# Sets and Symbols

## Whole numbers, integers, rational numbers, and irrational numbers

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Subsets of Real Numbers

The number of survey participants who declined to respond can be represented by the decimal 0.14141414... How would you write this decimal as a fraction?

By being able to write a repeating decimal as fraction, we know it is a rational number.

### Real Numbers

There are several types of real numbers. You are probably familiar with fractions, decimals, integers, whole numbers and even square roots. All of these types of numbers are real numbers. There are two main types of numbers: real and complex. We will address real numbers in this concept.

Real Numbers Any number that can be plotted on a number line. Symbol: \begin{align*}\mathbb{R}\end{align*} Examples: \begin{align*}8, 4.67, - \frac{1}{3}, \pi\end{align*}
Rational Numbers Any number that can be written as a fraction, including repeating decimals. Symbol: \begin{align*}\mathbb{Q}\end{align*} Examples: \begin{align*}-\frac{5}{9},\frac{1}{8},1.\overline{3},\frac{16}{4}\end{align*}
Irrational Numbers Real numbers that are not rational. When written as a decimal, these numbers do not end nor repeat. Example: \begin{align*}e, \pi, -\sqrt{2},\sqrt[3]{5}\end{align*}
Integers All positive and negative “counting” numbers and zero. Symbol: \begin{align*}\mathbb{Z}\end{align*} Example: -4, 6, 23, -10
Whole Numbers All positive “counting” numbers and zero. Example: 0, 1, 2, 3, ...
Natural Numbers All positive “counting” numbers. Symbol: \begin{align*}\mathbb{N}\end{align*} Example: 1, 2, 3, ...

A counting number is any number that can be counted on your fingers.

The real numbers can be grouped together as follows:

Now, let's do the following problems using the different subset of real numbers.

1. What is the most specific subset of the real numbers that -7 is a part of?

-7 is an integer.

1. List all the subsets that 1.3 lies in.

1.3 is a terminating decimal. Therefore, it is considered a rational number. It would also be a real number. As a fraction, we would write \begin{align*}1 \frac{3}{10}\end{align*} because the 3 is in the tenths position after the decimal.

1. True or False: \begin{align*} \frac{8}{3}\end{align*} is a rational number.

Yes, by definition, because it is written as a fraction.

### Examples

#### Example 1

Earlier, you were asked to write 0.14141414.... as a fraction.

Let's devise a step-by-step process.

Step 1: Set your repeating decimal equal to x. \begin{align*}x = 0.14141414\end{align*}

Step 2: Find the repeating digit(s).

In this case 14 is repeating.

Step 3: Move the repeating digits to the left of the decimal point and leave the remaining digits to the right.

\begin{align*}14.14141414\end{align*}

Step 4: Multiply x by the same factor you multiplied your original repeating decimal to get your new repeating decimal.

\begin{align*}14.14141414 = 100(0.14141414)\end{align*}

So, \begin{align*}100x = 14.14141414\end{align*}

Step 5: Solve your system of linear equations for x.

\begin{align*}(100x = 14.14141414) - (x = 0.14141414)\end{align*} yields:

\begin{align*}99x = 14\end{align*}so \begin{align*}x = \frac{14}{99}\end{align*}

#### Example 2

Write 0.327272727... as a fraction.

The 0.3 does not repeat. So, rewrite this as \begin{align*}0.727272727...-0.4\end{align*} Therefore, the fraction will be:

\begin{align*}\frac{72}{99}-\frac{4}{10}\\ \frac{8}{11}-\frac{2}{5}\\ \frac{40}{55}-\frac{22}{55}\\ \frac{18}{55}\end{align*}

#### Example 3

What type of real number is \begin{align*}\sqrt{5}\end{align*}?

\begin{align*}\sqrt{5}\end{align*} is an irrational number because, when converted to a decimal, it does not end nor does it repeat.

#### Example 4

List all the subsets that -8 is a part of.

-8 is a negative integer. Therefore, it is also a rational number and a real number.

#### Example 5

True or False: \begin{align*}-\sqrt{9}\end{align*} is an irrational number.

\begin{align*}-\sqrt{9}=-3\end{align*}, which is an integer. The statement is false.

### Review

What is the most specific subset of real numbers that the following numbers belong in?

1. 5.67
2. \begin{align*}-\sqrt{6}\end{align*}
3. \begin{align*}\frac{9}{5}\end{align*}
4. 0
5. -75
6. \begin{align*}\sqrt{16}\end{align*}

List ALL the subsets that the following numbers are a part of.

1. 4
2. \begin{align*}\frac{6}{9}\end{align*}
3. \begin{align*}\pi\end{align*}

Determine if the following statements are true or false.

1. Integers are rational numbers.
2. Every whole number is a real number.
3. Integers are irrational numbers.
4. A natural number is a rational number.
5. An irrational number is a real number.
6. Zero is a natural number.

Rewrite the following repeating decimals as fractions.

1. 0.4646464646...
2. 0.81212121212...
3. 0.35050505050...
4. 2.485485485485485...
5. 1.25141414141414...

To see the Review answers, open this PDF file and look for section 1.1.

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### Vocabulary Language: English

TermDefinition
inequality An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are $<$, $>$, $\le$, $\ge$ and $\ne$.
Real Number A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Repeating Decimal A repeating decimal is a decimal number that ends with a group of digits that repeat indefinitely. 1.666... and 0.9898... are examples of repeating decimals.
Subset A subset is a collection of numbers or objects within a larger set.
Terminating Decimal A terminating decimal is a decimal number that ends. The decimal number 0.25 is an example of a terminating decimal.
Whole Numbers The whole numbers are all positive counting numbers and zero. The whole numbers are 0, 1, 2, 3, ...