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Irrational Number Definition: What It Is and How To Use It

irrational number definition: Mathematical equation on graph paper and a pencil pointing to it

What are irrational numbers? If you find yourself unsure of the difference between real numbers, rational numbers, natural numbers, complex numbers, and even imaginary numbers, you are not alone. Math is one subject where it pays to be precise.

Counting With Whole Numbers and Natural Numbers

Before we give the irrational number definition, let's cover integers and whole and natural numbers.

Let's start with some basic examples. When you first learn to count, say, the number of fingers that you have, you use natural, or whole numbers: 1, 2, 3, 4, 5… and so on. These are natural numbers. Whole numbers are also numbers without fractions, but whole numbers include 0 while natural numbers do not.

Like whole numbers, integers don't have fractions. However, unlike whole numbers, they include negative numbers: -4, -3, 0, 14 are all examples of integers.

Definition of a Rational Number

A rational number can be expressed as a fraction.

As we get older, we learn that not everything is countable on our fingers. For example, 0.5 is a rational number. It is not a whole number, natural number, or integer, but it can be expressed as 1/2, which a fraction of two other integers: 1 is the numerator and 2 is the denominator. So, 0.5, or 1/2, is a rational number.

Rational numbers can also be divided into two types: those that have a terminating decimal expansion and those that have a repeating decimal expansion. A decimal expansion when a fraction is converted into a decimal number. The decimal expansion of 1/2 is 0.5. The decimal expansion of 1/3 is 0.333.

1/2, or 0.5, is terminating because it ends at the tenths digit.

1/3, or 0.333 is non-terminating because the “3” repeats forever.

What's an Irrational Number?

By definition then, an irrational number can’t be expressed as a fraction of two integers. For example, let’s look at the square root of 2. Even though we know the number exists, the decimal points go on forever, so we can't expand it fully.

Depending on whether you are able to write them as a fraction, some square roots and cube roots are rational and some are irrational.

Examples of Irrational Numbers

You may know of more irrational numbers than you think.

Pi is the ratio of a circle's circumference to its diameter. Its decimals go on infinitely long. You may know the first couple digits as 3.14.

Euler's Number E is the base for natural logarithms. It is approximated as 2.718.

Phi, or the golden ratio, is another frac that does not end when it’s in decimal point form. It is closest to 1.618.

The Irrational Number Definition and Beyond

You can think of all these definitions as subsets of what defines a number. Irrational numbers cannot be defined as the quotient of two integers. Rational numbers, on the other hand, can be split into rational numbers with repeating decimals and rational numbers with non-repeating decimals. Natural numbers and whole numbers are the smallest subsets, and 0 is considered a whole number but not a natural number.

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