Terminating and Non-Terminating Decimals: The decimal numbers are used to express both whole numbers and fractions. Depending on the type of digits after the decimal point, decimals can be categorised into several types such as repeating, non-repeating, ending, or un-ending (infinite digits after the decimal point). There are two types of decimals they are terminating and non-terminating decimals.

A terminating decimal is a decimal number with a finite number of digits after the decimal point. A non-terminating decimal will never end but may predictably repeat one or more values after the decimal point. Scroll down to know what is terminating decimal, what is non-terminating decimals, repeating decimals and more.

Terminating and Non-Terminating Decimals Definition

What is Terminating Decimal: Numbers with a fixed or finite number of digits following the decimal point are known as terminating decimals. In the same way that fractions represent the partial amount of a whole, decimal numbers represent a whole.

Example: \(0.2,\,0.125\) and \(0.35\)
These terminating decimals can be expressed in the form \(\frac{p}{q}.\)
Example: Express \(0.2\) in the form of \(\frac{p}{q}.\)
Solution: \(0.2 = \frac{{0.2 \times 10}}{{10}} = \frac{2}{{10}} = \frac{1}{5}\)

Non-terminating Decimal Definition: A non-terminating decimal has infinite decimal places. The digits after the decimal point will not terminate. A non-terminating decimal can be repeating or non-repeating. A non-terminating, non-repeating decimal is a decimal number with no repeating digits and continues indefinitely. The non-terminating and non-recurring or non-repeating decimals are irrational numbers. Because it’s an irrational number, this decimal can’t be stated as a fraction.

When we split a fraction expressed in decimal form, we receive any remainder. The decimal is non-terminating if the dividing technique does not result in a remainder equal to zero. In some circumstances, a single digit or a group of digits in the decimal component repeats. A sort of non-terminating repeating decimal is pure repeated decimals, which are also known as non-terminating repeating decimals. To symbolize these decimal numbers, a bar is put on the replicated portion.

Example: \(0.2857142857\) and \(0.3333….. = 0.\overline 3 \)

There are two forms of non-terminating decimal expansions, they are:
(i) Non-terminating recurring decimal expansion
(ii) Non-terminating non-recurring decimal expansion

Non-Terminating Repeating Decimal Expansion

A non-terminating decimal is a decimal with an infinite number of digits after the decimal point.
A non-terminating, recurring decimal is a decimal in which some digits after the decimal point repeat without terminating. A non-terminating, recurring decimal can be expressed as \(\frac{p}{q}\) form.
Example: \(0.666….\) or \(0.\overline 6 ,\,2.6666…\) or \(2.\overline 6 .\)
Express \(0.\overline 6 \) in the form of \(\frac{p}{q}\)
Here, \(0.\overline 6 = 0.6666…\)
Take, \(x = 0.6666…\)
\(10\,x = 6.6666…\) (Multiplying \(10\) on both sides)
\( \Rightarrow 10\,x = 6 + 0.6666…\)
\( \Rightarrow 10\,x = 6 + x…\) \(∵\left( {x = 0.666…} \right)\)
\( \Rightarrow 10\,x – x = 6\)
\( \Rightarrow 9x = 6\)
\( \Rightarrow x = \frac{6}{9} = \frac{2}{3}\)
Hence, \(0.\overline 6 = \frac{2}{3}\)

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Non-Terminating Non-Recurring Decimal Expansion

A non-terminating, non-recurring decimal in which the digits after the decimal point do not repeat and do not terminate.
Example: \(1.4142135623…,\,1.7320508075…\) and \(2.2360679774…\)
Such numbers are irrational numbers. They cannot be written in \(\frac{p}{q}\) form.

