1 A Review of Basics and Applications

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1 1 A Review of Basics and Applications Chapter 1, Section 1: Concepts: The number system. The real number line. Arithmetic. Applications. 1.1 The Real Number System Definition 1.1 The natural numbers: 1,2,3,4... They are also refereed to as the counting numbers. The integers:... 3, 2, 1,0,1,2,3... The rational numbers: r, where r and s are integers and s 0. Any rational number can s be expressed as a repeating or terminating decimal The irrational numbers: Any real number that does not belong to the any of the sets of numbers above. Example 1.2 (Real numbers) Match the following numbers with one of the number definitions above. Is it possible that some numbers fit more than one definition? (a) 5 (b) 0 (c) 3 4 (d) (e) π (f) where the decimal part never repeats. 1

2 1.2 Arithmetic In expressions without parentheses, are performed first from left to right. are performed last. In expressions with parentheses, do all computations the parentheses before doing any computations the parentheses. When dealing with parentheses within parentheses, begin with and work Theorem 1.3 (Distributive Law) For all real numbers a,b,c, a(b+c) = ab+ac and (b+c)a = ba+ca Example 1.4 (Real numbers) Evaluate each expression. Note that square brackets can be used instead of parenthesis. (a) 3 (5+(3 6)+2 (7 9)) (b) 5[ 3(3 9) 4] Square Roots and Principal Square Roots Definition 1.5 If x 2 = y, then x is a square root of y. If x 2 = y and x is non-negative, then x is the principal square root of y and we write x = y. Example 1.6 (Square Roots) All of the following are true. (a) 3 is a square root of 9. (b) 3 is a square root of 9. (c) 3 is the principal square root of 9. (d) 9 = 3 2

3 Example 1.7 (Do you understand square roots?) What is 4? (a) 2 (b) 2 (c) Both 2 and 2 (d) 16 (e) 16 (f) Both 16 and 16 Property 1.8 ab = a b Example 1.9 (Can you simplify square roots?) Simplify Pictures of Numbers: The Number Line Pictures are one of the best things about Geometry. If we are going to start to understand how to combine Algebra and Geometry, we need to find a way to draw a geometric picture of the numbers and variables that crop up in Algebra. Every real number corresponds to a point on the number line. Every point on the number line corresponds to a real number. 3

4 Number lines can be horizontal, vertical or any other direction. Most of the number lines we will use will be horizontal or vertical. of a larger number on a hori- a larger number Traditionally, a smaller number appears to the zontal number line. Traditionally, a smaller number appears on a vertical number line. Some things to know about the pictures you can draw with number lines: Points that are shaded correspond to numbers that you want to include. 2. Points that are not shaded correspond to numbers that you do not want to include. 3., [, or ] means that you include the number. 4., (, ) means that you do not include the number Your textbook allows you to use parentheses and brackets in the pictures that you draw on number lines. You are welcome to do this, but you also need to know the appropriate use for and on the number line. These are more useful when we move into higher dimensions, and your instructor is likely to use them in class and on exams. 4

5 1.5 Algebraic Notation for Number Line Pictures: Interval Notation Example 1.10 Find the interval that corresponds to the graph. A. (7, 5] B. [ 5,7) C. [7, 5) D. ( 5,7] E. ( 5,7) Example 1.11 Graph the interval ( 2, ) on a number line. Example 1.12 Graph the interval (, 2] on a number line The Union Operator, If you need to include values that are in one interval OR another, we use the union operator. For example, the interval notation for is [ 8, 5) [ 3, ). Example 1.13 (Do you understand?) Write the interval notation that corresponds to the following graph? 5

6 1.6 Distance on the Number Line: Absolute Value Undoubtedly, you have seen absolute values in other math classes. What you may not understand is why you have seen them. Do you know why we talk about absolute values? The absolute value notation is shorthand for the distance between two points on a number line. Definition 1.14 (Absolute Value - Geometric Definition) The absolute value of a number x, denoted x, is the distance between x and 0 on a number line. Example 1.15 Draw a picture that represents the definition of x when 1. x is positive. 2. x is negative. Keeping in mind that algebra and geometry go hand in hand, let s try to convert this geometric idea to algebraic symbols. If x is non-negative, then x =. If x is negative, then x =. 6

7 Definition 1.16 The distance between c and d on the number line is c d = d c. Example 1.17 (Real numbers) Find the distance between the following numbers. (a) 5,3 (b) 0, 2 (c) 3 4,5 Example 1.18 Use the number line to solve the absolute value equation x+2 = Negation If x is positive, then x is. If x is negative, then x is. The negative of 5 x is. The negative of x y equals. Example 1.19 (Do you understand negative numbers?) Which of the following is positive? (a) π 2 (b) 7 3 Example 1.20 (Do you understand negation?) Find the exact value. (a) (π 2) (b) ( 7 3) 7

8 1.8 Scientific Notation Example 1.21 (Do you Understand Scientific Notation) Express in ordinary notation. Example 1.22 (Do you Understand Scientific Notation) Express 1,906,183,200 in scientific notation. Example 1.23 (Do you Understand Scientific Notation) Express in scientific notation. Example 1.24 (Do you Understand Scientific Notation) Evaluate , 100, 000, 000. Express your final answer in scientific notation. 1.9 Applications Example 1.25 As you are tracking a deer in the woods, you first spot it 200 yards from where you are. Indicate this on a number line. As you follow the deer, it moves away from you another 500 yards. Then it turns, walks toward you 200 yards, and stops and stares at you. Mark these positions on the same number line. How far is the deer from you? Show this on the number line. Example 1.26 You have invested $1,000 on stocks. After 2 years their value is $1,075. What is your loss or gain at the end of the 2nd year? On the 4th year they are worth $989. What is your gain or loss at the end of 4th year? Example 1.27 Central Library in Lexington, KY has a Foucault pendulum. It is about 75 feet long. Find the time it takes to make one oscillation by taking the square root of (75/32) and multiply the result by 2π. Your answer will be in seconds. Example 1.28 A charge on a proton is approximately Coulomb. It contains 2 up Quarks and 1 down Quark. Each up Quark contain +2/3 of the total charge of the proton and each down Quark has -1/3 of the charge. What are the charges on a single up Quark and a single down Quark? 8