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Beginning Algebra
Tutorial 2: Symbols and Sets of Numbers

Learning Objectives

 After completing this tutorial, you should be able to: Know what a set and an element are.  Write a mathematical statement with an equal sign or an inequality. Identify what numbers belong to the set of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.  Use the Order Property for Real Numbers. Find the absolute value of a number.

Introduction

 Have you ever sat in a math class, and you swear the teacher is speaking some foreign language?  Well, algebra does have it's own lingo.  This tutorial will go over some key definitions and phrases used when specifically working with sets of numbers as well as absolute values.  Even though it may not be the exciting part of math,  it is very important that you understand the language spoken in algebra class.  It will definitely help you do the math that comes later.  Of course, numbers are very important in math. This tutorial helps you to build an understanding of what the different sets of numbers are.   You will also learn what set(s) of numbers specific numbers,  like -3, 0, 100, and even   (pi) belong to.  Some of them belong to more than one set.  I think you are ready to go forward.  Let's make you a  numeric set whiz kid (or adult).

Tutorial

 Sets and Elements

 A set is a collection of objects.  Those objects are generally called members or elements of the set.

 Roster Form

 Roster form just lists out the elements of a set between two set brackets.  For example,   {January, June, July}

 Equal =

 To notate that two expressions are equal to each, use the symbol = between them.

 Inequalities    Not Equal Read left to right a < b : a is less than b a < b : a is less than or equal to b a > b : a is greater than b a > b : a is greater than or equal to b

 Mathematical Statement

 A mathematical statement uses the equality and inequality symbols shown above.  It can be judged either true or false.

 Natural (or Counting) Numbers  N = {1, 2, 3, 4, 5, ...}

 Makes sense, we start counting with the number 1 and continue with 2, 3, 4, 5, and so on.

 Whole Numbers  {0, 1, 2, 3, 4, 5, ...}

 The only difference between this set and the one above is that this set not only contains all the natural numbers, but it also contains 0,  where as 0 is not an element of the set of natural numbers.

 Integers Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

 This set adds on the negative counterparts to the already existing whole numbers (which, remember, includes the number 0). The natural numbers and the whole numbers are both subsets of integers.

 Rational Numbers Q = {| a and b are integers and }

 In other words, a rational number is a number that can be written as one integer over another. Be very careful. Remember that a whole number can be written as one integer over another integer. The integer in the denominator is 1 in that case. For example, 5 can be written as 5/1. The natural numbers, whole numbers, and integers are all subsets of rational numbers.

 Irrational Numbers I = {x | x is a real number that is not rational}

 In other words, an irrational number is a number that can not be written as one integer over another.  It is a non-repeating, non-terminating decimal. One big example of irrational numbers is roots of numbers that are not perfect roots - for example or . 17 is not a perfect square - the answer is a non-terminating, non-repeating decimal, which CANNOT be written as one integer over another.  Similarly, 5 is not a perfect cube. It's answer is also a non-terminating, non-repeating decimal. Another famous irrational number is   (pi).  Even though it is more commonly known as 3.14, that is a rounded value for pi.  Actually it is 3.1415927... It would keep going and going and going without any real repetition or pattern. In other words, it would be a non terminating, non repeating decimal, which again, can not be written as a rational number, 1 integer over another integer.

 Real Numbers R = {x | x corresponds to point on the number line}

 Any number that belongs to either the rational numbers or irrational numbers would be considered a real number. That would include natural numbers, whole numbers and integers.

 Real Number Line

 Above is an illustration of a number line. Zero, on the number line, is called the origin.  It separates the negative numbers (located to the left of 0) from the positive numbers (located to the right of 0).  I feel sorry for 0, it does not belong to either group.  It is neither a positive or a negative number.

 Order Property for  Real Numbers

 Given any two real numbers a and b,  if a is to the left of b on the number line, then a < b. If a is to the right of b on the number line, then a > b.

 Absolute Value

 Most people know that when you take the absolute value of ANY number (other than 0)  the answer is positive.  But, do you know WHY? Well, let me tell you why! The absolute value of x, notated |x|, measures the DISTANCE that x is away from the origin (0) on the real number line.  Aha!  Distance is always going to be positive (unless it is 0) whether the number you are taking the absolute value of is positive or negative.   The following are illustrations of what absolute value means using the numbers 3 and -3:

 Example 1:  Replace ? with  <, >, or = .       3 ?  5

 Since 3 is to the left of 5 on the number line, then 3 < 5.

 Example 2:  Replace ? with  <, >, or = .       7.41 ?  7.41

 Since 7.41 is the same number as 7.41, then 7.41 = 7.41.

