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Non Terminating Decimal

Q: Is 0.999... really the same as 1?

Yes, it is! While it looks strange, $0.\overline{9}$ is a different way of writing the number $1$. One simple proof is that we know $1/3 = 0.\overline{3}$. If you multiply both sides by $3$, you get $3 \times 1/3 = 3 \times 0.\overline{3}$, which simplifies to $1 = 0.\overline{9}$.

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Ever divided two numbers and found the decimal answer just keeps going... and going? You've discovered a non-terminating decimal! Let's explore these endless numbers, learn their secrets, and see how to write them down without using a million digits or running out of paper.

Non Terminating Decimal — an original Algebra911 reference diagram defining non terminating decimal with its key formula and a worked example. — A Clear Guide to Non-Terminating Decimals

What Is a Non-Terminating Decimal?

A non-terminating decimal is a decimal number that continues forever without ending. Unlike some decimals that have a clear stopping point, these numbers stretch on towards infinity. For example, when you convert the fraction $1 / 4$ to a decimal, you get exactly $0.25$ . It stops. This is called a terminating decimal.

But what about a fraction like $1 / 3$ ? If you type $1 \div 3$ into a calculator, you'll see something like $0.333333333$ . The calculator runs out of space, but those 3s actually go on forever! This is a non-terminating decimal.

There are two main types of these endless decimals, which we will explore in this lesson:

Repeating Decimals: These decimals go on forever, but they do so in a predictable pattern. The decimal for $1 / 3$ is a perfect example, as the digit $3$ repeats endlessly.
Non-Repeating Decimals: These are the wild ones! They also go on forever, but they have no repeating pattern at all. A famous example is Pi ( $π$ ), which starts $3.14159 . . .$ and continues with no predictable sequence.

Terminating vs. Non-Terminating: How Can You Tell from a Fraction?

Believe it or not, you can often predict whether a fraction will become a terminating or a non-terminating decimal without even doing the division! The secret is hidden in the denominator (the bottom number of the fraction).

Here's the rule:

First, make sure the fraction is in its simplest form. For example, if you have $6 / 12$ , simplify it to $1 / 2$ .
Next, look at the denominator of the simplified fraction.
Find the prime factors of the denominator. (Prime factors are the prime numbers that multiply together to make the number).
If the only prime factors are $2$ s and/or $5$ s, the decimal will terminate.
If the denominator has any other prime factor (like $3$ , $7$ , $11$ , etc.), the decimal will be non-terminating and repeating.

Why does this work? Our number system is based on ten, and the prime factors of $10$ are $2$ and $5$ . So, fractions with denominators that are built from only $2$ s and $5$ s can be neatly converted into tenths, hundredths, thousandths, and so on, which are the basis of terminating decimals.

Original Fraction	Simplified Fraction	Denominator	Prime Factors of Denominator	Decimal Type
$3 / 15$	$1 / 5$	$5$	$5$	Terminating ( $0.2$ )
$7 / 8$	$7 / 8$	$8$	$2 \times 2 \times 2$	Terminating ( $0.875$ )
$12 / 50$	$6 / 25$	$25$	$5 \times 5$	Terminating ( $0.24$ )
$2 / 9$	$2 / 9$	$9$	$3 \times 3$	Non-Terminating ( $0.222 . . .$ )
$5 / 6$	$5 / 6$	$6$	$2 \times 3$	Non-Terminating ( $0.8333 . . .$ )

Getting to Know Repeating Decimals

A repeating decimal is a non-terminating decimal that has a digit, or a block of digits, that repeats over and over again in a pattern. The part that repeats is called the repetend.

Writing out $0.666666 . . .$ forever is impossible. To solve this, mathematicians created a special shorthand called bar notation or vinculum. We simply draw a horizontal line or bar over the digit or digits that repeat.

For $0.66666 . . .$ , the digit $6$ repeats. We write it as $0. \overset{―}{6}$ .
For $0.121212 . . .$ , the block of digits $12$ repeats. We write it as $0. \overset{―}{12}$ .
Sometimes the repeating part starts after a few other digits. For $0.8333 . . .$ , only the $3$ repeats. We write this as $0.8 \overset{―}{3}$ . The bar only goes over the repeating part.

