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}

The heights of the trophies are given as

π^{2},

8 \frac{1}{2},

90,

8.6,

and

8 . \overline{6}

inches. These numbers will be rewritten as decimal numbers and rounded to one decimal place to write them in ascending order. Note that

8.6

is already in decimal form. Start by expanding

8 . \overline{6} .

8 . \overline{6} = 8.6666 \dots

Now round it to the nearest tenth.

8 . \overline{6} \approx 8.7

The next step is to write

8 \frac{1}{2}

as a decimal. Recall that

\frac{1}{2}

is equal to

0.5 .

8 \frac{1}{2} = 8.5

Now, approximate

90

by using perfect squares. The two nearest perfect squares are

81

and

100 .

81 < 90 < 100

CalcRoot

Calculate root

9 < 90 < 10

Try to find a better approximation by using decimals. Choose several decimals bigger than

9

and less than

10 .

Then, calculate the square of each number and compare them with

90 .

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

Considering the squares of approximations,

90

is closer to

90.25 .

This means that

90

is closer to

9.5 .

Now, calculate the square of

π .

π^{2} = π \cdot π

Rewrite

Rewrite $π$ as $3.141592 \dots$

π^{2} = (3.141592 \dots) \cdot (3.141592 \dots)

Multiply

π^{2} = 9.869604 \dots

RoundDec

Round to $1$ decimal place(s)

π^{2} \approx 9.9

Finally, mark all the approximations on a 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.

8.5 8 \frac{1}{2} < < 8.6 8.6 < < 8.7 8 . \overline{6} < < 9.5 90 < < 9.9 π^{2}

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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

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