SAMPLE Fundamentals of Math
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Fundamentals of Mathematics Study Guide
2 nd Edition 11/26/2022
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Acknowledgements
We would like to thank the author, Matthew Stewart, for his patience, support, and expertise in contributing to this study guide; and our editor for her invaluable efforts in reading and editing the text. We would also like to thank those at Achieve whose hard work and dedication to fulfilling this project did not go unnoticed. Lastly, we would like to thank the Achieve students who have contributed to the growth of these materials over the years.
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Fundamentals of Mathematics
Table of Contents Chapter 1: Operations with Real Numbers
6 6 8
1.1 Operations
1.2 Properties of Real Numbers
1.3 Factors and Multiples
14
1.4 Operations with Integers
17
1.5 Order of Operations
21
Chapter 1 Summary
25 26 28 28 32 38 42 45 46 47 47 50 54 56 60 63 64 65 65 69 76 76 61
Chapter 1 Practice Quiz
Chapter 2: Fractions, Decimals, and Percentages
2.1 Introduction to Fractions
2.2 Operations with Fractions
2.3 Decimals
2.4 Percentages
Chapter 2 Summary
Chapter 2 Practice Quiz
Chapter 3: Beginning Algebra
3.1 Exponents
3.2 Square and Cube Roots
3.3 Expressions and Equations
3.4 Simplifying and Solving Linear Equations
3.5 Ratios and Proportions 3.6 Percent Change Chapter 3 Summary Chapter 3 Practice Quiz
Chapter 4: Intermediate Algebra
4.1 Introduction to Graphs
4.2 Graphing Linear and Quadratic Functions
4.3 Introduction to Functions
4.4 Evaluating Functions
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4.5 Multiplying Binomials
77
Chapter 4 Summary
79
Chapter 4 Practice Quiz
80 82 83 86 88 90 92 94 95 97 97 98 87
Chapter 5: Geometry
5.1 Properties of Triangles 5.2 Pythagorean Theorem
5.3 Perimeter and Area
5.4 Properties of Quadrilaterals
5.5 Similar Figures
5.6 Properties of Circles
Chapter 5 Summary
Chapter 5 Practice Quiz
Chapter 6: Data Analysis
6.1 Characteristics of a Good Graph
6.2 Line Graph
6.3 Pie Charts
101 101 102 103 107 108 112 112 113 114 116
6.4 Bar Graphs 6.5 Histograms 6.6 Scatter Plots
Chapter 6 Summary
Chapter 6 Practice Quiz
Chapter 7: Statistics
7.1 Mean
7.2 Median
7.3 Mode
Chapter 7 Summary
Chapter 7 Practice Quiz
117
Chapter 8: Logic
118 119 121
8.1 Truth Tables
8.2 Conditional Statements
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8.3 Counterexamples
122 123 124
Chapter 8 Summary
Chapter 8 Practice Quiz
Answer Key Chapter 1
126
Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8
127
129 130 132 134 136 136 138
Index
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Fundamentals of Mathematics Chapter 1: Operations with Real Numbers
Merriam Webster defines mathematics as, “the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations”. Pretty fancy talk for trying to say a scientific study of numbers and how numbers behave. Essentially all of mathematics from Algebra to Trigonometry to Calculus all gather their roots from the fundamental concepts defined by the real number system and the mathematical properties that follows. Sorry, academia talk again – basically, all of mathematics boils down to the real number system and its four primary operators: addition, subtraction, multiplication, and division. 1.1 Operations Let’s start by defining a few basic definitions so we are all on the same page. ● Addition – combining two or more numbers (or items) together to make a new total. ● Subtraction – taking one number away from another number. ● Multiplication – a set value combined with itself a set number of times (repeated addition) ● Division – splitting a number into an equal number of groups These four simple operators most of us initially learned in grade school are used in all domains of mathematics. Of course, the days of straightforward questions such as “what is 2+2?” are probably long gone. Today, most of what we encounter come in stories or worded questions like “If all red tags indicate the item is 20% off then how much will this red-tag shirt cost?” So, let’s begin by reviewing a few phrases you are probably already familiar with and remind ourselves what they mean.
