Irrational number
Irrational Numbers introduces students to numbers that cannot be expressed as a ratio of two integers — numbers whose decimal forms never terminate and never repeat. Famous examples include π (3.14159…) and √3 (1.7320508…). From OpenStax Prealgebra 2E, the key distinction: square roots of non-perfect squares are always irrational, while square roots of perfect squares (√4 = 2, √9 = 3) are rational. Understanding irrational numbers completes the real number system and explains why so many calculator results appear as endless decimals.
Key Concepts
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.
Common Questions
What is an irrational number?
An irrational number cannot be written as a ratio of two integers. Its decimal form never terminates and never repeats.
Is π irrational?
Yes. π = 3.14159… continues infinitely with no repeating pattern. It cannot be expressed as a simple fraction.
Is √3 irrational?
Yes. Since 3 is not a perfect square, √3 = 1.7320508… is irrational.
How do you tell if a square root is irrational?
If the number under the root sign is not a perfect square (1, 4, 9, 16, 25…), its square root is irrational.
Is 0.333… (repeating) irrational?
No. A repeating decimal is rational — 0.333… = 1/3. Irrational decimals never repeat.
What is the difference between rational and irrational numbers?
Rational numbers can be written as p/q where p and q are integers. Irrational numbers cannot — their decimals go on forever without repeating.