Rational and Irrational Numbers
Rational and Irrational Numbers
Definition: Rational number
A rational number (\(\mathbb{Q}\)) is any number which can be written as:
\[\cfrac{a}{b}\]where \(a\) and \(b\) are integers and \(b \ne 0\).
The following numbers are all rational numbers:
\[\cfrac{10}{1} \; ; \; \cfrac{21}{7} \; ; \; \cfrac{-1}{-3} \; ; \; \cfrac{10}{20} \; ; \; \cfrac{-3}{6}\]We see that all numerators and all denominators are integers.
This means that all integers are rational numbers, because they can be written with a denominator of \(\text{1}\).
Definition: Irrational numbers
Irrational numbers (\(\mathbb{Q}'\)) are numbers that cannot be written as a fraction with the numerator and denominator as integers.
Examples of irrational numbers:
\[\sqrt{2} \; ; \; \sqrt{3} \; ; \; \sqrt[3]{4} \; ; \; \pi \; ; \; \cfrac{1 + \sqrt{5}}{2}\]These are not rational numbers, because either the numerator or the denominator is not an integer.
This lesson is part of:
Algebraic Expressions Overview
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