Is 0.4 A Terminating Decimal

Written By Madhurima Das
Final Modified 17-ten-2022

Recurring Decimals: Check Types, Conversion and Solved Examples

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Recurring Decimals Notes: Recurring decimals are referred to as numbers that are uniformly repeated after the decimal. Decimal numbers are the standard grade of representing not-integer numbers. A decimal number can exist expressed as a fraction. A decimal number is a number in which a decimal bespeak separates the whole number part and the partial function. We can represent rational numbers and irrational numbers in the form of decimals. Some rational numbers produce recurring decimals later converting them into decimal numbers, but all irrational numbers produce recurring decimals after converting them into decimal form. Permit usa know virtually decimals, types of decimals, and more about recurring decimals.

In this article, nosotros have provided Recurring Decimals Notes. Students can view the notes and grasp the Concepts of the chapter Recurring Decimals in a better manner. Continue reading this article and learn everything about Recurring Decimals.

Recurring Decimals Notes: Decimal Numbers Definition

The numbers expressed in decimal forms are known as decimals. The decimals can too exist considered fractions only when the denominators are \(10, 100, 1000\), etc.

Therefore, nosotros tin depict decimal numbers as a number that has a decimal point followed past digits that show the fractional function. The number of digits in the decimal part determines the number of decimal places.
For example, \(0.ane, 2.03, 11.12\), etc., are the decimal numbers.

Recurring Decimals Notes: Decimal Numbers Types

There are two different types of decimal numbers. They are,

Terminating decimals
Non-terminating decimals
(a) Non-terminating recurring decimal
(b) Non-terminating non-recurring decimals

Terminating Decimal Numbers

The decimal numbers having finite numbers of digits after the decimal point are known as the terminating decimal numbers. Their number of decimal places is finite. These decimal numbers are called exact decimal numbers.

We can correspond these decimal numbers in \(\frac{p}{q}\) form where \(q \ne 0\), or nosotros can represent decimal numbers as rational numbers.

For case, \(2.3, 4.433, 13.34\) are the terminating decimal numbers.

\(two.three\) is represented as \(\frac{23}{ten}\), when \(p=23\) and \(q=10\) and the number of decimal places \(=1\).

\(4.433\) is represented as \(\frac{4433}{1000}\), when \(p=4433\) and \(q=1000\) and the number of decimal places \(=3\).

\(thirteen.34\) is represented as \(\frac{1334}{100}\), when \(p=1334\) and \(q=100\) and the number of decimal places \(=2\).

Non-Terminating Decimal Numbers

The decimal numbers having infinite numbers of digits later on the decimal bespeak are known as the non-terminating decimal numbers. For example, \(0.3333 \ldots, iv.43333 \ldots, 5.34672310 \ldots\), are examples of non-terminating decimal numbers.

We tin classify non-terminating decimal numbers into two types such equally recurring decimals and non-recurring decimals.

We tin can correspond recurring decimal numbers in \(\frac{p}{q}\) class where \(q \neq 0\), or we can correspond these decimal numbers every bit rational numbers. We cannot represent non-recurring decimals in \(\frac{p}{q}\) form. We know that the numbers that cannot exist represented in \(\frac{p}{q}\) grade where \(q \neq 0\) are known every bit irrational numbers. Thus, nosotros can say that not-terminating not-recurring decimals are irrational numbers.

Recurring Decimals

The decimal numbers having infinite numbers of digits afterwards the decimal bespeak, and the digits are repeated at equal intervals after the decimal point are known as the recurring decimal numbers.

For example, \(0.111…, four.444444…, v.232323…, 21.123123…\) etc., are the recurring decimals.

Period of Recurring Decimals

The repeating digit or the fix of repeating digits after the decimal point is called the period of recurring decimals.

For instance, in \(0.111…, ane\) is the menstruation as \(1\) is recurring or repeating after the decimal point infinitely.

Similarly, in \(4.444444…, 4\) is the flow.
In \(5.232323…, 23\) is the menses.
In \(21.123123…, 123\) is the period.

Periodicity of Recurring Decimals

The number of repeating digits afterwards the decimal point in a recurring decimal is called the periodicity of recurring decimals. Or, the number of digits in a period of a recurring decimal is the periodicity.

For example,
In \(iv.444444…, 4\) is the catamenia, and the periodicity is \(1\), every bit only one digit is repeating afterward the decimal indicate.
In \(5.232323…, 23\) is the period, and the periodicity is \(ii\).
In \(21.123123…, 123\) is the period, and the periodicity is \(3\).
Menstruation and periodicity are very crucial to know while converting the recurring decimal to fraction.

