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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.