# Definition, Meaning, and Examples of Terminating Decimals

Being able to identify and work with examples of terminating decimals is an important skill in studies of mathematics. There are so many different types of decimals that it is easy to become overwhelmed!

That’s why I put together this guide to help you understand the definition and meaning of terminating decimals. I will also share a collection of examples of terminating decimals that will help you understand the difference between non-terminating and terminating decimal numbers!

It’s time to master your understanding of numbers in decimal form and *terminate *your confusion!

## What Is a Terminating Decimal?

A terminating decimal number is a number in decimal form where the digits to the right of the decimal point stop and have a definitive ending.

When you picture a decimal number, there are generally two parts:

- the whole number before the decimal point
- the numbers after the decimal point

Since terminating decimal numbers end, there are a specific number of digits to the right of the decimal point.

If you think about your knowledge of the real numbers in our number system, rational numbers are numbers that can be written as a fraction with an integer numerator and denominator. Terminating decimal numbers are considered rational numbers because they can be written as a fraction in this way.

One helpful trick I share to help my students remember this terminating decimal definition is that the word “terminate” literally means “to put a stop to”.

To help you understand this terminating decimal definition, consider this example of a terminating decimal in the real-world:

*Imagine that you just ordered an 8 slice pizza. If you eat 2 of the slices, what fraction of the pizza did you eat?*

As you can see in the image above, 2 out of 8 slices is the same as the fraction \(\frac{1}{4}\). And the decimal representation of \(\frac{1}{4}\) is 0.25.

Notice that this decimal number has a fixed number of digits, ending with the number 5. Because of this, we can say that this is an example of a terminating decimal number!

## How to Identify a Terminating Decimal

There are a few key interesting facts that all terminating decimal numbers have in common. Using these can help you identify whether a decimal number terminates or not.

A given decimal number that terminates:

- will have a finite number of digits the the right of the decimal point
- must be a rational number that can be expressed in fraction form (irrational numbers are not terminating decimal numbers)

There are a few other interesting facts that can help you identify terminating decimal numbers that I will explain below!

## Examples of Terminating Decimals

In order to help you develop an understanding of terminating decimal numbers, it is best to look at a variety of examples. Whether you are working with algebraic expressions, or solving equations, there are many scenarios where you may encounter decimals.

The following are examples of decimal numbers, fractions, and square roots that all contain a decimal representation that terminates.

### Terminating Decimal Number Examples

Whether you are completing a simple calculation, conducting measurements, or describing amounts, your reality is made up of more than just whole numbers. Because of this, you will encounter decimal numbers from time to time.

Each example of a terminating decimal below will involve a simple decimal number that has a finite number of decimal places. Take note of the last digit in each decimal so that you can get used to seeing decimal numbers that end!

- 0.25
- 1.301
- 0.125
- -3.875

Notice that these terminating decimal examples include cases where zero, a positive whole number, and a negative whole number are to the left of the decimal point. Regardless, each example has a finite number of digits. Because of this, these numbers are also considered rational numbers since they can be written in fraction form.

### Examples of Fractions that Terminate

Speaking of fractions, there are plenty of examples of fractions that can be considered terminating decimal numbers when written in decimal form.

One of the most interesting facts about fractions is that if the denominator of a fraction is a factor of 2 or 5, then the given fraction will be a terminating decimal number when written in decimal form.

For example, consider each of the following fractions:

- In the fraction \(\frac{1}{8}\), the denominator of the fraction is a factor of 2, so we know that this fraction will terminate.
- \(\frac{7}{25}\) has a denominator that is a factor of 5, so we know that this fraction will terminate as well!

Pretty neat trick, isn’t it? As it turns out, the numerator does not play a role in whether or not the fraction terminates in decimal form. You can do this any number of times to test it out!

By contrast, if the denominator of a fraction is *not* a factor of 2 or 5, then the fraction will *not* terminate.

Consider the following fractions, taking note of their denominators:

- In the fraction \(\frac{1}{7}\), the denominator of the fraction is
*not*factor of 2 or 5, so we know that this fraction will*not*terminate. It turns out that this fraction is equal to 0.1428571429… written is decimal form. - \(\frac{3}{11}\) also has a denominator that is not a factor of 2 or 5. Therefore, we know that this fraction will also not terminate. In fact, the decimal representation of this fraction is 0.2727272727…, which is a non-terminating repeating decimal number!

Looking at the fractional values of the denominator tells us quite a bit about whether a given fraction will terminate or not! Each of these examples of fractions showcases how you predict the type of decimal numbers that you are working with!

### Square Root Examples

There are a few different possibilities for the decimal equivalent of a square root. The decimal representation of some square roots terminate with a fixed number of digits. For example, finding the square root of perfect square numbers such as 1, 4, 9, 16, and 25 will each produce a whole number:

- \(\sqrt{1}=1\)
- \(\sqrt{4}=2\)
- \(\sqrt{9}=3\)
- \(\sqrt{16}=4\)
- \(\sqrt{25}=5\)

Since there are a finite number of digits after the decimal, we say that these square roots are examples of terminating decimals.

By contrast, there are many square root examples where the decimal representation has an infinite number of digits. For example, consider the square root of 2, 3, 5, and 7:

- \(\sqrt{2}=1.414213562…\)
- \(\sqrt{3}=1.7320508076…\)
- \(\sqrt{5}=2.2360679775…\)
- \(\sqrt{7}=2.6457513111…\)

You will notice that the decimal representation for each square root above has a whole number to the left of the decimal, and a string of seemingly random digits to the right of the decimal point. Because of this, these are all non-terminating decimal examples. Since there are no repeating digits, these examples are also considered non-terminating, non-repeating decimals.

In fact, if a number is not a perfect square, its square root will always be a non-terminating decimal number!

## What Is a Non-Terminating Decimal?

We say that terminating decimal numbers have a finite number of decimal places and that there is a defined end digit or end term. By contrast, non-terminating decimals have an infinite number of digits after the decimal point. The simplest form definition is that non-terminating decimal numbers do not end.

To help you remember this definition, remember that the word “terminate” literally means “bring to an end.” So it makes sense that *non*-terminating means* does not *come to an end.

When it comes to non-terminating decimals, it is possible to see the numbers after the decimal point repeat in some sort of pattern. These cases are referred to as repeating decimal numbers because a specific set of digits repeats without end.

A non-terminating decimal example that has a repeating pattern is the fraction \(\frac{2}{7}\), which has a non-terminating recurring decimal expansion of 0.285 714 285 714… . Notice that the “285714” group of digits repeats indefinitely!

It is also possible for non-terminating decimals to be *non-*repeating. Non-repeating decimal numbers do not have a repeated term or sequence of digits. For example, the number \(\sqrt{5}\) has a non-recurring decimal representation of 2.2360679775… . Notice that there are an infinite number of terms without a repeating pattern!

A famous non-terminating decimal example that you have likely heard of is Pi (written as π). Perhaps the most famous of the irrational numbers, Pi is a decimal number that does *not* end and does not have repeating digits.

## Understanding Examples of Terminating Decimals

Whether you are just learning about terminating decimals for the first time, or you are working with them during a more complex high school problem, it is important to have a strong understanding of each type of decimal numbers.

Take some time to familiarize yourself with the key differences between the types of decimals. It will take practice, but getting to know each decimal definition will help you feel more comfortable in your studies of mathematics!

I am hoping that you find some of the tips and tricks from throughout this post helpful. I am truly amazed by some of the patterns that exist in mathematics, and I hope that you find them equally amazing!

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