + 4i, 2 + (-3)i, 0 + 2i, 3 + 0i Real Numbers – all the numbers on the REAL NUMBER LINE – include all RATIONAL NUMBERS and IRRATIONAL NUMBERS Rational Numbers – ratios of integers decimals that terminate or repeat Examples: 0.50 = 1 2 , -.75 = 3 , 4 43 33 0.43 = , 0.33 = 100 100 Irrational Numbers Examples: , e, 5 √ , 47 √ Decimal representations for these numbers never terminate and never repeat Integers – Zero, Counting Numbers and their negatives {…-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, …..} Whole Numbers – Counting Numbers and Zero {0, 1, 2, 3, 4, 5, 6, 7, …..} Counting Numbers {1, 2, 3, 4, 5, 6, 7, …..} Scottsdale Community College Page 223 Intermediate Algebra
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Scottsdale Community College Interm
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Table of Contents Lesson 1a - Intro
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Lesson 1a - Introduction to Functio
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Lesson 1b - Linear Equations In the
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Lesson 1b - Linear Equations Mini-L
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Lesson 4b - More on Logarithms This
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Lesson 4b - More on Logarithms Mini
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Appendix A: You Try Answers Lesson
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Appendix A: You Try Answers You Try
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Intermediate Algebra – Student Workbook – Second Edition 2013

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