Figure 1.1.1: Common Verbal Operator Expressions Addition Subtraction Multiplication Division Combined Difference Multiplied Divided by Increased Decreased Product Into More than Less than Times Per Sum Minus Of Quotient Together Reduced Twice Out of Plus Fewer than Thrice Ratio of Added to Take away Split
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Fundamentals of Mathematics Recognizing these key words can help us translate what we are hearing/asking in our everyday lives to turn our questions into actionable problems. For example, the manager knows that the restaurant has 24 tables in total and there will be four waitresses on staff today. How many tables per waitress will there be? By recognizing the number per number (or object per person) phrasing here helps us identify this is a division problem, since we know _____ per _____ means to divide. Practice 1.1 Directions: Circle the word (or words) in the questions below that indicate what operator is being mentioned. Then state what the operation would be if you were to actually answer the question. 1. Martin used to have four vehicles, but that has been reduced by one ever since he gave his oldest car to his grandson to drive. How many cars does Martin have now? 3. According to a recent survey by the U.S. Census Bureau there were almost 9.5 times more female nurses than male nurses. If there were 330,000 male nurses in the U.S. in 2016 how many nurses were female? 4. Brett and Christina decided to finally tackle their weekend project of getting the rotting boards on their back-deck replaced. They already had 10 pieces of lumbar in their garage and Brett brought home 15 more from the hardware store. How many pieces of lumbar do they have in total? 5. Emilio had gone to the gym 3 times this week, but his wife Alexandra went twice as many times as he did. She was determined to win their diet bet. How many times did Alexandra go to the gym? 2. Phyllis needed to remember to stop by the bank on her way home from work to get her $10 bill split in half so she could give each of her girls their weekly allowance.
Answer Key on Page 126
Visually we also need to recognize that each operator has different symbols to indicate what the problem is asking. Like the above phrases, most of them you probably already know, but take a moment to review figure 1.1.2 to refresh yourself.
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Figure 1.1.2: Operator Symbols Addition Subtraction Multiplication Division + − or or × ( ) * or ÷ /
1.2 Properties of Real Numbers A quick Google search asking for a definition of Real numbers will yield many results; however, they ultimately all draw the same conclusion.
Real numbers have a value of a continuous quantity. Collectively these values can be represented as a distance along a line.
Let's take a moment to unpack that statement, starting with "continuous quantity". To the average human most vales we interact with daily are rather fixed (e.g., $1.99, 20%, ¼ cup, ...), but numbers exists beyond these finite values. It is possible for a value to go on forever - like pi for instance. π≈3. 141592653589793238462643383279502884197169399375105820974… Some continuous values may even form repeatable patterns. 1 3 ≈0. 33333333333333333333333333333333333333333333333333333333… In essence, pretty much any number you can think of is a Real number, but because Real numbers contain so many types of numbers, we can further categorize them into parts to make them more discernible.
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Fundamentals of Mathematics Figure 1.2.1 The Real Number System
Noting that the larger sets of numbers also include their small number counterparts. For example, by definition, we know -7 is an Integer it is also included in the set of Rational, Irrational, and Real Numbers.
Figure 1.2.2 Number Classification Name Consists Of
Examples 5, 27, 134, 2641 0, 1, 2, 3, 4 − 5, − 1, 0, 2, 4 − 3, 0, 1 2 , 7 9 , 7.5 π, 2, ℮, 2 4 3 Any number
Symbol
ℕ
Natural
The counting numbers
---
Whole
Natural numbers and zero
Natural numbers, zero, and opposite natural numbers Natural numbers, zero, opposite natural numbers, and any number that results in a finite decimal or a repeating decimal. Non-repeating and non-terminating decimals
ℤ
Integer
ℚ
Rational
---
Irrational
ℝ
Real
All rational and irrational numbers
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Fundamentals of Mathematics Now let's look at the second sentence in our definition: Collectively, these values can be represented as a distance along a line. We organize Real numbers on a straight line with values placed in equal segments along its length; this line, known as a number line , can be extended infinitely in any direction. Values are placed numerically left to right with smaller quantities on the left and extending to larger quantities on the right.
Figure 1.2.3 A Generic Number Line
There is no set number line because a number line is just a visual tool to help users compare values and carry out operations like addition, subtraction, multiplication, and division.
Figure 1.2.4 Types of Number Line
Practice 1.2.1 Directions: Identify the following numbers as natural, whole, integer, rational, irrational, or real - place a checkmark in each category that could be used to describe the number.
