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.

### Watch This

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.

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

### 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*} and \begin{align*}b\end{align*}, if \begin{align*}a = b^2\end{align*}, then \begin{align*}b = \sqrt{a}\end{align*}. - 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*}
- \begin{align*}A \sqrt{a} \ \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
- \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*}

- 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 1-7, 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.