Expressing Fractions as Decimal Numbers

Fractions can be expressed as decimal numbers, which can help us better understand their values on the number line. Let’s look at an example of this:

The fraction \frac { 1 }{ 2 } can also be written as 0.5. On a number line, 0.5 is halfway between 0 and 1.

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The fraction \frac { 1 }{ 4 } is 0.25 and \frac { 3 }{ 4 } is 0.75.

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Decimal numbers can either terminate after a finite number of decimal places or recur (repeat) indefinitely.

Terminating Decimals

Some fractions, when converted to decimal form, have a finite number of decimal places. For example:

  • \frac{1}{2} = 0.5 (1 decimal place)
  • \frac{3}{4} = 0.75 (2 decimal places)
  • \frac{1}{8} = 0.125 (3 decimal places)

Recurring Decimals

Other fractions, when converted to decimal form, have an infinitely repeating pattern of digits. For example:

  • \frac{1}{3} = 0.\overline{3} (the digit 3 repeats indefinitely)
  • \frac{2}{3} = 0.\overline{6} (the digit 6 repeats indefinitely)

To show a repeating pattern, a dot is placed above the repeating digit or group of digits.

Rational and Irrational Numbers

Every fraction is a rational number, which can be expressed in the form \frac{a}{b}, where a and b are integers and b \neq 0. Rational numbers include integers, as they can be written as fractions with a denominator of 1 (e.g., 2 = \frac{2}{1}, 5 = \frac{5}{1}, -3 = -\frac{3}{1}).

On the other hand, irrational numbers cannot be expressed as fractions. Their decimal representations go on forever without repeating. Some examples of irrational numbers are \pi, \sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt{7}.

Examples of Converting Fractions to Decimals

  • \frac{2}{7} = 0.\overline{285714}
  • \frac{4}{11} = 0.\overline{36}
  • \frac{2}{3} = 0.\overline{6}

In these examples, the overline notation indicates which digits or groups of digits are repeating in the decimal representation of the fraction.

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