What Are Rational and Irrational Numbers?

Rational numbers are numbers that can be expressed as the ratio of two integers, whereas irrational numbers cannot be expressed as a ratio of integers. Irrational numbers expressed as decimals generate an unending series of numbers after the decimal point that never falls into a repeating pattern.

Because a ratio can written as a fraction, any rational number can be expressed as a fraction. For example, the number 7 can be written as 7/1. The number 1.25 can be written as 5/4. The repeating decimal .333... can be written as 1/3. These are all rational numbers. Complex fractions, such as 97/98, often result in a long series of numbers after the decimal point, but the series must ultimately terminate or fall into a repeating pattern. The fact that the number can be expressed as the ratio of two whole numbers means it is, by definition, rational.

Irrational numbers, such a pi, cannot be written as the ratio of two whole numbers. While the fraction 7/22 is often used as a substitute for pi, it is only an approximation. Pi is the ratio of the circumference of any circle to its diameter. Though it has been calculated out to billions of decimal places, no consistently repeating pattern has ever been found. Though irrational numbers may seem like rarities, there are an infinite number of them. Taking the square root of any number that is not a perfect square always yields an irrational number.