• Recognize irrational numbers in a list of numbers
• Simplify irrational numbers to their lowest terms
• Add, subtract, multiple and divide irrational numbers

Defining and Identifying Numbers That Are Irrational

We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. Irrational numbers are numbers that cannot be expressed as a fraction of two integers.

Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point.

So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.

irrational number

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.

Let’s summarize a method we can use to determine whether a number is rational or irrational. If the decimal form of a number,

• stops or repeats, the number is rational.
• does not stop and does not repeat, the number is irrational.

Identify each of the following as rational or irrational:

1. [latex]0.58\overline{3}[/latex]
2. [latex]0.475[/latex]
3. [latex]3.605551275\dots[/latex]

Show Solution

1. [latex]0.58\overline{3}[/latex] The bar above the [latex]3[/latex] indicates that it repeats. Therefore, [latex]0.58\overline{3}[/latex] is a repeating decimal, and is therefore a rational number.
2. [latex]0.475[/latex] This decimal stops after the [latex]5[/latex], so it is a rational number.
3. [latex]3.605551275\dots[/latex] The ellipsis [latex](\dots)[/latex] means that this number does not stop. There is no repeating pattern of digits. Since the number doesn’t stop and doesn’t repeat, it is irrational.

Another collection of irrational numbers is based on the special number, pi, denoted by the Greek letter [latex]\pi[/latex], which is the ratio of the circumference of the diameter of the circle.

 

Any multiple or power of [latex]\pi[/latex] is an irrational number.

Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the rational part, and the irrational number the irrational part. It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number.