Chapter 1: Tutorial 2: Symbols and Sets of Numbers

  Learning Objectives


 
After completing this tutorial, you should be able to:

  1. Know what a set and an element are. 
  2. Write a mathematical statement with an equal sign or an inequality.
  3. Identify what numbers belong to the set of natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. 
  4. Use the Order Property for Real Numbers.
  5. 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 < ba is less than or equal to b
a > b a is greater than b
a > ba 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: Replace ? with <, >, or = . 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: Replace ? with <, >, or = . 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.