Non Terminating Decimal Examples (The Ultimate List)

Whether you realize it or not, non terminating decimal examples are everywhere in math. They can be quite scary when you happen to come across one! There is something unnerving about seeing a long string of seemingly random numbers to the right of the decimal point. 

But just what exactly are non terminating decimal numbers? And how do you recognize non terminating decimal examples when you see them? 

Let’s get into it so that you can terminate your confusion once and for all! See what I did there?

Non-Terminating Decimal Definition

Simply put, non terminating decimals are decimal numbers that do not have an end. An easy way to remember this definition is 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 you look to right of the decimal point in a non-terminating decimal number, you will see a series of decimal digits that does not end.

Pretty scary, right? No end!?

There are a few specific types of decimals that can be grouped to make it easier to understand non terminating decimal numbers. For example:

• Irrational Numbers: numbers that cannot be expressed as a fraction of two integers.
• Repeating decimal numbers: numbers that have repeating digits that form some sort of pattern.
• Non-repeating decimal numbers (or non-recurring decimal numbers): numbers that do not have a set of digits that repeat in some sort of pattern.

Let’s take a look at a few common non terminating decimal examples from each of these categories to simplify things!

Non Terminating Decimal Examples

When it comes to decimal numbers that do not have an end, there are an infinite number of such decimals. Some are quite famous, while others are more obscure and harder to come across. 

Non terminating decimal examples can take many forms. You may come across them when evaluating a square root, finding the decimal representation of a fraction, or calculating the area of a circle. 

This list of non terminating decimal examples will help you understand the possible cases where you could encounter a decimal number that does not have an end term to the right of the decimal point.

Pi: The Most Famous Non Terminating Decimal Example

Pi (written as π) is potentially one of the most famous numbers there is. It has been depicted in movies and TV shows, and it even has it’s own day named after it!

Even non-mathematicians know at least a few digits of pi, but few know its meaning or the reason it is an example of a non-terminating decimal number.

Pi is considered an irrational non-terminating decimal number because it cannot be written as the ratio of two integer numbers. To calculate an accurate estimation of pi, one must look at the ratio of the circumference of a circle to its diameter. The result is a value that is non terminating and non-repeating. 

This means that not only does pi not end, but it also has no group of digits to form any sort of repeated pattern. There are just an infinite number of digits to the right of the decimal point.

Because of this, it is common for mathematicians to round pi to 3.14. However, rounding pi in this way can be problematic as it can lead to inaccuracies in complex calculations.

Euler’s Number

While it isn’t as famous amongst non-mathematicians as Pi, Euler’s number has many applications that range from compound interest, to probability, to calculus.

Named after Swiss mathematician Leonhard Euler, Euler’s number is approximately equal to 2.71828 and is commonly known as e. Like Pi, Euler’s number is a non-terminating decimal number that is also a non-repeating decimal number. This means that e has no end term nor any single block of digits that is repeated to the right of the decimal point.

Also like Pi, e is an irrational number, meaning that it cannot be expressed as a fraction of two integer numbers. 

Square Root Examples

Another common place that you will see non terminating decimal examples is when calculating square roots. 

Not every square root will be non-terminating. Perfect roots will result in whole numbers when a square root is performed. For example, calculating the square root of 4 will result in 2 – a whole number. This is also the case for 9, 16, 25, and any other perfect square number. Since these numbers do not contain decimals that continue without end, they are considered examples of terminating decimal numbers.

By comparison, a number such as the square root of 2, or \(\sqrt{2}\), is a different story! The decimal equivalent of the square root of 2 turns out to be 1.4142135624… You will notice that \(\sqrt{2}\) has an infinite number of decimal places and does not have a repeating block of digits. 

This is also the case for square roots such as \(\sqrt{3}\), \(\sqrt{5}\), and many more!

These types of square root numbers are non-terminating, non-repeating, and irrational numbers.

Fractions

Decimal numbers are often seen as a different form of a fraction. Because of this, it makes sense that there are some examples of fractions that will produce non-terminating decimal numbers as well!

There are many non terminating decimal examples that are fractions. For example, the fraction \(\frac{1}{3}\) can be written in decimal form as 0.3333…  Notice that the digit 3 forms a repeating pattern. Because of this, 0.3333… in fraction form is considered a repeating non-terminating decimal number. 

By contrast, the fraction \(\frac{2}{7}\) can be written as 0.285714285714…. This is an interesting example of a decimal number that has a repeating block of digits that forms a repeating pattern to the right of the decimal point. Notice that the block of “285714” is repeated without end. As such, \(\frac{2}{7}\) is considered a repeating non-terminating decimal number when it is written in decimal form.

There are also examples of improper fractions that are non terminating when written in decimal form. Consider the fraction \(\frac{12}{7}\) for example. In decimal form, this fraction is equal to 1.7142857142857 …

You will notice that when this improper fraction is written in decimal form, it also forms a repeating block of digits. As such, \(\frac{12}{7}\) is considered a repeating non-terminal decimal number.

Understanding Non Terminating Decimal Examples

In order to fully understand the concept of a non terminating decimal number, it is important to look at many different examples and types of decimals. As you saw with the non terminating decimal examples above, there are many different ways for a decimal number to “not end’.

Whether it is a repeating pattern, or no pattern at all, non terminating decimal numbers are a pretty bizarre idea! You may think to yourself “so, wait … the decimal keeps going forever? Without end?” But with some practice, this idea becomes a bit easier to process.

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