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Fundamentals of Mathematics I

Sophisticated engines could not even happen until Maxwell’s use of dif- ferential equations in order to stop the engines of that time from flying apart, stopping, or oscillating wildly, so the mathematics here starts with advanced calculus. Today’s engines are far more sophisticated. Their designs require the solutions of complex non-linear partial differential equations and very advanced work with linear algebra. Today a major focus is on autonomous machines, machines that can do routine and even non-routine tasks without human control. They will do the most repetitive jobs, for example automating the assembly line and the most dangerous jobs | Fundamentals of Mathematics I Kent State Department of Mathematical Sciences Fall 2008 Available at http www.math.kent.edu ebooks 10031 book.pdf August 4 2008 Contents 1 Arithmetic 2 1.1 Real Numbers. 2 1.1.1 Exercises 1.1 . 7 1.2 Addition . 7 1.2.1 Exercises 1.2 . 12 1.3 Subtraction. 12 1.3.1 Exercises 1.3. 19 1.4 Multiplication . 19 1.4.1 Exercises 1.4. 23 1.5 Division. 23 1.5.1 Exercise 1.5. 28 1.6 Exponents . 28 1.6.1 Exercises 1.6. 31 1.7 Order of Operations . 31 1.7.1 Exercises 1.7 . 33 1.8 Primes Divisibility Least Common Denominator Greatest Common Factor . 34 1.8.1 Exercises 1.8. 40 1.9 Fractions and Percents. 40 1.9.1 Exercises 1.9 . 50 1.10 Introduction to Radicals . 51 1.10.1 Exercises 1.10 . 53 1.11 Properties of Real Numbers . 53 1.11.1 Exercises 1.11 . 57 2 Basic Algebra 58 2.1 Combining Like Terms . 58 2.1.1 Exercises 2.1 . 60 2.2 Introduction to Solving Equations . 60 2.2.1 Exercises 2.2. 65 2.3 Introduction to Problem Solving . 66 2.3.1 Exercises 2.3. 71 2.4 Computation with Formulas . 71 2.4.1 Exercises 2.4 . 76 3 Solutions to Exercises 77 1 Chapter 1 Arithmetic 1.1 Real Numbers As in all subjects it is important in mathematics that when a word is used an exact meaning needs to be properly understood. This is where we will begin. When you were young an important skill was to be able to count your candy to make sure your sibling did not cheat you out of your share. These numbers can be listed 1 2 3 4 . . They are called counting numbers or positive integers. When you ran out of candy you needed another number 0. This set of numbers can be listed 0 1 2 3 . . They are called whole numbers or non-negative integers. Note that we have used set notation for our list. A set is just a collection of things. Each thing in the collection is called an element or member the set. When we describe a set by listing its elements we enclose the list in curly braces . In notation 1 2 3 . the ellipsis . means that the list goes on forever in the same pattern. So for