# Rational and Irrational Numbers

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Rational and Irrational Numbers
CCSS.MATH.CONTENT.8.NS.A.1

## Rational and Irrational Numbers – Definition

In our daily lives, we encounter various types of numbers, from the price of an item to measuring distances. Some of these numbers can be expressed as a fraction of two integers, known as rational numbers, while others cannot and are known as irrational numbers.

## Understanding Rational and Irrational Numbers – Differences

Rational numbers include fractions, integers, and perfect squares, while examples of irrational numbers include non-perfect square roots and the number pi ($\pi$).

Rational numbers are neat - you can write them as simple fractions. They include whole numbers and decimals that end or repeat. Irrational numbers are different. They can't be written as simple fractions and their decimals go on forever without repeating. They're like the endless, unique numbers you get from square roots and special math constants.

## Rational and Irrational Numbers – Exercises

Let’s look at how to classify numbers as rational or irrational and locate them on a number line.

Is the number $\frac{8}{12}$ rational or irrational?
Simplify $\frac{8}{12}$ to its lowest terms.
How would you describe the decimal expansion of $\frac{2}{3}$?
Is the number $\sqrt{5}$ rational or irrational?
What about the number $0.25$? Is it rational or irrational?
Describe the decimal expansion of $\pi$ (pi).

## Rational and Irrational Numbers – Summary

Key Learnings from this Text:

• Rational numbers can be expressed as fractions with both numerator and denominator as integers.
• The decimal representation of rational numbers either ends or repeats.
• Irrational numbers cannot be written as a simple fraction, and their decimal form does not repeat or terminate.
• Identifying and classifying numbers correctly is essential for mathematical calculations and real-world applications.
Category Definition Examples
Rational Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. $\frac{1}{2}$, $0.75$, $\frac{4}{5}$, $2$, $-3$
Irrational Numbers that cannot be expressed as a simple fraction. Their decimal expansion is non-repeating and non-terminating. $\sqrt{2}$, π, $\sqrt{3}$, $5.63094…$

This video is helpful for more information on the different Types of Numbers you will encounter in life!

## Rational and Irrational Numbers – Frequently Asked Questions

What is a simple example of a rational number?
Can all fractions be considered rational numbers?
What makes a number irrational?
Are all square roots irrational numbers?
Can irrational numbers be estimated?
How do you know if a decimal is rational or irrational?
Is the number $0$ rational or irrational?
Why can't $\pi$ (pi) be a rational number?
Are decimals that don't repeat always irrational?
Is every real number either rational or irrational?

### TranscriptRational and Irrational Numbers

Are you ready for the math joke of the day? How do rational and irrational numbers communicate? Well, before we can answer that joke, let's make sure we know what these types of numbers are. Rational numbers can be written as a fraction with two integers, where the denominator is not equal to zero. If there is a decimal, it either terminates, which means to end, or the decimal repeats. Here are some examples of rational numbers. Irrational numbers cannot be written as a fraction, and their decimals are non-repeating and non-terminating. Here are some examples of irrational numbers. So, how do rational and irrational numbers communicate? Rational numbers get to the point, while irrational numbers just go on forever... and ever... and ever! Now that that's answered, let's practice identifying some numbers as either rational or irrational. Is five-eighths a rational or irrational number? Since the number is written as a fraction using two integers, it is a rational number. Even if we divide five by eight, it would result in this, which is still rational since the decimal terminates at the thousandth place. Is the square root of ten considered rational or irrational? It is equal to this value and because it goes on forever and there is no repeating pattern, it is an irrational number. Pause the video here and sort this list of numbers into two categories: rational and irrational. Pi is an irrational number because it is a decimal that is non-terminating and non-repeating. One-sixth is a rational number since it is a fraction comprised of two integers. The number negative nine and two-hundredths is also considered rational because we have a decimal that terminates. Evaluating the square root of seven results in a non-repeating, never-ending decimal, making it an irrational number. Negative radical four is equal to negative two, so it is a rational number. The last number is a non-repeating decimal that does not have an end as noted by the ellipsis, or the three dots, which makes it irrational. Let's review. Numbers can be classified as either rational or irrational. In order to identify which category they belong to, you must determine if they can be written as a fraction, and then observe their decimal numbers to see if they repeat or terminate. Both types of numbers are commonly seen in the real world, even if you are looking to bake a new recipe for double chocolate chip cookies or build a new world in a video game you may encounter rational and irrational numbers. Math will always have its rational side where everything is in harmony, but with irrational numbers, it's always an endless adventure!

## Rational and Irrational Numbers exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Rational and Irrational Numbers.
• ### Find the rational numbers.

Hints

A rational number:

• can be written as a fraction
• can be a decimal which terminates or repeats.

A decimal which terminates is rational.

A decimal which does not terminate is irrational.

There are $3$ correct answers here.

Solution

Rational numbers are:

• $\mathbf{\frac{1}{4}}$
• $\mathbf{\frac{5}{6}}$
• $2.56$
The others are non-terminating decimals, and are therefore irrational numbers.

• ### Find the irrational numbers.

Hints

$\sqrt{9} = 3$ and $-3$

Therefore, this is a rational number.

If the square root does not terminate or repeat it is irrational.

An example of an irrational number is $\sqrt{7}$.

As a decimal it is $2.6457513......$ and does not terminate.

There are 3 irrational numbers here.

Solution

The irrational numbers are:

• $\sqrt{3}$
• $\sqrt{5}$
• $\sqrt{2}$
They do not terminate.

$\sqrt{4} = 2$ and $-2$

$\sqrt{9} = 3$ and $-3$

They both terminate and therefore they are rational.

• ### Order the irrational numbers.

Hints

First, work out the square roots so you have all the choices as non-terminating decimals.

This will make it easier to determine the smallest value when they are in the same format.

Line up the decimals when they are all in the same format as it is easier to see the values.

For example, like this.

Order the numbers from smallest to largest.

Solution
1. $1.05729.....$
2. $\sqrt{2} = 1.4142...$
3. $1.939572....$
4. $\sqrt{7} = 2.645...$
5. $2.804382...$
• ### Which irrational number fits?

Hints

Start by matching up the non-terminating decimals.

This will make it easier to see what is left.

For example, if we were looking for an irrational number between $7$ and $8$, we could pair it with $7.3498234$.

When you have matched up the non-terminating decimals, work out the square roots to see if any fit.

For example, an irrational number between $4$ and $5 = \sqrt{17}$.

$\sqrt{17} = 4.12310....$

Solution

Irrational numbers between:

• $2$ and $3 = \sqrt{5} = 2.23606...$
• $3$ and $4 = \sqrt{11} = 3.31662...$
• $4$ and $5 = 4.826936...$
• $5$ and $6 = 5.0403729...$
• ### Find the fraction and decimal pairs.

Hints

To match these up we divide the fraction.

For example, $\frac{1}{3} = 1\div3 = 0.3333...$

To convert a fraction to a decimal we divide the numerator by the denominator.

Solution

We find the decimal by dividing the fraction.

Numerator divided by denominator.

For example, $\frac{13}{99} = 13\div99 = 0.1313...$

• ### Solve the equation.

Hints

Start by solving the equation, like this.

Solve the equation to get the value of $n$.

Calculate the decimal of $\sqrt{11}$ and decide if it is rational or irrational.

Solution

The $3$ answers are:

• $\sqrt{11}$
• $3.3166...$
• Irrational