2.10: Properties of Rational Numbers versus Irrational Numbers
What if you wanted to identify a number like \begin{align*}\sqrt{2}\end{align*}
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CK12 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*}
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*}
b) 2.8956 can be written as \begin{align*}2 \frac{8956}{10000}\end{align*}
c) \begin{align*}\pi = 3.141592654 \ldots\end{align*}
d) \begin{align*}\sqrt{6} = \sqrt{2} \ \times \sqrt{3}\end{align*}
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*}
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*}
b.) \begin{align*}0.25 \overline{38}\end{align*}
Classify Real Numbers
We can now see how real numbers fall into one of several categories.
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*}
d) Irrational
e) Rational (It simplifies to \begin{align*}\frac{6}{9}\end{align*}
Watch this video for help with the Examples above.
CK12 Foundation: Irrational Numbers
Vocabulary
 The square root of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers \begin{align*}a\end{align*}
a and \begin{align*}b\end{align*}b , if \begin{align*}a = b^2\end{align*}a=b2 , then \begin{align*}b = \sqrt{a}\end{align*}b=a√ .  A square root can have two possible values: a positive value called the principal square root, and a negative value (the opposite of the positive value).
 A perfect square is a number whose square root is an integer.
 Some mathematical properties of square roots are:

\begin{align*}\sqrt{a} \ \times \sqrt{b} = \sqrt{ab}\end{align*}
a√ ×b√=ab−−√ 
\begin{align*}A \sqrt{a} \ \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
Aa√ ×Bb√=ABab−−√  \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{ \frac{a}{b}}\end{align*}
 \begin{align*}\frac{A\sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}

\begin{align*}\sqrt{a} \ \times \sqrt{b} = \sqrt{ab}\end{align*}
 Square roots of numbers that are not perfect squares (or ratios of perfect squares) are irrational numbers. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.
 Computing a square root on a calculator will produce an approximate solution since the calculator only shows a finite number of digits after the decimal point.
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*}
Practice
For questions 17, classify the following numbers as an integer, a rational number or an irrational number.
 \begin{align*}\sqrt{0.25}\end{align*}
 \begin{align*}\sqrt{1.35}\end{align*}
 \begin{align*}\sqrt{20}\end{align*}
 \begin{align*}\sqrt{25}\end{align*}
 \begin{align*}\sqrt{100}\end{align*}
 \begin{align*}\sqrt{\pi^2}\end{align*}
 \begin{align*}\sqrt{2\cdot 18}\end{align*}
 Write 0.6278 as a fraction.
 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*}
 Use the marked points on the number line and identify each proper fraction.
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
Description
Learning Objectives
Here you'll learn how to differentiate between rational and irrational numbers. You'll also learn how to classify and order real numbers.