1. − 8 2. π≈3. 14… 3. 5 4. 0 5. 1 2
Natural
Whole Integer Rational Irrational
Real
Answer Key on Page 126
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Fundamentals of Mathematics Chapter 2: Fractions, Decimals, and Percentages
Now that have you have a thorough understanding of integers and order of operations, you may have noticed some difficulty when dividing certain numbers. For instance, 18 divided by 6 or 10 divided by 2 may be relatively simple, but something like 24 divided by 5 causes some trouble: 5 doesn't go into 24 evenly--the quotient is somewhere between 4 and 5! (We would, therefore, say that 24 is not divisible by 5.) This uncertainty with the divisibility is definitely a conundrum, but this type of division is not far-fetched from what people experience in the real world. Such as a recipe calling for one and a half pound of beef or a pizzaiolo needing to cut the pizza into eighths. 2.1 Introduction to Fractions From our previous chapter, we know Integers are all the counting positive and negative counting numbers . However, between any two integers (..., − 3, − 2, − 1, 0, 1, 2, 3, ...) (such as 0 and 1 or and ) is an entirely new set of numbers (rational) known as − 5 − 4 fractions. When we discuss fractions, we will refer to a number such as one-fourth as a 1 4 ( ) fraction rather than the decimal notation 0.25. The term fraction, however, simply tells us how many parts of a whole we have - whether it is written as a decimal, percentage, or with traditional fraction notation. Fractions are written in the form , where and and are both real numbers. The top ≠0 number is referred to as the numerator , and the bottom number is the denominator ( ) ( ) (remember "D" for the number down below). 34 ←← The line seen between the numerator and denominator is treated as the division operator (like ). Let's take a look at this relationship. Imagine a circle divided into six equal parts (the ÷ denominator), and we shaded in one part (the numerator), then we are left with one-sixth of the circle (the shaded portion).
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Equivalent Fractions Some fractions may look different, but are really the same ( equivalent fractions ), for example:
2 4
4 8
“One-Half”
“Two-Quarters”
“Four-Eights”
1 2
You should always simplify your fractions into their smallest forms; however, unless the problem explicitly asks you not to.
Increase Fractions
Simplify Fractions
Divide both the number and denominator by the same values (number must divide into both sets of numbers evenly).
Multiply both the numerator and denominator by the same values.
1 2 = 1 2 × × 3 3 = 3 6
1 2 6 8 = 1 2 6 8 ÷ ÷ 4 4 = 4 7
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Chapter 3: Beginning Algebra Our world is full of numbers from clocks, to temperatures, sports odds, and stock figures. Well, Algebra is all about determining unknown quantities of numbers. You might know you can throw a ball at 40 mph, but do you know how far that ball will travel at that rate? Or if a home improvement project calls for 35 linear yards of lumber and you can only purchase increments in then how many pieces do you need? 2×4 With algebra, you can answer all of these questions, using what you know about numbers already to solve for the unknown. Let's start with a quick review of exponents and square roots (or cube roots) first though before we dive in. 3.1 Exponents Formally speaking, an exponent is defined as the quantity representing the power to which a given number or expression is to be raised, usually expressed as a raised symbol beside the number or expression (e.g., the 3 in or 5^3). The symbol "^" is referred to as the carrotkey 5 3 and often the icon used on a calculator to indicate the following number is an exponent. In layman's terms though an exponent is just shorthand for repeated multiplication. For example, rather than write something like we would write instead. The 2×2×2×2×2×2×2 2 7 larger number of the exponent is known as the base, and the exponent is the smaller number (visually smaller not necessarily size wise) written as a superscript. Where and are integers and is the base and is the exponent.
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Properties of Exponents Negatives and Exponents
Negative Base
Case #2 −5 4 =− 1×5×5×5 =− 625 Case #1 (− 5) 4 =(− 5)×(− 5)×(− 5)×(− 5)= 625
Exponents only apply to the immediate base. If a negative sign is written in front of the base, it is only included in the repeated multiplication if and only if the entire base (number and sign) is included inside a parenthesis.
Negative Exponents
4 −2 = 4 1 2 = 4× 1 4 = 1 1 6
Raising a number to a negative power is the same as taking 1 over the number raised to the positive power.
Powers of 0 or 1
Power of Zero
Power of One
Any base raised to the 0 power is 1, but watch out for negatives. and 24 0 = 1 −24 0 =− 1 Practice 3.1.1 Directions: Simplify the following expressions. 1. (− 7) 3
Any base raised to the 1 power is the base. and 4 1 = 4 −27 1 = −27
2. − 2 4 4. (− 3) 5 6. −42 0 8. (− 5) 0 10. (− 2) −1
3. −5 2 5. 508 0 7. −4 0 9. 3 −1
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