1.1: 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.
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James Sousa: Identifying Sets of Real Numbers
Guidance
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 complex (imaginary) numbers in the Quadratic Functions chapter.
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*} |
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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:
Example A
What is the most specific subset of the real numbers that -7 is a part of?
Solution: -7 is an integer.
Example B
List all the subsets that 1.3 lies in.
Solution: 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.
Example C
True or False: \begin{align*} \frac{8}{3}\end{align*} is a rational number.
Solution: Yes, by definition, because it is written as a fraction.
Intro Problem Revisit How do we 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 mulitplied 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*}
What about 0.327272727... ? 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*}
Guided Practice
1. What type of real number is \begin{align*}\sqrt{5}\end{align*}?
2. List all the subsets that -8 is a part of.
3. True or False: \begin{align*}-\sqrt{9}\end{align*} is an irrational number.
Answers
1. \begin{align*}\sqrt{5}\end{align*} is an irrational number because, when converted to a decimal, it does not end nor does it repeat.
2. -8 is a negative integer. Therefore, it is also a rational number and a real number.
3. \begin{align*}-\sqrt{9}=-3\end{align*}, which is an integer. The statement is false.
Vocabulary
- Subset
- A set of numbers that is contained in a larger group of numbers.
- Real Numbers
- Any number that can be plotted on a number line.
- Rational Numbers
- Any number that can be written as a fraction, including repeating decimals.
- Irrational Numbers
- Real numbers that are not rational. When written as a decimal, these numbers do not end nor repeat.
- Integers
- All positive and negative “counting” numbers and zero.
- Whole Numbers
- All positive “counting” numbers and zero.
- Natural Numbers or Counting Numbers
- Numbers than can be counted on your fingers; 1, 2, 3, 4, ...
- Terminating Decimal
- When a decimal number ends.
- Repeating Decimal
- When a decimal number repeats itself in a pattern. 1.666..., 0.98989898... are examples of repeating decimals.
Practice
What is the most specific subset of real numbers that the following numbers belong in?
- 5.67
- \begin{align*}-\sqrt{6}\end{align*}
- \begin{align*}\frac{9}{5}\end{align*}
- 0
- -75
- \begin{align*}\sqrt{16}\end{align*}
List ALL the subsets that the following numbers are a part of.
- 4
- \begin{align*}\frac{6}{9}\end{align*}
- \begin{align*}\pi\end{align*}
Determine if the following statements are true or false.
- Integers are rational numbers.
- Every whole number is a real number.
- Integers are irrational numbers.
- A natural number is a rational number.
- An irrational number is a real number.
- Zero is a natural number.
Rewrite the following repeating decimals as fractions.
- 0.4646464646...
- 0.81212121212...
- 0.35050505050...
- 2.485485485485485...
- 1.25141414141414...
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inequality
An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .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, ...Image Attributions
Here you'll learn how to identify the subsets of real numbers and place a real number into one of these subsets.