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3. Real Numbers
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eCourses /
Pre-Algebra /
Chapter 5
3. 

Real Numbers

This lesson delves into the intricate world of real numbers, breaking them down into rational and irrational types. It explains how rational numbers can be expressed as fractions, while irrational numbers cannot be neatly represented. Number sets are introduced as a way to categorize these numbers, making it easier to understand their properties and relationships. Practical applications of understanding real numbers are vast. For instance, rational numbers are often used in financial calculations, while irrational numbers come into play in scientific computations and engineering. Understanding these concepts is crucial for anyone involved in mathematics, finance, science, or engineering.
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Lesson Settings & Tools
12 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Real Numbers
Slide of 12
In some situations, it is useful to group the objects to see the relationships in a clear way. Every object can be classified according to a characteristic. For instance, toys can be grouped according to colors. People can be grouped according to jobs or ages. In this lesson, the concept sets will be introduced especially to classify numbers.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Playing with Sets

The applet shows five different circles that represent number sets. Those sets are natural numbers whole numbers integers rational numbers and irrational numbers . Consider the numbers in these sets, then enlarge and arrange these circles to show the relationships between the sets.
Five Circles That Represents the Number Sets
  • Is there any set that is completely contained in another set? Think about the numbers that belong more than one number set.
  • Are there any sets that do not share any number with the other sets?
Discussion

Introduction to Sets

The concept of set is used in daily life to refer simply to a group of objects. In mathematics, these objects are defined according to some properties. Now, sets will be defined based on some characteristics and the different ways to describe them.

Concept

Set

A set is a collection of well-defined unique objects called elements. Sets are often illustrated by a Venn diagram, but there are several other ways to visualize them.

Verbal Description All negative integers greater than
Roster Notation
Set-Builder Notation

Each way of describing a set will be discussed and a corresponding example will be shown.

Verbal Description

A set can be described in words as long as there is no ambiguity of whether any particular object belongs to the set. For example, the set of delicious dishes is not a valid description because delicious is not well-defined. However, the set of desserts with chocolate is valid because including chocolate is an objective description.

Listing Method or Roster Notation

In listing method or roster notation, the elements of a set can be shown within curly brackets and separated by commas. The order of the elements is not relevant.
Furthermore, the elements of a set are only considered once and their original order can be disregarded. In the example below, a set is created with the numbers of a given list.
Although the number occurs twice in the list, it is written only once in the set .

Set-Builder Notation

In set-builder notation, sets are described by stating the common characteristics of its elements. To do this, a variable followed by or is written. Then, the common characteristics of the elements are defined. Finally, the whole notation is enclosed in curly brackets.
The symbols and are read as such that. The above example can be read as the set of all numbers such that is an odd integer greater than negative
Discussion

Visualizing Sets in a Diagram

It is also possible to show a set visually by writing all the elements inside a closed shape. This method is called Venn Diagram.

Concept

Venn Diagram

A Venn diagram is a way of illustrating sets. Here, a rectangle representing the universal set, usually denoted as or is drawn. Inside the rectangle circles are typically drawn to represent sets. If the sets have elements in common, circles that represent sets must intersect. In the example, sets and are illustrated.

A venn diagram that shows sets A and B intersecting inside the universal set U.
The universal set contains all of the elements in a given collection. For example, a universal set can be all positive integer numbers that are less than Consider also two example sets and
The above sets will now be illustrated with a Venn diagram. Since the number belongs to both and it is placed in the overlapping region. Also, since the numbers and are included in neither set, they are written outside the circles but inside the rectangle that represents the universal set.
Sets A and B inside the universal set U with their elements
In this example the circles have the same size even though the sets do not have the same number of elements. This means that Venn diagrams are generally not size proportional.
Pop Quiz

Identifying the Elements in a Venn Diagram

Examine the following Venn Diagram. Determine whether the number belongs to the sets shown or list the elements of the set shown in the roster notation.

Venn Diagram Elements
Discussion

Subset

If all the elements of a set are contained in another set then is a subset of This relationship can be written as A Venn diagram visualizes the expression.

A Venn diagram with Set A as a subset of Set B inside a universal set U.

As an example, consider the set All of its subsets are shown in the table.

