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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.

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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, \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.

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 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}<s , 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.

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