Terminating and Non-Terminating Decimals Representation

Terminating Decimals Representation: Let us say we are looking for \(2.665\) on the number line. We know it’s somewhere between \(2\) and \(3.\)

So, let’s take a closer look at the number line between \(2\) and \(3.\) Assume we divide it into ten equal sections and label each division point as shown in the diagram.
The first mark to the right of \(2\) will be \(2.1.\) The second will be \(2.2,\) and so on. You could be having trouble noticing the points of division in the figure between \(2\) and \(3.\)
Take a magnifying lens and gaze at the portion between \(2\) and \(3\) to get a clear view of the same. It will resemble what you see in the illustration. \(2.665\) is now halfway between \(2.6\) and \(2.7.\) So let’s concentrate on the area in the figure between \(2.6\) and \(2.7.\) Imagine dividing this into ten equal portions once more. The first mark will be \(2.61.\) The second mark will be \(2.62,\) and so on. We magnify this to view it clearly, as indicated in the image. Again, \(2.665\) is in the range of \(2.66\) and \(2.67.\) Let’s concentrate on this section of the number line and visualise dividing it into ten equal sections once more. We enlarge it to see it more clearly, as shown in the diagram. The first mark corresponds to \(2.661,\) the second to \(2.662,\) and so on. In these subdivisions, \(2.665\) is the fifth mark.

The technique of successive magnification refers to the visualisation of numerical representations on a number line using a magnifying glass.

Non-Terminating Decimals Representation: Let’s try visualizing a real number’s position (or representation) on the number line using a non-terminating recurring decimal expansion. We can visualize the position of the number on the number line by looking at appropriate intervals through a magnifying glass at successive magnifications.

Example:
Visualise the number \(5.3\overline 7 \) on the number line up to \(5\) decimal places, or \(5.3777.\)
We proceed by consecutive magnification and decreasing the lengths of the segments of the number line where \(5.3\overline 7 \) is placed once more. To begin, we can see that \(5.3\overline 7 \) is halfway between \(5\) and \(6.\) The next stage is to find \(5.3\overline 7 ,\), which is somewhere between \(5.3\) and \(5.4.\) To see the representation more clearly, we break this section of the number line into \(10\) equal portions and use a magnifying glass to see that \(5.3\overline 7 \) is between \(5.37\) and \(5.38.\)

To see \(5.3\overline 7 \) more clearly, split the space between \(5.37\) and \(5.38.\) into ten equal pieces again, and use a magnifying glass to see that \(5.3\overline 7 \) is between \(5.377\) and \(5.378.\) To visualise \(5.3\overline 7 ,\) divide the region between \(5.377\) and \(5.378\) into ten equal portions and envision the representation of \(5.37\) as shown in the diagram. It’s worth noting that \(5.3\overline 7 \) is closer to \(5.3778\) than \(5.3777.\)

Terminating Decimal Example: The numbers that terminate after a few digits after the decimal point are known as terminating decimals. \(0.62,\,3.156,\,13.435\) are few examples for terminating decimals.

Non-Terminating Decimal Example: When expressing a fraction in decimal form, we obtain some remainder when we divide it. If the division process does not result in a remainder equal to zero, the decimal is referred to be non-terminating.
\(\pi = \frac{{22}}{7} = 3.142857142857143…,\,0.1414141414…,\,2.565656…\) are few examples for non-terminating decimals.

Now students can easily find if they are asked what among the following terminating decimal is or if they are asked what non-terminating repeating decimal is.

Solved Examples – Terminating and Non-Terminating Decimals

Q.1. Show that \(0.333… = 0.\overline 3 \) can be expressed in the form \(\frac{p}{q},\) where \(p\) and \(q\) are integers and \(q \ne 0.\)
Ans: Since we do not know what \(0.\overline 3 \) is, let us call it \(x\) and so
\(x = 0.3333…\)
Now, \(10x = 10 \times \left( {0.333…} \right) = 3.333…\)
Now, \(3.333… = 3 + 0.333… = 3 + x,\) since \(x = 0.333…\)
Therefore, \(10x = 3 + x\)
Solving for \(x,\) we get \(9x = 3,\) i.e., \(x = \frac{1}{3}\)
Hence, \(0.\overline 3 = \frac{1}{3}.\)

