Algebra I – Preliminaries and Basic Operations - Quick Guides | Mathematics

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Study GuideAlgebra I–Preliminaries and Basic OperaƟons1.PreliminariesBefore you begin learning algebra, it’s important to feel comfortable with some basic math ideas.These ideas come frompre-algebra, and they help you understand how algebra works.In this section, you will review:•Differenttypes of numbers•Ways to write multiplication•Common math symbolsUnderstanding these basics will make learning algebra much easier.Categories of NumbersIn algebra, we work with several different kinds of numbers. Each category has its own meaning.Let’s look at them one by one.Natural Numbers (Counting Numbers)Natural numbers are the numbers we use when counting objects.Examples:1, 2, 3, 4, 5, ...These are sometimes calledcounting numbersbecause they start from 1 and go upward.Whole Numbers

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Study GuideWhole numbers includeall natural numbers plus zero.Examples:0, 1, 2, 3, 4, ...So the only difference betweenwhole numbers and natural numbersis thatwhole numbersinclude 0.IntegersIntegers include:•positive numbers•negative numbers•zeroExamples:...,−3,−2,−1, 0, 1, 2, 3, ...Integers are important in algebra because they allow us to work withboth positive and negativevalues.Negative IntegersNegative integers are numbersless than zero.Examples:−3,−2,−1These numbers are often used to represent things likeloss, decrease, or movement below zero.Positive IntegersPositive integers are simply thenatural numbers.

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Study GuideExamples:1, 2, 3, 4, ...They representquantities greater than zero.Rational NumbersRational numbers are numbers thatcan be written as a fraction.A fraction has the form:a / bwhere:•ais an integer•bis a natural number (and not zero)Examples:3/27/8Even numbers like5are rational because they can be written as:5 = 5/1Rational numbers also include:•terminating decimals(like 0.5 or 0.25)•repeating decimals(like 0.333...)This is because they can all be written as fractions.Irrational NumbersIrrational numbers are numbers thatcannot be written as fractions.

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Study GuideTheir decimal form:•never ends•never repeatsExamples include:√3πThese numbers continue forever without forming a repeating pattern.Even NumbersEven numbers are integers that aredivisible by 2.Examples:...,−6,−4,−2, 0, 2, 4, 6, ...If a number can be divided by 2 without leaving a remainder, it is even.Prime NumbersAprime numberis a natural number that hasexactly two factors:1 and itself.Example:19 is a prime number because it can only be divided by:1 and 19But 21 isnotprime because it can be divided by:1, 3, 7, and 21Important facts:

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Study Guide•2 is the only even prime number•All other even numbers are divisible by 2 and thereforenot primeThe first 10 prime numbers are:2, 3, 5, 7, 11, 13, 17, 19, 23, 29Odd NumbersOdd numbers are integers thatcannot be divided by 2 evenly.Examples:...,−5,−3,−1, 1, 3, 5, ...When divided by 2, they always leave a remainder of1.Composite NumbersAcomposite numberis a natural number that hasmore than two factors.Examples:4, 6, 8, 9, ...For example:8 can be divided by:1, 2, 4, and 8Since it has more than two factors, it is composite.Square NumbersAsquare numberis the result of multiplying a numberby itself.This is also called raising a number to thesecond power.

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Study GuideExamples:2 × 2 = 43 × 3 = 9The first six square numbers are:1, 4, 9, 16, 25, 36Cube NumbersAcube numberis the result of multiplying a numberby itself twice.This means raising a number to thethird power.Examples:2 × 2 × 2 = 83 × 3 × 3 = 27The first six cube numbers are:1, 8, 27, 64, 125, 216Ways to Show MultiplicationIn mathematics, multiplication can be written in several different ways.For example, multiplying4 and 3can be written as:

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Study GuideAll of these mean thesame thing.Multiplication with Numbers and VariablesWhen one value is a number and the other is a variable, multiplication can be written as:In algebra, we usually write this simply as:4aThis means4 multiplied by a.Multiplication with VariablesWhen multiplying two variables, the multiplication can be written as:But in algebra, we usually write it simply as:This meansa multiplied by b.Common Math SymbolsIn algebra, you will often see certain symbols. Each one has a specific meaning.

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Study GuideHere are some of the most common ones.SymbolMeaning=is equal to≠is not equal to>is greater than<is less than≥is greater than or equal to≤is less than or equal to≯is not greater than≮is not less than≱is not greater than or equal to≰is not less than or equal to≈is approximately equal toThese symbols help uscompare numbers and expressionsin algebra.2. Quiz: Preliminaries1.QuestionAn integer is a whole number.Answer Choices•always true•sometimes true•never trueCorrect Answer:always trueWhy This Is CorrectAn integer is defined as a whole number that can be positive, negative, or zero.

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Study GuideExamples of integers include−3,−1, 0, 2, and 7.Because integers do not include fractions or decimals, the statement that an integer is a wholenumber is always true.2.QuestionThe decimal form of a rational number terminates.Answer Choices•always true•sometimes true•never trueCorrect Answer:sometimes trueWhy This Is CorrectA rational number can be written as a fraction (x/y) where x and y are integers and (y≠0).Some rational numbers have terminating decimals, while others have repeating decimals.Examples:•(½= 0.5) (terminating)•(⅓= 0.333…) (repeating)Because rational numbers can either terminate or repeat, the statement is sometimes true.3.QuestionIf the decimal name of a number never ends but has a repeating pattern, then it is a rational number.

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Study GuideAnswer Choices•always true•sometimes true•never trueCorrect Answer:always trueWhy This Is CorrectA repeating decimal means the digits continue forever but follow a repeating pattern.All repeating decimals can be written as fractions, which makes them rational numbers.Example:(0.333… =⅓)Since every repeating decimal can be expressed as a fraction, the statement is always true.4.QuestionIrrational numbers can be expressed in the form (x/y) where x is an integer and y is a natural number.Answer Choices•always true•sometimes true•never trueCorrect Answer:never trueWhy This Is Correct

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