Recurring Decimals: Backdrop

These decimal numbers are pure periodic. Information technology means after the decimal indicate, the digits/digit are repeating in an equal interval.
We can write recurring decimals by putting a bar sign or dots over the digits repeating afterwards the decimal indicate.
We tin write recurring decimals in the form of rational numbers.

Conversion of a Rational Number to a Recurring Decimal

Permit us see how to convert the fractions to recurring decimals.

A rational number in its standard form has a terminating decimal representation if its denominator has only \(two\) or \(5\) or both as factors. However, if the denominator has some other factors than \(ii\) and \(5\), then the rational number is a repeating decimal or non-terminating recurring decimal.

Repeating Decimal case, consider \(\frac{{ten}}{iii}\)

We got \(\frac{10}{iii}=3.333 \ldots\)

In \(3.333 \ldots .\), the period is \(iii\), and the periodicity is \(i \).

Now consider \(\frac{ane}{7}\).

\(\frac{one}{7}=0.142857 \ldots\)

In \(0.142857 \ldots .\), the catamenia is \(142857\), and the periodicity is \(half dozen\).

Conversion of Recurring Decimals to Rational Numbers

Let us know how we can convert the recurring decimals into rational numbers.

Case 1: All digits are repeating subsequently the decimal point.

Example: \(0.33333 \ldots\)
We demand to catechumen the decimal in \(\frac{p}{q}\) course.
Permit u.s. say \(x=0.33333 \ldots…(1)\)
Here, periodicity is \(one\). That is, there is only ane digit that is repeating. So, we volition multiply it by \(x\). The periodicity decides the ability of \(10\) with which the decimal number has to exist multiplied.
Now, \(ten 10=3.33333 \ldots…(ii)\).
Now, subtracting \((1)\) from \((2)\) we accept,
\(x ten-ten=3.33333 \ldots .-0.33333 \ldots\)
\(\Rightarrow 10 10-x=3\)
\(\Rightarrow 9 ten=3\)
\(\Rightarrow x=\frac{three}{9}\) or, \(x=\frac{1}{iii}\)
\(\therefore 0.33333 \ldots=0 . \overline{iii}=\frac{i}{3}\)

Instance 2: If after the decimal indicate, one or more digits are constant and the digit/digits after are repeating.

Say if the number is \(2.13636363 \ldots\)
We demand to catechumen the decimal in \(\frac{p}{q}\) form.
Allow usa say \(x=2.13636363 \ldots…(ane)\)
Here, periodicity is \(2 \). So, we volition multiply the decimal by \(100\) .
Now, \(100 ten=213.636363 \ldots…(2)\)
Now, subtracting \((1)\) from \((2)\) we take,
\(100 x-x=213.636363 \ldots-2.13636363 \ldots \).
\(\Rightarrow 99 x=211.v\)
\(\Rightarrow x=\frac{211.v}{99}\)
To remove the decimal point in the numerator, multiply both numerator and denominator by \(10\).
\(\Rightarrow x=\frac{2115}{99 \times 10}=\frac{2115}{990}\)
Nosotros tin reduce this to standard form.
\(x = \frac{{2115 \div 5}}{{990 \div 5}} = \frac{{423}}{{198}}\)
\( = \frac{{423 \div three}}{{198 \div iii}} = \frac{{141}}{{66}}\)
\(=\frac{141 \div 3}{66 \div 3}=\frac{47}{22}\)
\(\therefore 2.13636363 \ldots=2.1 \overline{36}=\frac{47}{22}\)

Solved Example Questions on Recurring Decimals

Find some recurring decimals examples with solutions below:

Q.1: Select the recurring decimals from the following.
\(22.4666666 \ldots ., one.444444 \ldots, ane.4,v.67432145 \ldots\)
Ans:
Recurring decimal numbers are pure periodic. It means after the decimal point, the digits/digit are repeating in an equal interval.
\(1.4\) is a terminating decimal number and \(5.67432145….\) is a non-recurring and non-terminating decimal.
Therefore, the recurring decimal numbers are \(22.4666666….\), and \(1.444444…\)

Q.2: Convert \(ii . \overline{vi}\) into a fraction.
Ans: Given, \(ii . \overline{half dozen}\)
\(2.\overline{half-dozen}=two.6666666 \ldots \ldots\)
Nosotros need to catechumen the decimal in \(\frac{p}{q}\) form.
Let us say \(10=2.6666666 \ldots…(1)\)
At that place is only i digit that is repeating. So, we volition multiply it by \(x\).
Now, \(10 x=26.66666 \ldots…(two)\)
Now, subtracting \((1)\) from \((two)\) we have,
\(10 x-x=26.66666 \ldots .-two.666666\)
\(\Rightarrow 9 x=24\)
\(\Rightarrow x=\frac{24}{9}\) or, \(x=\frac{8}{3}\)