Q.2. Express \(0.37\) in the form of \(\frac{p}{q}.\)
Ans: The given decimal is \(0.37\)
\(0.37 = \frac{{0.37 \times 100}}{{100}} = \frac{{37}}{{100}}\)
Hence, \(0.37 = \frac{{37}}{{100}}.\)

Q.3. Classify the following as terminating and non-terminating decimals:
\(0.567,\,7.36363636…,\,1.234565432…,\,6.78\) and \(9.03892657853164…\)
Ans: \(0.567,\,6.78\) are the terminating decimals, and \(7.36363636…,\,1.234565432…,\,9.03892657853164…\) are the non-terminating decimals.

Q.4. Determine whether \(0.7265\) is a terminating or non-terminating decimal and Justify the answer.
Ans: The given decimal is \(0.7265\)
\(0.7265 = \frac{{0.7265 \times 10000}}{{10000}} = \frac{{7265}}{{10000}}\)
It’s a terminating decimal because the number that terminates after four decimal places and can be expressed in the form of \(\frac{p}{q}.\)

Q.5. Show that \(0.2353535… = 0.2\overline {35} \) can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \ne 0.\)
Ans: Let \(x = 0.2\overline {35} .\) Over here, note that \(2\) does not repeat, but block \(35\) repeats. Since two digits are repeating, we multiply \(x\) by \(100\) to get:
\(100x = 23.53535…\)
So, \(100x = 23.3 + 0.23535… = 23.3 + x\)
Therefore, \(99x = 23.3\)
i.e., \(99x = \frac{{233}}{{10}},\) which gives \(x = \frac{{233}}{{990}}\)
Hence, \(0.2\overline {35} = \frac{{233}}{{990}}.\)

Summary

In this article, we learned about the definition of terminating and what is non-terminating decimal, the terminating and the non-terminating decimals representation, the terminating decimal example, and the non-terminating decimal example solved examples of the terminating and the non-terminating decimals and FAQs on the terminating and the non-terminating decimals.

The learning outcome of this article is we learned how to represent terminating and non-terminating decimals on the number line and how to covert terminating and non-terminating but recurring decimals into rational number form.

FAQs on Terminating and Non-Terminating Decimals

Q1. What are terminating and non-terminating decimals?
Ans: Terminating decimal: Numbers with a fixed or finite number of digits following the decimal point are terminating decimals.
Non-terminating decimal: When expressing a fraction in decimal form, we obtain some remainder when dividing it. If the division process does not result in a remainder equal to zero, the decimal is referred to be non-terminating.

Q2. How do you know if a number is terminating or non-terminating?
Ans: When we divide the number, if we get the remainder as \(0,\) it is a terminating decimal. If the division process does not result in a remainder equal to zero, the decimal is non-terminating.

Q3. Is 0.5 terminating or non-terminating?
Ans: The numbers that come to an end after a few repetitions after the decimal point are known as terminating decimals.
Therefore, \(0.5\) is a terminating decimal.

Q4. In what way can you determine a non-terminating decimal?
Ans: There are two forms of non-terminating decimal expansions, they are
(i) Non-terminating recurring decimal expansion
(ii) Non-terminating non-recurring decimal expansion
Non-terminating recurring decimal expansion:
A non-terminating decimal is a decimal with an infinite number of digits after the decimal point.
A non-terminating, recurring decimal is a decimal in which some digits after the decimal point repeat without terminating. A non-terminating, recurring decimal can be expressed as \(\frac{p}{q}\) form.
Non-terminating non-recurring decimal expansion:
A non-terminating, non-recurring decimal in which the digits after the decimal point do not repeat and do not terminate.

Q5. What is the non-terminating but repeating number?
Ans: A decimal fraction that will never end but will predictably repeat one or more values after the decimal point.
Example: \(3.454545…\) and \(0.234324324…\)

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