Example 1

Convert the fraction $1 / 3$ into a repeating decimal using long division and write it with bar notation.

Step 1: Set up the long division. The numerator, $1$ , goes inside the division box, and the denominator, $3$ , goes outside.

Step 2: Since $3$ cannot go into $1$ , we add a decimal point and a zero. We also place a decimal point in our answer.

\begin{array}{r} 0. \\ 3 1.0 \end{array}

Step 3: Divide $10$ by $3$ . It goes in $3$ times ( $3 \times 3 = 9$ ). Write $3$ in the answer and subtract $9$ from $10$ .

\begin{array}{r} 0.3 \\ 3 1.0 \\ - \underset{―}{0.9} \\ 0.1 \end{array}

Step 4: We have a remainder of $1$ . Bring down another zero. We are dividing $10$ by $3$ again! This will give us another $3$ in the answer and another remainder of $1$ . We can see a pattern forming. The remainder will always be $1$ , and we will keep getting $3$ s in our answer.

Conclusion: The decimal is $0.333 . . .$ . Using bar notation, we write this as $0. \overset{―}{3}$ .

How Do You Convert Any Fraction to a Decimal?

The most reliable method to convert any fraction into a decimal is long division. It works every time, whether the decimal terminates or repeats. The process is always to divide the numerator by the denominator.

You just need to know when to stop:

You stop if the remainder becomes $0$ . This means you have a terminating decimal.
You stop if a remainder repeats. As soon as you see a remainder that you've seen before, you know that the digits in your answer will start to repeat from that point on.

Example 2

Convert the fraction $5 / 11$ to a decimal.

Step 1: Set up the long division with $5$ inside the box and $11$ outside.

\begin{array}{r} 115 \end{array}

Step 2: Add a decimal point and zeros. Divide $50$ by $11$ . It goes in $4$ times ( $4 \times 11 = 44$ ).

\begin{array}{r} 0.4 \\ 11 5.0 \\ - \underset{―}{4.4} \\ 0.6 \end{array}

Step 3: The remainder is $6$ . Bring down another zero to make $60$ . Divide $60$ by $11$ . It goes in $5$ times ( $5 \times 11 = 55$ ).

\begin{array}{r} 0.45 \\ 11 5.00 \\ - \underset{―}{4.4} \\ 0.60 \\ - \underset{―}{0.55} \\ 0.05 \end{array}

Step 4: The remainder is $5$ . Bring down a zero to make $50$ . Notice that our remainder is now $5$ , which is where we started! This means the pattern is about to repeat. If we divide $50$ by $11$ again, we will get another $4$ , then a remainder of $6$ , and so on.

Conclusion: The sequence of digits $45$ will repeat forever. So, $5 / 11 = 0.454545 . . .$ , which we write as $0. \overset{―}{45}$ .

The Other Kind: Non-Terminating, Non-Repeating Decimals

So far, we've seen that fractions (which are called rational numbers) turn into decimals that either terminate or repeat. But what about the decimals that go on forever without a pattern? These special numbers are called irrational numbers.

You cannot write an irrational number as a simple fraction. They represent exact values, but their decimal representations are infinite and non-repeating. You have probably already met the most famous irrational number of all!

Example 3

Look at the decimal expansions of two famous irrational numbers, Pi ( $π$ ) and the square root of 2 ( $\sqrt{2}$ ).

Pi ( $π$ ): This number is the ratio of a circle's circumference to its diameter. It is used in all calculations involving circles. Its decimal form begins:

π \approx 3.141592653589793 . . .

The Square Root of 2 ( $\sqrt{2}$ ): This is the number that, when multiplied by itself, equals $2$ . It is the length of the diagonal of a square with side lengths of $1$ . Its decimal form begins:

\sqrt{2} \approx 1.414213562373095 . . .

As you can see, the digits in both numbers go on and on, but there is no block of digits that ever starts to repeat in a predictable cycle. You will learn much more about these fascinating numbers as you continue in your math journey!