Subset With No Elements
Subsets With One Element
Subsets With Two Elements
Subset With Three Elements
It is worth noting that all the subsets of except are called proper subsets. A proper subset of a set is a subset with fewer elements than and is denoted by
Other than that the empty set is a subset of all sets. Additionally, every set is always a subset of itself and it is denoted by This fact is illustrated with set

Extra

 Another Example
Actually, sets and subsets do not necessarily include just the numbers all the time. Remember that any well-defined object can be an element of a set. For instance, consider the following sets written in verbal description.
Since all the oceans and lakes are also water bodies of the world, both the sets and are the subsets of set
Pop Quiz

Identifying the Subsets

Determine whether the given set is a subset of set or not.

Given set A and set B
Discussion

Number Sets

A number set is a collection of numbers that allows different types of numbers to be placed in various categories. Listed are some of the most common number sets.

  • Natural Numbers, also called counting numbers, are the numbers used for counting. The set of natural numbers is denoted by
  • Whole Numbers are the natural numbers in addition to zero. The set of whole numbers is denoted by
  • Integer Numbers are the whole numbers in addition to their opposites. The set of integer numbers is denoted by
  • Rational Numbers are the numbers that can be expressed as a ratio between two integers. The set of rational numbers is denoted by
  • Irrational Numbers are the numbers that cannot be expressed as the ratio between two integers. The set of irrational numbers can be written as
  • Real Numbers include rational numbers in addition to irrational numbers. The set of real numbers is denoted by

Notice that the elements of some numbers sets are also the elements of another number set. For instance, all natural numbers are also integers. This means that natural numbers are the subset of integers. With these in mind, consider the applet at the beginning of the lesson once again. The relations between the number sets can be shown as the following way.

Number Sets in Venn Diagram

In the following table, each set will be shown by using the roster notation.

Number Set Roster Notation
Natural Numbers
Whole Numbers
Integer Numbers
Rational Numbers
Irrational Numbers
Real Numbers
Discussion

Real Numbers

The combination of rational and irrational numbers makes the set of real numbers represented by Represented by a graph, the set of real numbers is formed by continuous values marked on a number line.
Real numbers on a number line
Real numbers are the numbers that can be found in the real world. For instance, natural numbers are used for counting objects, rational numbers are used for representing fractions, irrational numbers are used for calculating the square root of a number, integers for measuring temperature, and so on. These different types of numbers make a collection of real numbers.
Pop Quiz

Identifying the Sets of Numbers

Recall that a number can belong to more than one number set. For instance, is a whole number, but it is an integer, a rational number, and a real number at the same time. Consider the given numbers in the following applet and determine all the number sets that include the number.

Random Numbers
Example

Sorting the Trophies

Vincenzo is a professional athlete and captain of the an amputee football team in San Francisco — Bayside Flyers. Since he became the team captain, the Flyers have won five different trophies. Vincenzo wants to display the trophies in order of height from shortest to tallest.

A boy in a wheelchair sits on the bottom left. In front of him, there is a table with five trophies. Above him, there is a clock, and above the trophies is a picture.
The heights of the trophies are    and  inches. Help him to sort these heights from shortest to tallest.

Hint

Round all the numbers to one decimal place and plot them on a number line.

Solution
The heights of the trophies are given as    and  inches. These numbers will be rewritten as decimal numbers and rounded to one decimal place to write them in ascending order. Note that is already in decimal form. Start by expanding
Now round it to the nearest tenth.
The next step is to write as a decimal. Recall that is equal to
Now, approximate by using perfect squares. The two nearest perfect squares are and
CalcRoot

Calculate root

Try to find a better approximation by using decimals. Choose several decimals bigger than and less than Then, calculate the square of each number and compare them with
Approximation Square of Approximation Comparison
Approximation is low
Approximation is low
Approximation is low
Approximation is low
Approximation is high
Considering the squares of approximations, is closer to This means that is closer to Now, calculate the square of
Rewrite

Rewrite as

Multiply

Multiply

RoundDec

Round to decimal place(s)

Finally, mark all the approximations on a number line.
Number Line
The order of the rounded numbers in decimal form can be seen from the number line. Now Vincenzo can properly sort the heights of the trophies from shortest to tallest.
Closure

Beyond Real Numbers

In this lesson, it was mentioned that natural numbers, whole numbers, integers, rational numbers, and irrational numbers are real numbers. Considering that vast range, the set of real numbers must be the biggest number set ever. Right?🤔

Numbers that are not real.

