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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

What if you wanted to identify a number like \begin{align*}\sqrt{2}\end{align*}? Would you classify it as rational or irrational? After completing this Concept, you'll be able to decide which category numbers like this one fall into.

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CK-12 Foundation: 0210S Irrational Numbers (H264)

Guidance

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, \begin{align*}\sqrt{49}\end{align*} is rational because it equals 7, but \begin{align*}\sqrt{50}\end{align*} can’t be reduced farther than \begin{align*}5 \sqrt{2}\end{align*}. That factor of \begin{align*}\sqrt{2}\end{align*} is irrational, making the whole expression irrational.

Example A

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

a) 23.7

b) 2.8956

c) \begin{align*}\pi\end{align*}

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

Solution

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

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

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

d) \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, \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 \begin{align*}\frac{36}{11}\end{align*}.

Example B

Express the following decimals as fractions.

a.) 0.439

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

Solution:

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

b.) \begin{align*}0.25 \overline{38}\end{align*} can be expressed as \begin{align*}\frac{25}{100} + \frac{38}{9900}\end{align*}, which is equivalent to \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.

Example C

Classify the following real numbers.

a) 0

b) -1

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

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

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

Solution

a) Integer

b) Integer

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

d) Irrational

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

Watch this video for help with the Examples above.

CK-12 Foundation: Irrational Numbers

Guided Practice

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

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

Solution:

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

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

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

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

This means that the ordering is:

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

Explore More

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

  1. \begin{align*}\sqrt{0.25}\end{align*}
  2. \begin{align*}\sqrt{1.35}\end{align*}
  3. \begin{align*}\sqrt{20}\end{align*}
  4. \begin{align*}\sqrt{25}\end{align*}
  5. \begin{align*}\sqrt{100}\end{align*}
  6. \begin{align*}\sqrt{\pi^2}\end{align*}
  7. \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. \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

Answers for Explore More Problems

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

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Vocabulary

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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