 Example 3:  Replace ? with  <, >, or = .       2.5 ?  1.5

 Since 2.5 is to the right of 1.5 on the number line, then 2.5 > 1.5.

 Example 4:  Is the following mathematical statement true or false?          2 > 7

 Since 2 is to the left of 7 on the number line, then 2 < 7.  Therefore, the given statement is false.

 Example 5:  Is the following mathematical statement true or false?          5 > 5

 Since 5 is the same number as 5 and the statement includes where the two numbers are equal to each other, then this statement is true.

 Example 6:  Write the sentence as a mathematical statement.  2 is less than 5.

 Reading it left to right we get: 2 is less than 5 2  <  5

 Example 7:  Write the sentence as a mathematical statement.  10 is less than or equal to 20.

 Reading it left to right we get: 10 is less than or equal to 20 10  <  20

 Example 8:  Write the sentence as a mathematical statement.  -2 is greater than -3.

 Reading it left to right we get: -2 is greater than -3 -2  >  -3

 Example 9:  Write the sentence as a mathematical statement.  0 is greater than or equal to -1.

 Reading it left to right we get: 0 is greater than or equal to -1 0  >  -1

 Example 10:  Write the sentence as a mathematical statement.  5 is not equal to 2.

 Reading it left to right we get: 5 is not equal to 2

 Example 11:   List the elements of the following sets that are also elements of the given set {-4, 0, 2.5,  , ,, 11/2, 7} Natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

 Natural numbers? The numbers in the given set that are also natural numbers are {, 7}. Note that  simplifies to be 5, which is a natural number.

 Whole numbers? The numbers in the given set that are also whole numbers are {0, , 7}.

 Integers? The numbers in the given set that are also integers are {-4, 0,, 7}.

 Rational numbers? The numbers in the given set that are also rational numbers are {-4, 0, 2.5, , 11/2, 7}.

 Irrational numbers? The numbers in the given set that are also irrational numbers are {,  }. These two numbers CANNOT be written as one integer over another.  They are non-repeating, non-terminating decimals.

 Real numbers? The numbers in the given set that are also real numbers are {-4, 0, 2.5, , ,, 11/2, 7}.

 Example 12:  Replace ? with  <, >, or = .        |-2.5|   ?   |2.5|

 Since |-2.5| = 2.5 and |2.5| = 2.5, then the two expressions are equal to each other: |-2.5|   =   |2.5|

 Example 13:  Replace ? with  <, >, or = .           -3   ?   |3|

 First of all, |3| = 3 . Since -3 is to the left of 3 on the number line, then -3  <  |3|.

 Example 14:  Replace ? with  <, >, or = .           4 ? |-1|

 First of all, |-1| = 1 Since 4 is to the right of 1 on the number line, then 4  >  1.

Practice Problems

 These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice. To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problems 1a - 1c: Replace ? with  < , > , or = .

 1a. 5 ?  0 (answer/discussion to 1a)

 1c. -2   ?   2 (answer/discussion to 1c)

Practice Problems 2a - 2b: Is the following mathematical statement true or false?

 2a.   -3  <  -3 (answer/discussion to 2a) 2b.    2 > 4 (answer/discussion to 2b)

Practice Problems 3a - 3c: Write each sentence as a mathematical statement.

 3a.    - 4 is less than 0. (answer/discussion to 3a) 3b.    3 is not equal to -3. (answer/discussion to 3b)

 3c.  5 is greater than or equal to -5. (answer/discussion to 3c)

Practice Problems 4a - 4f: List the elements of the following set that are also elements of the given set: {-1.5, 0, 2, }

 4a.  Natural numbers (answer/discussion to 4a) 4b.  Whole numbers (answer/discussion to 4b)

 4e.  Irrational numbers (answer/discussion to 4e) 4f.  Real numbers (answer/discussion to 4f)

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:

 http://www.purplemath.com/modules/numtypes.htm This webpage goes over the different types of numbers: natural, whole, integers, rational, irrational, and real. http://www.mathleague.com/help/integers/integers.htm#positiveandnegativeintegers This webpage will help you with positive and negative integers. http://www.mathleague.com/help/integers/integers.htm#thenumberline This webpage helps you with the number line. http://www.mathleague.com/help/integers/integers.htm#comparingintegers This webpage helps you with comparing two integers. http://www.purplemath.com/modules/absolute.htm This webpage helps you with absolute value. http://www.mathleague.com/help/integers/integers.htm#absolutevalueofaninteger This webpage helps you with absolute value of an integer. http://www.math.utah.edu/online/1010/line/ This webpage helps you with ordering of integers and absolute value. http://www.eduplace.com/math/mathsteps/7/b/ This webpage helps you with absolute value.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

Last revised on July 22, 2011 by Kim Seward.