Irrational numbers
An irrational number is a number that cannot be written in the form of a common fraction of two integers. It is part of the set of real numbers alongside rational numbers. It can also be defined as the set of real numbers that are not rational numbers.
When an irrational number is expanded in decimal form, it is a non-terminating decimal that does not repeat. Note that a non-terminating decimal that repeats is a rational number, not an irrational number.
Examples
The following are a few of the more commonly known irrational numbers:
| π | = | 3.14159... |
| e | = | 2.71828... |
 |
= | 1.41421... |
No matter the number of decimal places we calculate these values to, there will always be another digit after it, hence the term non-terminating decimal.
Properties of irrational numbers
As a subset of real numbers, irrational numbers share the same properties as the real numbers. Below are some of the properties of irrational numbers as they relate to their rational counterpart.
- The sum of an irrational number and a rational number is irrational.
- The product of an irrational number and a rational number is irrational, as long as the rational number is not 0.
- Two irrational numbers may or may not have a least common multiple.
- Irrational numbers are not closed under addition, subtraction, multiplication, and division. This is in contrast to rational numbers which are closed under all these operations.
In regards to the last bullet point, the property of closure, this means that operations involving only the set of irrational numbers can result in numbers that are members of different sets, such as rational numbers:
Addition and subtraction
Addition and subtraction of irrational numbers can result in either an irrational number or a rational number. Whenever operations between two irrational numbers can result in a number that is not irrational, it is not closed under that operation.
Examples
Addition:
(rational)
Subtraction:
(rational)
Multiplication and division
Irrational numbers are also not closed under multiplication and division. In both cases, it is possible for irrational numbers undergoing these operations to result in a rational number.
Examples
Multiplication:
(rational)
Division:
(rational)
Did you know??
There are more irrational numbers than there are rational numbers. Even though there are an infinite number of both types of numbers, we still know that there are more irrational numbers than rational ones. One way to think about this is that within even the relatively small set of natural numbers, the square root of all natural numbers that are not perfect squares (1, 4, 9, 16, etc), are irrational numbers. Having only listed the first 4 perfect squares, we've already reached the natural number 16. Between 1 and 16, there are 12 natural numbers the square root of which are irrational numbers. Furthermore, irrational numbers are non-terminating and non-repeating, so imagine adding many decimal places to each natural number along with all the combinations of digits we can use for each of the decimal places, and you can start to imagine just how many more irrational numbers there are!