Learning the difference between rational and irrational numbers doesn’t have to be complicated, let’s find out how together.

Author

Amber Watkins

Published

August 2, 2023

Author

Amber Watkins

Published

August 2, 2023

Author

Amber Watkins

Published

August 2, 2023

Key takeaways

- Rational numbers can be written as fractions and ratios.
- The most common rational numbers are positive and negative integers.
- The most common irrational number used in math is Pi.

Table of contents

Rational numbers are the most common numbers we see in the world around us. We use rational numbers on our speed limit signs, in our recipes, and on our shoe labels to show what size we wear.

Irrational numbers are not as common, but they are very important! The most common irrational number is Pi! Without that very long, never-ending number we wouldn’t be able to calculate the area of a circle.

Rational numbers can be defined as any number or value that can be written as a fraction or a ratio. Any number that can not be written as a fraction is irrational. Let’s review some examples to help us identify rational vs irrational numbers.

**1. ****Integers:** all integers are rational numbers. Integers include all real numbers both positive and negative. Since all integers can be written as a fraction, they are all rational numbers.

**Example: Is 3 a rational number?**

3 can be written as 3/1 or 6/2.

Since it can be written as a ratio and fraction, it is a rational number.

**2. Repeating decimals: All repeating decimals are rational numbers. Repeating decimals have numbers after the decimal place that repeat. Even though it may be difficult to change repeating decimals into fractions, it is possible, so they are considered rational numbers.**

**Example: Is .66666666 a rational number?**

*.66666666 can be written as ⅔.*

*Since it can be written as a ratio and fraction, it is a rational number.*

**3. Non-repeating decimals that are finite: All decimals that are finite (come to an end) are rational numbers. Finite decimals can be written as fractions, so they are rational numbers.**

**Example: is 5.25 a rational number?**

*5.25 can be written as 525/100. *

*Since it can be written as a ratio and fraction, it is a rational number.*

**4. Perfect square roots: **All perfect square roots are rational numbers.** Perfect square roots always have a whole number as the answer. Since the square roots are positive whole numbers, they are rational and therefore make the square root rational.**

**Example: Is √25 a rational number?**

*The square root of 25 is 5. Five can be written as 5/1. *

*Since it can be written as a ratio and fraction, it is a rational number.*

**1. Non-repeating, non-terminitating decimals: **All decimals that do no repeat and continue indefinitely are the most common irrational numbers. **Decimals that never end can not be written as a fraction without rounding. For this reason they are considered to be irrational, not rational.**

**Example: Is 5.432698762 a rational number?**

*No, it is an irrational number because it can’t be written as a fraction.*

Did you know?

The first 12 digits of Pi are 3.14159265358 and it continues on forever.

**2. Non-perfect square roots: **All square roots that are not perfect squares are classified as irrational. You can enter the square root of the numbers 7, 12, or 18 into a** calculator, but the answer you get will not be rational. This is an indication that the square root of 7, 12, and 18 are irrational numbers.**

**Example: Is √12 a rational number?**

*The √12 is 3.4610161514 which can not be made into a fraction.*

*No, it is an irrational number.*

As you can see from the examples, the primary difference between rational and irrational numbers is rational numbers can be written as fractions, irrational numbers can not. Numbers in the form of decimals and square roots can be classified as rational and irrational numbers, so we have to be extra careful when checking. Now it’s your turn to practice!

**Yes**

.25 is a rational number because it can be written as the fraction 1/4

**Yes**

√100 is a rational number because the square root of 100 is 10 and 10 can be written as 10/1 so it is a rational number, so √100 is also a rational number

**No**

4.986432 is an irrational number because it is a non-repeating decimal and it can’t be put in the form of a fraction or ratio

Toggle Content

We know learning the difference between rational and irrational numbers is complex so we’ve provided a few frequently asked questions many students or parents have when they start working with these numbers.

The primary difference between rational numbers and irrational numbers is whether the numbers can be written as fractions. In order to determine whether a number is rational or irrational, you must check to see if the number can be written as a fraction.

The most common irrational number is Pi or 3.14159265358. Some examples of rational numbers are 5, 10, 3/4, and .80. For more examples of rational and irrational numbers see our math practice app.

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Lesson credits

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

Amber Watkins

Amber is an education specialist with a degree in Early Childhood Education. She has over 12 years of experience teaching and tutoring elementary through college level math. "Knowing that my work in math education makes such an impact leaves me with an indescribable feeling of pride and joy!"

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