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.
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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).
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Tutorial
A set is a collection of objects.
Those objects are generally called members or elements of the set. |
Roster form just lists out the elements of a set between two set
brackets. For example,
{January, June, July}
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To notate that two expressions are equal to each, use the symbol
= between them. |
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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 |
A mathematical statement uses the equality and inequality symbols
shown above. It can be judged either true or false. |
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Natural (or Counting) Numbers
N = {1, 2, 3, 4, 5, ...}
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Makes sense, we start counting with the number 1 and continue with
2, 3, 4, 5, and so on. |
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Whole Numbers
{0, 1, 2, 3, 4, 5, ...}
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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. |
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Integers
Z = {..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4,
5, ...}
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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. |
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Rational Numbers
Q = { |
a and b are integers and  }
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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. |
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Irrational Numbers
I = {x | x is a real number that
is not rational}
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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. |
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Real Numbers
R = {x | x corresponds to point
on the number line}
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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. |
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Real Number Line
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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
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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.
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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:
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 Example
1: Replace ? with <, >, or = .
3 ? 5 |
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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 |
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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 |
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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
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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
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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
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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
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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
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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. |
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{, 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}.
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Integers?
The numbers in the given set that are also integers are
{-4, 0,, 7}.
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{-4, 0, 2.5, , 11/2,
7}.
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{, }.
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}.
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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|
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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 = .
Practice
Problems 2a - 2b: Is the following mathematical statement true or false?
Practice
Problems 3a - 3c: Write each sentence as a mathematical statement.
Practice
Problems 4a - 4f: List the elements of the following set that are also
elements of the given set: {-1.5, 0, 2, ,  }
Need Extra Help on these Topics?
Last revised on July 22, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.
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