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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, \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.

 

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 \begin{align*}23 \frac{7}{10}\end{align*}, so it is rational.

b) 2.8956

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

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

\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}\end{align*}

 \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*}.

Expressing Decimals as Fractions 

Express the following decimals as fractions.

a.) 0.439

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*}

\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.

Classify the following real numbers.

a) 0

Integer

b) -1

Integer

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

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

Irrational

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

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

 

Example

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

Example 1

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

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*}

Review 

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

Review (Answers)

To view the Review 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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