Exploring Real Numbers: Rational, Irrational Numbers and Number Sets

Verbal Description	All negative integers greater than $- 5 .$
Roster Notation	${- 1, - 2, - 3, - 4}$
Set-Builder Notation	${x ∣ x is a negative integer greater than - 5}$

Verbal Description

All negative integers greater than

- 5 .

Roster Notation

{- 1, - 2, - 3, - 4}

Set-Builder Notation

{x ∣ x is a negative integer greater than - 5}

Subset With No Elements	${} or \emptyset$
Subsets With One Element	${1}, {2}, {3}$
Subsets With Two Elements	${1, 2}, {2, 3}, {1, 3}$
Subset With Three Elements	${1, 2, 3}$

Subset With No Elements

{} or \emptyset

Subsets With One Element

{1}, {2}, {3}

Subsets With Two Elements

{1, 2}, {2, 3}, {1, 3}

Subset With Three Elements

{1, 2, 3}

Number Set	Roster Notation
Natural Numbers	${1, 2, 3, \dots}$
Whole Numbers	${0, 1, 2, 3, \dots}$
Integer Numbers	${\dots, - 2, - 1, 0, 1, 2, \dots}$
Rational Numbers	${\dots, - \frac{3}{2}, - 2, - 1, 0, 1, 2, \frac{5}{9}, \dots}$
Irrational Numbers	${π, e, 2, \dots}$
Real Numbers	${\dots, - \frac{3}{2}, - 2, - 1, 0, 1, 2, \frac{5}{9}, π, e, 2, \dots}$

Number Set

Roster Notation

Natural Numbers

{1, 2, 3, \dots}

Whole Numbers

{0, 1, 2, 3, \dots}

Integer Numbers

{\dots, - 2, - 1, 0, 1, 2, \dots}

Rational Numbers

{\dots, - \frac{3}{2}, - 2, - 1, 0, 1, 2, \frac{5}{9}, \dots}

Irrational Numbers

{π, e, 2, \dots}

Real Numbers

{\dots, - \frac{3}{2}, - 2, - 1, 0, 1, 2, \frac{5}{9}, π, e, 2, \dots}

Approximation	Square of Approximation	Comparison
$9.1$	$9.1 \times 9.1 = 82.81$	Approximation is low
$9.2$	$9.2 \times 9.2 = 84.64$	Approximation is low
$9.3$	$9.3 \times 9.3 = 86.49$	Approximation is low
$9.4$	$9.4 \times 9.4 = 88.36$	Approximation is low
$9.5$	$9.5 \times 9.5 = 90.25$	Approximation is high

Approximation

Square of Approximation

Comparison

9.1

9.1 \times 9.1 = 82.81

Approximation is low

9.2

9.2 \times 9.2 = 84.64

Approximation is low

9.3

9.3 \times 9.3 = 86.49

Approximation is low

9.4

9.4 \times 9.4 = 88.36

Approximation is low

9.5

9.5 \times 9.5 = 90.25

Approximation is high

Select the options which represent a correct relationship between the number sets.

We will find which of the given subset relations are correct. We will have a look at each option one at a time. Let's remember several elements of the number sets in a table.

Number Set	Notation	Elements in the Roster Form
Natural Numbers	N	{1,2,3, ... }
Whole Numbers	W	{0,1,2,3, ... }
Integers	Z	{...,-2,-1,0,1,2, ... }
Rational Numbers	Q	{...,-3/2,-2,-1,0,1,2,5/9, ... }
Irrational Numbers	R \ Q	{ π,e,sqrt(2),...}
Real Numbers	R	{...,-3/2,-2,-1,0,1,2,5/9,π,e,sqrt(2), ... }

Let's use the information in the table to create a Venn diagram to see the relations between these number sets.

Great! Note that natural numbers N are a subset of the whole numbers W, integers Z, rational numbers Q, and real numbers R. N ⊂ R ✓ See that the set of real numbers is the biggest of the given sets. This means that the set of real numbers is not a subset of any of the other sets. This makes the relation R ⊂ Q false. In other words, the set of rational numbers is a subset of real numbers, not the opposite. R & ⊂ Q * & ⇓ Q & ⊂ R ✓ Also, the set of irrational numbers is a subset of just the real numbers. It is not a subset of rational numbers. Other than that there is no common element between the sets of rational numbers and irrational numbers. (R\ Q) & ⊂ Q * & ⇓ (R\ Q) & ⊂ R ✓ Now consider the set whole numbers W. According to diagram, it is a subset of integers Z, rational numbers Q, and real numbers R. W ⊂ Z ✓ However, the set of natural numbers is a subset of whole numbers because whole numbers include 0 in addition to all elements of natural numbers. W & ⊂ N * & ⇓ N & ⊂ W ✓ Finally, let's see the given relations as correct and incorrect! lc N ⊂ R & ✓ R ⊂ Q & * (R \ Q) ⊂ Q & * W ⊂ Z & ✓ W ⊂ N & *

