Rational and Irrational Numbers

Real numbers are a broad category of numbers that include both rational and irrational numbers. We can write all real numbers in decimal form (e.g. 3.1416).

Rational Numbers

Rational numbers are numbers that can be expressed as \dfrac{m}{n}, where m and n are integers with n\neq 0. The division by zero is not allowed. Rational numbers are commonly referred to as fractions. Some examples of rational numbers are:

\dfrac{3}{4}, -\dfrac{2}{5}, \dfrac{3}{17}, \dfrac{7}{20}, \dfrac{2}{3}

Integers are also considered rational numbers because any integer m can be written as \dfrac{m}{1}, where both m and 1 are integers.

Fractions can be simplified by dividing both the numerator and the denominator by their common factors. For example, the fraction \dfrac{12}{15} can be simplified to \dfrac{4}{5} by dividing both numbers by 3.

Equivalent fractions represent the same value, such as \dfrac{12}{15}, \dfrac{4}{5}, \dfrac{8}{10}, \dfrac{28}{35}, and \dfrac{492}{615}.

Decimal representations of rational numbers

Rational numbers can be expressed as decimals that either terminate or repeat. For example:

1.25, 1.75, 3.35, 1.825, 0.3225 (terminating decimals)

1.3333333\dots, 0.72727272\dots, 15.23232323\dots (repeating decimals)

Terminating decimals can easily be converted into fractions:

  • 1.25=\dfrac{5}{4}
  • 0.5=\dfrac{1}{2}
  • 0.75=\dfrac{3}{4}
  • 0.1275=\dfrac{51}{400}

You can convert repeating decimals to fractions by writing the repeating digits as the numerator over the same number of 9s. For example:

0.\dot{3}=\dfrac{1}{3}

To express a repeating decimal like 0.\dot{3}125\dot{5} as a fraction:

x = 0.312531253125\dots

10000x = 3125.31253125\dots

9999x = 3125

x = \dfrac{3125}{9999}

So, 0.\dot{3}12\dot{5} is equivalent to \dfrac{3125}{9999} as a fraction.

Fractions can also be converted into decimals using long division.

Irrational Numbers

Irrational numbers are numbers that cannot be written as a fraction of two integers. In other words, they cannot be written as \dfrac{m}{n}, where m and n are integers. Some examples of irrational numbers are:

\sqrt{2}, \sqrt{3}, \sqrt{5}, \sqrt[3]{2}, \sqrt[5]{3}, \pi, e, -\sqrt{7}

Decimal expansions for irrational numbers are non-repeating and infinite.