Well, there is what is called an imaginary number Imaginary numbers combined with real numbers form a set of complex numbers

Number Sets in Venn Diagram
These number sets will be explored in later courses.



Level 1
Level 2
Level 3
Real Numbers
Exercises

Consider the following verbal descriptions of some sets. Write each set in roster form.

The roster form lists all elements in a set within curly brackets and separates them with commas. We are asked to list all integers greater than -3 and less than 5. These numbers are to the right side of -3 and to the left side of 5 in a number line.

Now that we identified the set, let's list its elements. -2, -1, 0, 1, 2, 3, 4 We can now write the roster form of set A. A={ -2, -1, 0, 1, 2, 3, 4 } Although we can write the elements of a set in any order, writing them in ascending order is a good practice. We just need to be careful to write each once.


We now need to find the whole numbers that are less than 10. Recall that whole numbers start from 0 and increase by one unit indefinitely. The whole numbers less than 10 are to the right side of 0 included and to the left side of 10 in a number line.

Now that we identified the numbers in the set, let's write them down. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Finally, let's enclose them within curly brackets and add its name. Remember that the order of the elements can be disregarded. B={ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Consider the following sets which are given in roster notation. Choose the best option that represents each set in set-builder notation.

Consider the given set. C={ 2, 3, 5, 7, 11, 13, 17, 19} This set contains odd and even numbers starting from 2 and continuing up to 19. Yet, we need a more specific property to describe these numbers. After going deeper, we can see that these numbers are the first prime numbers up to 19 because the only factors of each are 1 and the numbers themselves.

C contains the first prime numbers less than or equal to 19

We can now use this statement and a variable to represent the elements of set C. In this case, let's use the variable x. We can then say that x is a prime number such that x is less than or equal to 19. Let's write the set-builder notation of C. C={ x | x is a prime number, x ≤ 19 }


Let's look at the given set. D={ ... , -4, -2, 2, 4,  ... } The numbers in this set are integers. These numbers are also even numbers and the set does not contain 0. This means that D contains even integers different than 0.

D contains even integers different than 0.

Let's now use this statement and the variable x to write the set-builder notation. In this case, x is an even integer different than 0. D={ x | x is an even integer , x ≠ 0 }

Consider the following sets.
List the elements that are common in both sets and
List all the elements of sets and together.

We want to identify the common elements between sets A and B. We are given set A in set-builder notation. Let's first write set A into its listing form. A={x | xis an even whole number less than9} ⇕ A={0, 2, 4, 6, 8} A common element is an element that is in both sets. In other words, it is an element that belongs to A and B at the same time. Let's identify these common elements. A&={0, 2, 4, 6, 8} B&={ 4, 5, 6, 7, 8} The common elements are 4, 6, and 8. { 4, 6, 8}


This time we will write all the elements that belong to sets A and B. Recall that each element in a set is unique, so if any element exists in both sets, we will write them once. A&={0, 2, 4, 6, 8} B&={ 4, 5, 6, 7, 8} Great! Let's now write all elements that are in both sets. {0, 2, 4, 5, 6, 7, 8}

Consider the following Venn Diagram.

Venn Diagram
 Write set in roster form.
Write the elements that are common for both sets and in roster form.
What is the number of elements in set

We are asked to list the elements of set A. Let's only consider the elements inside the red circle representing set A.

Let's keep in mind that the elements in the overlapping section of A and B are also part of A. We can now list the elements of A. A={-12,7,-8,-5,15,-23,54} Recall that the order is not important while listing the elements of a set.

This time we will list only the elements that are common in the sets A and B. Let's now examine only the overlapping region.

We can see from the diagram that only two elements belong to both sets. {54, -23 }

Finally, we will find the total number of elements in the universal set U. We will list all the elements that we see inside the bigger rectangle.

Note that we should also consider the elements that do not belong to sets A and B but are still inside set U. U={& -12,7,-8, -5,15,54,-23,-13,10, & 67, 98,9,-21,-43,-2,25,0,6,17} There are a total of nineteen elements in the universal set.

