What is an Irrational Number?
A number that cannot be written as a common fraction. Every irrational number can be expressed as a non-repeating, non-terminating decimal.
An irrational number is a number that cannot be written as a simple fraction, or ratio, of two integers. This means you can't express it as something like \(\Large\frac{3}{4}\) or \(-\Large\frac{2}{5}\). Instead, irrational numbers are expressed as decimals that never end and never repeat.
For example:
- π (pi) ≈ 3.14159265… (and it keeps going, without repeating)
- √2 (the square root of 2) ≈ 1.41421356…
- e ≈ 2.7182818…
Even though these numbers go on forever, they still represent exact values in math. We often round them when doing calculations, but their true decimal form never ends or follows a pattern.
How are irrational numbers different from rational numbers?
Rational numbers can be expressed as a fraction (like \(\Large\frac{1}{2}\) or \(-\Large\frac{3}{4}\)), and their decimal forms either end or repeat (like 0.75 or 0.333…). Irrational numbers do neither.
When Do Students Learn About Irrational Numbers?
Students are introduced to irrational numbers when they begin working with square roots of non-perfect square numbers, pi, and non-repeating decimals, usually starting in middle school and continuing through high school.
Grades 6–8 – Introduction to Irrational Numbers
Students learn to identify irrational numbers and compare them to rational numbers. They work with square roots and decimals that don’t repeat or terminate.
Grades 9+ – Applying Irrational Numbers in Algebra and Geometry
Students use irrational numbers in calculations involving the Pythagorean Theorem, pi in geometry, and solving quadratic equations.