Ramsha's mother wants a rug that covers an area of $30$ square feet. She likes three rugs in different shapes. The dimensions of the rugs are given. Which rug should she purchase?

We will find the rug which fits in a 30 square feet area. There are three options to decide the most appropriate one. Since we know the dimensions of each rug, we will calculate their areas one at a time. Let's start with the first one.

The first rug is a rectangle. We will multiply its width and length to find its area. Then, we will round the result to the nearest integer.

The area of the first rug is about 31 square feet. Now, we will calculate the second rug's area. Notice that its width and length are equal. This means that it is a square.

We can find its area by taking square of its side length, 5.5 feet. Then, we will round the obtained value to the nearest integer.

The area of the second rug is about 30 square feet. It looks like it fits Ramsha's mom's room. Still, we need to calculate the third rug's area. Notice that it is a circle.

Let's recall the formula for the area of a circle. Area=2 π r In this formula, the numbers 2 and π are constants but r represents the radius of the circle which is the half of the diameter. r=9/2 Now that we know the value of r, we can calculate the area of the third rug. Remember to round the obtained value to the nearest integer.

The area of the third rug is about 28 square feet. Let's see all areas at the same time! c Rugs & Areas First Rug & ≈ 31 Second Rug & ≈ 30 Third Rug & ≈ 28 Among these three rugs, it looks that Ramsha's mom can choose the second one.

Ramsha likes reading newspapers. She noticed some statistics in her school's monthly newspaper.

Three Balls of Volleyball, Basketball, and Tennis

There are $26$ students who play volleyball, $35$ students who play basketball, $13$ students who play volleyball and basketball, $8$ students who play tennis and basketball, $18$ students who play tennis and volleyball, $5$ students who play all three sports, and $6$ students who do not play any sport.

How many students play only tennis and volleyball?

How many students play only tennis and basketball?

How many students play only basketball?

Let's first draw a Venn diagram representing the given situation.

According to school reports, we know the following information about the numbers of students who play these three sports. r|c Volleyball & 26 Basketball& 35 Volleyball and Basketball& 13 Tennis and Basketball& 8 Tennis and Volleyball& 18 All Sports& 5 No Sports& 6 Now we will illustrate these numbers in the Venn Diagram. From the results, we know that 5 students play all three sports. This will be written in the intersection of all three circles. We also know that 6 students like none of the sports. We write this number outside of the circles.

Great! Now we will consider the students who play tennis and volleyball. We know that 18 students play both tennis and volleyball. This number includes the students that play all three sports. This means that the number of students that only play tennis and volleyball must be 18-5=13.

There are 13 students who play only tennis and volleyball.

This time we will find the number of students that play only tennis and basketball. We know that 8 students play both tennis and basketball. This number includes the students that play all three kind of sports. This means that the number of students that only play tennis and basketball must be 8-5=3.

There are 3 students who play only tennis and basketball.

Finally, we will find the number of students that play only basketball. Let's first find the number of students that play only volleyball and basketball. We know that 13 students play both volleyball and basketball. This means that 13-5=8 students play only volleyball and basketball.

Now consider the number of students inside the circle which represents the students who play basketball. There are 5 students who play all three sports. There are 8 students who play only volleyball and basketball and 3 students who play only basketball and tennis. Let's add these numbers up! 5+8+3=16 Recall that there are 35 students who play basketball. So, the rest of the students play only basketball. 35-16=19 The number of students that play only basketball is 19.

Real Numbers

Catch-Up and Review

Playing with Sets

Introduction to Sets

Set

Verbal Description

Listing Method or Roster Notation

Set-Builder Notation

Visualizing Sets in a Diagram

Venn Diagram

Identifying the Elements in a Venn Diagram

Subset

Extra

Identifying the Subsets

Number Sets

Real Numbers

Identifying the Sets of Numbers

Sorting the Trophies

Hint

Solution

Beyond Real Numbers

Real Numbers

Recommended exercises

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