Lesson 3: Identifying Irrational Numbers

Property

An irrational number is a number that cannot be written as the ratio of two integers. Its decimal form does not stop and does not repeat.
Famous examples include $\pi$ and the square roots of numbers that are not perfect squares.

Examples

• The number $\pi$ is a famous irrational number, beginning with $3.14159...$ and continuing infinitely without repetition.
• The square root of 3, $\sqrt{3}$, is irrational because 3 is not a perfect square. Its decimal form is $1.7320508...$.
• A decimal that continues without a pattern, such as $67.121231234...$, is an irrational number.

Explanation

Irrational numbers cannot be written as a simple fraction. Their decimal representations are infinite and non-repeating, meaning they go on forever without any predictable pattern. Think of them as the 'wild' numbers on the number line.

Property

A decimal is rational if it terminates (ends) or repeats a specific block of digits (e.g., $0.\overline{12}$). A decimal is irrational if it is non-terminating AND non-repeating, even if it appears to have a pattern (e.g., $0.121121112...$).

Examples

Property

When a positive integer is not a perfect square, its square root is an irrational number. An irrational number cannot be written as the ratio of two integers, and its decimal form does not terminate or repeat.

Examples