Q.3: Catechumen \(0 . \overline{5}\) into fraction.
Ans: Given, \(0 . \overline{5}\)
\(0 . \overline{5}=0.5555555 \ldots \ldots\)
We need to catechumen the decimal in \(\frac{p}{q}\) form.
Let united states of america say \(x=0.5555555 \ldots \ldots \ldots…(1)\)
There is but i digit that is repeating. So, we will multiply information technology by 10.
At present, \(10 ten=five.555555 \ldots \ldots…(two)\)
Now, subtracting \((ane)\) from \((ii)\) we accept,
\(10 x=five+0.55555 \ldots\)
\(\Rightarrow 10 x=5+x\)
\(\Rightarrow 10 x-10=v\)
\(\Rightarrow ix x=five\)
\(\Rightarrow x=\frac{five}{ix}\)

Q.4: Convert \(3.ii \overline{45}\) into a rational number.
Ans: Given, \(3.two \overline{45}\)
\(three.ii \overline{45}=3.2454545 \ldots \ldots\)
Nosotros need to catechumen the decimal in \(\frac{p}{q}\) form.
Let united states say \(10=three.2454545 \ldots..\)
There are ii digits that are repeating. And then, we will multiply it past \(100\).
At present, \(100\,x = 324.54545….\)
\( \Rightarrow 100\,x – ten = 324.54545….. – 3.2454545……\)
\( \Rightarrow 99\,x = 321.three\)
\(\Rightarrow 10=\frac{3213}{99 \times ten}=\frac{3213}{990}=\frac{1071}{330}=\frac{357}{110}\)
\(\therefore 3.2 \overline{45}=\frac{357}{110}\)

Q.5: Convert \(iv.2 \overline{4}\) into a fraction.
Ans: Given, \(4.2 \overline{4}\)
\(4.ii \overline{4}=4.244444 \ldots…\)
We need to convert the decimal in \(\frac{p}{q}\) course.
Let us say \(x=iv.244444 \ldots…\)
In that location is only one digit that is repeating. So, we will multiply it by \(10\).
Now, \(x ten=42.4444 \ldots…\)
\(\Rightarrow 10 x-10=42.4444 \ldots…-4.244444 \ldots…\)
\(\Rightarrow ix ten=38.2\)
\(\Rightarrow x=\frac{382}{9 \times x}=\frac{382}{90}=\frac{191}{45}\)

Summary

This article covered the definition of decimal, types of decimals, and recurring decimals. We have discussed that all non-terminating decimals are non recurring decimals. Nosotros have learned the conversion of recurring decimals to fractions.

FAQs on Equivalent Decimals

Often asked questions related to decimals is listed every bit follows:

Q.1: What is recurring and not-recurring decimal?
Ans: The decimal numbers having infinite numbers of digits after the decimal point, and the digits are repeated at equal intervals after the decimal point are known every bit the recurring decimal numbers. Non-recurring decimals tin can non exist represented as \(\frac{p}{q}\) form where \(q \neq 0\). You can find further details in the article above.

Q.2: What is a recurring decimal called?
Ans: A recurring decimal is called a repeating decimal, as this decimal number is purely periodic. It means after the decimal bespeak, the digits/digit are repeating in an equal interval.

Q.3: Explain recurring decimal with an instance.
Ans: The decimal numbers having infinite numbers of digits after the decimal point, and are repeated at equal intervals, are known as the recurring decimal numbers.
\(0.111…, iv.444444…, 5.232323…, 21.123123…\) are examples of recurring decimals.

Q.4. Is \(0.iv\) a terminating decimal?
Ans: \(0.4\) can be represented equally \(\frac{4}{ten}\), and in that location is a finite number after the decimal betoken that is \(iv\). Therefore, \(0.4\) is a terminating decimal number.

Q.5. Is \(2 / 3\) a terminating or recurring decimal?
Ans: If \(\frac{2}{iii}\) is converted to the decimal number, we will get \(0.6666 \ldots\) Hither, a digit after the decimal point is repeating infinitely. Hence, \(\frac{2}{iii}\) is not a terminating decimal; it is a recurring decimal.

We hope this detailed article on recurring decimals is helpful to you. If you lot accept any queries on this page, ping us through the annotate box beneath and we volition become back to you as before long as possible.