Common Mistakes to Avoid

Working with non-terminating decimals can be tricky at first. Here are a few common pitfalls to watch out for:

Incorrect Bar Placement: A very common error is putting the bar over the wrong digits. Remember, the bar goes only over the repetend (the part that actually repeats). For $1 / 6 = 0.1666 . . .$ , the correct notation is $0.1 \overset{―}{6}$ , not $0. \overset{―}{16}$ . The $1$ does not repeat, so it should not be under the bar.
Giving Up on Long Division Too Soon: Some fractions have very long repeating patterns. The fraction $1 / 7$ becomes $0. \overset{―}{142857}$ . You have to perform six steps of division before the remainder repeats! If your division seems to be going on for a while, persevere until you either get a remainder of $0$ or a remainder you've already had.
Confusing 'Long' with 'Infinite': A decimal like $0.1234567$ is long, but it stops. It is a terminating decimal. A non-terminating decimal doesn't just have many digits; it has an infinite number of them. Don't assume a decimal is non-terminating just because it doesn't fit on your calculator screen.
Forgetting to Simplify the Fraction: The denominator rule (checking for prime factors of 2 and 5) only works if the fraction is fully simplified. For the fraction $9 / 12$ , the denominator $12$ has a prime factor of $3$ , suggesting a repeating decimal. But if you simplify $9 / 12$ to $3 / 4$ , the denominator is $4$ (prime factors $2 \times 2$ ), which correctly tells you it will be a terminating decimal ( $0.75$ ).

Quick Summary: Your Cheat Sheet

Here are the most important ideas to remember about non-terminating decimals:

Decimal = Numerator ÷ Denominator

Terminating Decimal: A decimal that ends (e.g., $0.5$ , $1.375$ ).
Non-Terminating Decimal: A decimal that goes on forever.
Two Types of Non-Terminating Decimals:
1. Repeating: Has a pattern that repeats forever (e.g., $0. \overset{―}{3}$ ). These come from rational numbers (fractions).
2. Non-Repeating: Has no pattern (e.g., $π$ ). These are called irrational numbers.
The Fraction Rule: To predict the decimal type, simplify the fraction. If the denominator's prime factors are only $2$ s and $5$ s, it terminates. Otherwise, it repeats.
Bar Notation: A line placed over the repeating digits is used as a shorthand (e.g., $0.166 . . . = 0.1 \overset{―}{6}$ ).
The Method: Long division is the key to converting any fraction to its decimal form.

Frequently Asked Questions

Can a number be both terminating and non-terminating?

No, a number must be one or the other. A terminating decimal has a finite number of digits, while a non-terminating decimal has an infinite number of digits. They are direct opposites.

Is every non-terminating decimal a repeating decimal?

No. While many of the non-terminating decimals you see in this grade come from fractions and are repeating, there is another category called irrational numbers. Famous examples like Pi ( $π$ ) and $\sqrt{2}$ are non-terminating and non-repeating.

How do I know when to stop doing long division?

You can stop dividing when one of two things happens. If your remainder becomes 0, the decimal terminates and you are done. If you get a remainder that you have had before, you know the decimal will repeat from that point on.

Is 0.999... really the same as 1?

Yes, it is! While it looks strange, $0. \overset{―}{9}$ is a different way of writing the number $1$ . One simple proof is that we know $1 / 3 = 0. \overset{―}{3}$ . If you multiply both sides by $3$ , you get $3 \times 1 / 3 = 3 \times 0. \overset{―}{3}$ , which simplifies to $1 = 0. \overset{―}{9}$ .

What is the official name for the bar over the repeating numbers?

The bar used in repeating decimal notation is called a vinculum. It's a general mathematical symbol for grouping, but in this context, it specifically indicates the repetend.

Can you turn a repeating decimal back into a fraction?

Yes, you can! There is a cool algebraic method for converting any repeating decimal back into its fraction form. It's a slightly more advanced topic that you will likely learn in pre-algebra or algebra.

Why do only prime factors of 2 and 5 in the denominator make a terminating decimal?

Our number system is base-10, which is built on powers of 10 (like 10, 100, 1000). The prime factors of 10 are $2$ and $5$ . A fraction with only 2s and 5s in the denominator can always be scaled up to have a denominator that is a power of 10, which is the definition of a terminating decimal.