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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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Practice Properties of Rational Numbers versus Irrational Numbers
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Properties of Rational Numbers versus Irrational Numbers

What if you wanted to identify a number like $\sqrt{2}$ ? 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.

### 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, $\sqrt{49}$ is rational because it equals 7, but $\sqrt{50}$ can’t be reduced farther than $5 \sqrt{2}$ . That factor of $\sqrt{2}$ 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) $\pi$

d) $\sqrt{6}$

Solution

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

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

c) $\pi = 3.141592654 \ldots$ We know from the definition of $\pi$ that the decimals do not terminate or repeat, so $\pi$ is irrational.

d) $\sqrt{6} = \sqrt{2} \ \times \sqrt{3}$ . 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, $3. \overline{27} = 3.272727272727 \ldots$ 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 $\frac{36}{11}$ .

#### Example B

Express the following decimals as fractions.

a.) 0.439

b.) $0.25 \overline{38}$

Solution:

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

b.) $0.25 \overline{38}$ can be expressed as $\frac{25}{100} + \frac{38}{9900}$ , which is equivalent to $\frac{2513}{9900}$ .

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) $\frac{\pi}{3}$

d) $\frac{\sqrt{2}}{3}$

e) $\frac{\sqrt{36}}{9}$

Solution

a) Integer

b) Integer

c) Irrational (Although it’s written as a fraction, $\pi$ is irrational, so any fraction with $\pi$ in it is also irrational.)

d) Irrational

e) Rational (It simplifies to $\frac{6}{9}$ , or $\frac{2}{3}$ .)

Watch this video for help with the Examples above.

### 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 $a$ and $b$ , if $a = b^2$ , then $b = \sqrt{a}$ .
• 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:
• $\sqrt{a} \ \times \sqrt{b} = \sqrt{ab}$
• $A \sqrt{a} \ \times B \sqrt{b} = AB \sqrt{ab}$
• $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{ \frac{a}{b}}$
• $\frac{A\sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}$
• 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.

$\frac{100}{99} \qquad \frac{\sqrt{3}}{3} \qquad -\sqrt{.075} \qquad \frac{2\pi}{3}$

Solution:

Since $-\sqrt{.075}$ is the only negative number, it is the smallest.

Since $100>99$ , $\frac{100}{99}>1$ .

Since the $\sqrt{3} , then $\frac{\sqrt{3}}{3}<1$ .

Since $\pi>3$ , then $\frac{\pi}{3}>1 \Rightarrow \frac{2\pi}{3}>2$

This means that the ordering is:

$-\sqrt{.075}, \frac{\sqrt{3}}{3}, \frac{100}{99}, \frac{2\pi}{3}$

### Practice

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

1. $\sqrt{0.25}$
2. $\sqrt{1.35}$
3. $\sqrt{20}$
4. $\sqrt{25}$
5. $\sqrt{100}$
6. $\sqrt{\pi^2}$
7. $\sqrt{2\cdot 18}$
8. Write 0.6278 as a fraction.
9. Place the following numbers in numerical order, from lowest to highest. $\frac{\sqrt{6}}{2} \qquad \frac{61}{50} \qquad \sqrt{1.5} \qquad \frac{16}{13}$
10. Use the marked points on the number line and identify each proper fraction.

### Vocabulary Language: English

approximate solution

approximate solution

An approximate solution to a problem is a solution that has been rounded to a limited number of digits.
Irrational Number

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Perfect Square

Perfect Square

A perfect square is a number whose square root is an integer.
principal square root

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.