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# Properties of Rational Numbers versus Irrational Numbers

## Differentiate between numbers that can be written as a fraction and numbers that can't be

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Properties of Rational Numbers versus Irrational Numbers

### Properties of Rational versus Irrational Numbers

Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, 49\begin{align*}\sqrt{49}\end{align*} is rational because it equals 7, but 50\begin{align*}\sqrt{50}\end{align*} can’t be reduced farther than 52\begin{align*}5 \sqrt{2}\end{align*}. That factor of 2\begin{align*}\sqrt{2}\end{align*} is irrational, making the whole expression irrational.

#### Identifying Rational and Irrational Numbers

Identify which of the following are rational numbers and which are irrational numbers.

a) 23.7

23.7 can be written as 23710\begin{align*}23 \frac{7}{10}\end{align*}, so it is rational.

b) 2.8956

2.8956 can be written as 2895610000\begin{align*}2 \frac{8956}{10000}\end{align*}, so it is rational.

c) π\begin{align*}\pi\end{align*}

π=3.141592654\begin{align*}\pi = 3.141592654 \ldots\end{align*} We know from the definition of π\begin{align*}\pi\end{align*} that the decimals do not terminate or repeat, so π\begin{align*}\pi\end{align*} is irrational.

d) 6\begin{align*}\sqrt{6}\end{align*}

6=2 ×3\begin{align*}\sqrt{6} = \sqrt{2} \ \times \sqrt{3}\end{align*}. We can’t reduce it to a form without radicals in it, so it is irrational.

#### Repeating Decimals

Any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, 3.27¯¯¯¯¯=3.272727272727\begin{align*}3. \overline{27} = 3.272727272727 \ldots\end{align*} This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as 3611\begin{align*}\frac{36}{11}\end{align*}.

#### Expressing Decimals as Fractions

Express the following decimals as fractions.

a.) 0.439

0.439 can be expressed as 410+3100+91000\begin{align*}\frac{4}{10} + \frac{3}{100} + \frac{9}{1000}\end{align*}, or just 4391000\begin{align*}\frac{439}{1000}\end{align*}. Also, any decimal that repeats is rational, and can be expressed as a fraction.

b.) 0.2538¯¯¯¯¯\begin{align*}0.25 \overline{38}\end{align*}

0.2538¯¯¯¯¯\begin{align*}0.25 \overline{38}\end{align*} can be expressed as 25100+389900\begin{align*}\frac{25}{100} + \frac{38}{9900}\end{align*}, which is equivalent to 25139900\begin{align*}\frac{2513}{9900}\end{align*}.

#### Classify Real Numbers

We can now see how real numbers fall into one of several categories.

License: CC BY-NC 3.0

If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its simplest form is one, then it is an integer. If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)

If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is irrational.

Classify the following real numbers.

a) 0

Integer

b) -1

Integer

c) π3\begin{align*}\frac{\pi}{3}\end{align*}

Irrational (Although it's written as a fraction, π\begin{align*}\pi\end{align*} is irrational, so any fraction with π\begin{align*}\pi\end{align*} in it is also irrational.)

d) 23\begin{align*}\frac{\sqrt{2}}{3}\end{align*}

Irrational

e) 369\begin{align*}\frac{\sqrt{36}}{9}\end{align*}

Rational (It simplifies to 69\begin{align*}\frac{6}{9}\end{align*}, or 23\begin{align*}\frac{2}{3}\end{align*}.)

### Example

Place the following numbers in numerical order, from lowest to highest.

#### Example 1

1009933.0752π3\begin{align*} \frac{100}{99} \qquad \frac{\sqrt{3}}{3} \qquad -\sqrt{.075} \qquad \frac{2\pi}{3}\end{align*}

Since .075\begin{align*}-\sqrt{.075}\end{align*} is the only negative number, it is the smallest.

Since 100>99\begin{align*}100>99\end{align*}, 10099>1\begin{align*}\frac{100}{99}>1\end{align*}.

Since the 3<s\begin{align*}\sqrt{3}, then 33<1\begin{align*}\frac{\sqrt{3}}{3}<1\end{align*}.

Since π>3\begin{align*} \pi>3\end{align*}, then π3>12π3>2\begin{align*}\frac{\pi}{3}>1 \Rightarrow \frac{2\pi}{3}>2\end{align*}

This means that the ordering is:

.075,33,10099,2π3\begin{align*}-\sqrt{.075}, \frac{\sqrt{3}}{3}, \frac{100}{99}, \frac{2\pi}{3}\end{align*}

### Review

For questions 1-7, classify the following numbers as an integer, a rational number or an irrational number.

1. 0.25\begin{align*}\sqrt{0.25}\end{align*}
2. 1.35\begin{align*}\sqrt{1.35}\end{align*}
3. 20\begin{align*}\sqrt{20}\end{align*}
4. 25\begin{align*}\sqrt{25}\end{align*}
5. 100\begin{align*}\sqrt{100}\end{align*}
6. π2\begin{align*}\sqrt{\pi^2}\end{align*}
7. 218\begin{align*}\sqrt{2\cdot 18}\end{align*}
8. Write 0.6278 as a fraction.
9. Place the following numbers in numerical order, from lowest to highest. 6261501.51613\begin{align*}\frac{\sqrt{6}}{2} \qquad \frac{61}{50} \qquad \sqrt{1.5} \qquad \frac{16}{13}\end{align*}
10. Use the marked points on the number line and identify each proper fraction.
License: CC BY-NC 3.0

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.10.

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### Vocabulary Language: English

TermDefinition
approximate solution An approximate solution to a problem is a solution that has been rounded to a limited number of digits.
Irrational Number An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Perfect Square A perfect square is a number whose square root is an integer.
principal square root The principal square root is the positive square root of a number, to distinguish it from the negative value. 3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root.

### Image Attributions

1. [1]^ License: CC BY-NC 3.0
2. [2]^ License: CC BY-NC 3.0

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