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The Complete Answer: What is an Irrational Number?

Irrational Numbers are numbers that cannot be written as fractions. Irrational Numbers have two things special about their decimal forms. The first is that Irrational Numbers have decimals that do not terminate, meaning they never end. The second is that Irrational Numbers have decimals that will never repeat in pattern. This means that all integers, whole numbers, and natural numbers are not Irrational Numbers, they are instead Rational Numbers. The most common examples of Irrational Numbers are π, √2, √3, and e.

Common Core Standard: 8.NS.A

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A Short Explanation of the Irrational Numbers Definition

Well exactly what is an irrational number in math anyway? Rational and Irrational Numbers are defined by looking at their decimals. Rational Numbers are numbers that can be written as a fraction. This includes whole numbers, terminating decimals, and repeating decimals because you can write them all as fractions. Irrational Numbers are numbers that cannot be written as a fraction. Typically Irrational Numbers are numbers like pi and square roots.

6 Easy Steps for Completing any Irrational Numbers Example

Determine if the number can be written as a fraction.
If the number is a whole number it is a Rational Number.
If the number is a terminating decimal it is a Rational Number.
If the number is a repeating decimal it is a Rational Number.
If the number is a non-repeating decimal it is an Irrational Number.
If the number is a non-terminating decimal it is an Irrational Number.

Rational Numbers Practice Problems Quiz

A Short Video on our Rational and Irrational Numbers Worksheet

Watch our free video on how to solve Rational and Irrational Numbers. This video shows how to solve problems that are on our free Rational and Irrational Numbers worksheet that you can get by submitting your email above.

Watch the free Rational and Irrational Numbers video on YouTube here: Rational and Irrational Numbers Worksheet

Video Transcript:

This video is about our rational and irrational numbers worksheet with answers. You can get the rational or irrational worksheet we use in this video for free by clicking on the link in the description below.

Before we do a couple practice problems on our rational and irrational numbers worksheet, I want to go over what exactly a rational number is and what exactly an irrational number is. Now a rational number is any number that you can write as a ratio of two numbers. In other words, any number that you can write as a fraction.

The first type of rational numbers are whole numbers. The reason whole numbers are rational is because every whole number can be written as a fraction. For example if we have the whole number of six, all whole numbers technically have this one underneath of them, because it’s like saying 6 divided by one. We typically do not write this one because obviously 6 divided by one is just six so you do not have to write it because it doesn’t change the number. When you see a whole number, it doesn’t matter what the whole number is, there is a divided by one or a fraction one underneath of it.

The second type of rational number our terminating decimals. Now a terminating decimal is any decimal that stops or ends. If you look at one point seven five, this clearly has an end point. It’s right here after the 5. You can write this as a fraction because you can rewrite 1.75 as a mixed number. It would be 1 and 75 over 100 which would be the unsimplified version but it just proves that you can write it as a fraction.

The third type of rational numbers are repeating decimals. Most commonly you will see point three repeating which is one-third or 0.6 repeating which would be 2/3. It’s any decimal that repeats the same pattern over and over again.

Irrational numbers practice problems are numbers that cannot be written as fractions. The easiest way to remember what an irrational number is, is that it’s any non repeating and non terminating decimal. The most common examples of irrational numbers examples are pi, because it goes on forever. Unless a square root is a perfect square, it will be an irrational number. You can see that both of these decimals do not follow a pattern and the dots indicate that they go on forever so they never end. These are what makes a number irrational.

Moving on to some rational number practice problems on a rational and irrational numbers worksheet. Now the directions for this part of the worksheet just say to identify whether it’s a rational number or an irrational number number. Three gives us negative 8 which is a whole number. Now we know negative 8 can be re-written as negative 8 over 1 and negative 8 over 1 is a fraction, which means it is rational number. Four gives us the cube root of 64. In order to simplify this we have to find what number times what number times what number equals 64. In this case that number is 4. Because it’s 4 times 4 times 4, 4 is a whole number which means the cube root of 64 is rational number.

Eight gives us 4 times the square root of 2. Now we already know that the square root of 2 is an irrational number. This is equal to 1.4142 and it goes on forever. When you multiply four times the square root of 2, you will get five point six five six eight zero and it goes on forever. Our decimal never repeats and because of the dots that means it goes on forever. That means this will be an irrational number. The square root of 77 is not a perfect square and when you do the square root of 77 in a calculator you get eight point seven seven four nine six that goes on forever. This is neither repeating or terminating so it’s non-repeating, non-terminating which means it is irrational. Try all the practice problems by downloading the free rational vs irrational numbers worksheet above.