What is an irrational number?
An irrational number is a number that cannot be written as a fraction of two whole numbers. Its decimal expansion goes on forever without repeating, like pi or the square root of 2.
Rational numbers can be written as fractions, such as 1/2, 3/4 or 7. Their decimals either stop, like 0.25, or repeat a pattern, like 0.333... Irrational numbers have decimals that never end and never settle into a repeating pattern.
Famous examples include pi (3.14159...), the square root of 2 (1.41421...), Euler's number e (2.71828...) and the golden ratio (1.61803...). The square roots of numbers that are not perfect squares, such as 3, 5 and 7, are all irrational.
More irrational than rational
Although we use rational numbers more often, there are in a sense far more irrational numbers. Georg Cantor showed in the 1800s that irrational numbers cannot be listed one by one, while rational numbers can.
Did You Know?
Together, rational and irrational numbers make up the real numbers, which include every point on the number line.