Set includes the letters in the word Consider the subsets of set

Select the options that are subsets of
How many subsets does set have?

At first, we will list the elements of the given set B. Recall that the elements of a set should be written only once which means we will disregard the duplicates of the letters E and T for this case. L-E-T-T-E-R ⇓ B={ L,E,T,R } Set B has four elements. Recall that all elements in a subset are also elements of the original set. Now, we will list all the subsets starting with the ones having zero element, followed by one, two, three, and four elements at most since we have maximum of four elements for this case.

Subsets with Zero-Element

The only subset with zero elements of any set is the empty set. We can write this using the empty set notation. ∅ or { }

Subsets with One-Element

There are four elements in our original set. We can use each of these to form the single element subsets. {L}, {E}, {T}, {R}

Subsets with Two-Elements

Next, we will write the subsets including only two-elements. { L, E }, { L, T }, { L, R }, {E, T }, {E, R }, {T, R}

Subsets with Three-Elements

After that, we will write all the three-element subsets. {L, E, T }, {L, E, R }, {L, T, R }, {E, T, R }

Subsets with Four-Elements

The original set has four elements, so we can only form one subset having four elements: the original set itself. { L, E, T, R }

Summary

Let's see all these subsets in a table.

Subsets Including n Elements
n=0 { }
n=1 {L}, {E}, {T}, {R}
n=2 { L, E }, { L, T }, { L, R }, {E, T }, {E, R }, {T, R}
n=3 {L, E, T }, {L, E, R }, {L, T, R }, {E, T, R }
n=4 { L, E, T, R }

Now, we will consider the given options and see whether they match with the obtained subsets. l { } & ✓ {E,T} & ✓ {T,R } & ✓ {E,B} & * {E,E,T} & * {L,T,R} & ✓ {L,E,T,R} & ✓ {L,E,T,T,E,R} & * Notice that the given set B does not include the letter B. Therefore, it does not include the subset {E,B}. Also, since the elements in the sets {E,E,T} and {L,E,T,T,E,R} are not written only once, they do not even represent a set itself.


Let's consider the table we created in Part A of the subsets of B.

Subsets Number of Subsets
n=0 { } 1
n=1 {L}, {E}, {T}, {R} 4
n=2 { L, E }, { L, T }, { L, R }, {E, T }, {E, R }, {T, R} 6
n=3 {L, E, T }, {L, E, R }, {L, T, R }, {E, T, R } 4
n=4 { L, E, T, R } 1

Let's add all the number of subsets. 1+4+6+4+1=16 Set B has a total of 16 subsets. This applies to any set with four elements. A shortcut for calculating the number of subsets of a set with n elements is with the formula 2^n. In this case, n is 4. 2^4=16

Determine which of the following numbers are irrational.

We will determine which of the given numbers are irrational. Let's recall the definition of irrational numbers.

An irrational number is a number that is not rational. So, an irrational number cannot be written as ab where a and b are integers and b≠0.

According to this definition, - 109 is a rational number because it is a fraction. -10/9= -10/9 → Rational Notice that 0 is also a rational number since it can be written as a fraction by choosing a nonzero integer for the denominator. For instance, 03=0. 0= 0/3 → Rational Next, let's recall the fact that the decimal form of an irrational number neither terminates nor repeats. In other words, if a decimal number terminates or repeats then it is rational. This means that both 5.70 and 0.13 are rational. 5.70 → Rational 0.13 → Rational We will now have a look at π. It is also one of the famous irrational number because its expanded form never terminates. π =3.14159265359... → Irrational Note that the cube root of any integer that is not a perfect cube is irrational. This information helps us examine sqrt(71). Notice that 71 is not a perfect cube which makes sqrt(71) an irrational number. sqrt(71) → Irrational Last, we will examine -sqrt(16). Recall that the square root of any whole number is rational if the radicand is a perfect square. Since 16 is a perfect square, we can calculate this root.

This means that -sqrt(16) is rational. Let's see the types of the given numbers as rational or irrational. - 109 &→ Rational 0 &→ Rational 5.70 &→ Rational 0.13 &→ Rational π &→ Irrational sqrt(71) &→ Irrational -sqrt(16) &→ Rational

Real Numbers

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