**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**

A **set** is a collection of objects.

Those objects are generally called **members** or **elements** of the set.

**Roster Form**

{January, June, July}

**Equal**

**=**

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**

*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.

**{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.

*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.**

*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.**

*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.

*R* = {*x *| *x* corresponds to point
on the 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.**

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:**

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

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

**Since 2 is to the left of 7 on the number line, then 2 < 7. **

**Therefore, the given statement is false.**

2 is less than 5.

**2 is less than 5**

**2 < 5**

10 is less than or equal to 20.

**10 is less than or equal to 20**

**10 < 20**

-2 is greater than -3.

**-2 is greater than -3**

**-2 > -3**

0 is greater than or equal to -1.

**0 is greater than or equal to -1**

**0 >**

5 is not equal to 2.

**5 is not equal to 2**

{-4, 0, 2.5, , ,, 11/2, 7}

Natural numbers, whole numbers, integers, rational numbers, irrational
numbers, and real 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.

The numbers in the given set that are also whole numbers are

{0, , 7}.

The numbers in the given set that are also integers are

{-4, 0,, 7}.

The numbers in the given set that are also rational numbers are

{-4, 0, 2.5, , 11/2,
7}.

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.

The numbers in the given set that are also real numbers are

{-4, 0, 2.5, , ,,
11/2, 7}.

**|-2.5| = |2.5|**

First of all, |3| = 3 .

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

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 = .

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)

(answer/discussion to 3a)

3c. 5 is greater than or equal to -5.

(answer/discussion to 3c)

(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)

4c. Integers

(answer/discussion to 4c)

(answer/discussion to 4c)

4d. Rational numbers

(answer/discussion to 4d)

(answer/discussion to 4d)

4e. Irrational numbers

(answer/discussion to 4e)

(answer/discussion to 4e)

4f. Real numbers

(answer/discussion to 4f)

(answer/discussion to 4f)

** Need Extra Help on these Topics?**

**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.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.