Recurring Decimals Notes: Recurring decimals are referred to as numbers that are uniformly repeated after the decimal. Decimal numbers are the standard form of representing non-integer numbers. A decimal number can be expressed as a fraction. A decimal number is a number in which a decimal point separates the whole number part and the fractional part. We can represent rational numbers and irrational numbers in the form of decimals. Some rational numbers produce recurring decimals after converting them into decimal numbers, but all irrational numbers produce recurring decimals after converting them into decimal form. Let us know about decimals, types of decimals, and more about recurring decimals.

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Recurring Decimals Notes: Decimal Numbers Definition

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

Therefore, we can describe decimal numbers as a number that has a decimal point followed by digits that show the fractional part. The number of digits in the decimal part determines the number of decimal places.
For example, \(0.1, 2.03, 11.12\), etc., are the decimal numbers.

Recurring Decimals Notes: Decimal Numbers Types

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

1. Terminating decimals
2. 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 represent these decimal numbers in \(\frac{p}{q}\)  form where \(q \ne 0\), or we can represent decimal numbers as rational numbers.

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

\(2.3\) is represented as \(\frac{23}{10}\), 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 \(=3\).

\(13.34\) 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 after the decimal point are known as the non-terminating decimal numbers. For example, \(0.3333 \ldots, 4.43333 \ldots, 5.34672310 \ldots\), are examples of non-terminating decimal numbers.

We can classify non-terminating decimal numbers into two types such as recurring decimals and non-recurring decimals.

We can represent recurring decimal numbers in \(\frac{p}{q}\) form where \(q \neq 0\), or we can represent these decimal numbers as rational numbers. We cannot represent non-recurring decimals in \(\frac{p}{q}\) form. We know that the numbers that cannot be represented in \(\frac{p}{q}\) form where \(q \neq 0\) are known as irrational numbers. Thus, we can say that non-terminating non-recurring decimals are irrational numbers.

Recurring Decimals

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 as the recurring decimal numbers.

For example, \(0.111…, 4.444444…, 5.232323…, 21.123123…\) etc., are the recurring decimals.

Period of Recurring Decimals

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

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

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

Periodicity of Recurring Decimals

The number of repeating digits after 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 \(4.444444…, 4\) is the period, and the periodicity is \(1\), as only one digit is repeating after the decimal point.
In \(5.232323…, 23\) is the period, and the periodicity is \(2\).
In \(21.123123…, 123\) is the period, and the periodicity is \(3\).
Period and periodicity are very crucial to know while converting the recurring decimal to fraction.

Recurring Decimals: Properties

1. These decimal numbers are pure periodic. It means after the decimal point, the digits/digit are repeating in an equal interval.
2. We can write recurring decimals by putting a bar sign or dots over the digits repeating after the decimal point.
3. We can write recurring decimals in the form of rational numbers. 

Conversion of a Rational Number to a Recurring Decimal

Let 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 \(2\) or \(5\) or both as factors. However, if the denominator has some other factors than \(2\) and \(5\), then the rational number is a repeating decimal or non-terminating recurring decimal.

Repeating Decimal example, consider \(\frac{{10}}{3}\)

We got \(\frac{10}{3}=3.333 \ldots\)

In \(3.333 \ldots .\), the period is \(3\), and the periodicity is \(1 \).

Now consider \(\frac{1}{7}\).

\(\frac{1}{7}=0.142857 \ldots\)

In \(0.142857 \ldots .\), the period is \(142857\), and the periodicity is \(6\).

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 after the decimal point.

Example: \(0.33333 \ldots\)
We need to convert the decimal in \(\frac{p}{q}\) form.
Let us say \(x=0.33333 \ldots…(1)\)
Here, periodicity is \(1\). That is, there is only one digit that is repeating. So, we will multiply it by \(10\). The periodicity decides the power of \(10\) with which the decimal number has to be multiplied.
Now, \(10 x=3.33333 \ldots…(2)\).
Now, subtracting \((1)\) from \((2)\) we have,
\(10 x-x=3.33333 \ldots .-0.33333 \ldots\)
\(\Rightarrow 10 x-x=3\)
\(\Rightarrow 9 x=3\)
\(\Rightarrow x=\frac{3}{9}\) or, \(x=\frac{1}{3}\)
\(\therefore 0.33333 \ldots=0 . \overline{3}=\frac{1}{3}\)

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

Say if the number is \(2.13636363 \ldots\)
We need to convert the decimal in \(\frac{p}{q}\) form.
Let us say \(x=2.13636363 \ldots…(1)\)
Here, periodicity is \(2 \). So, we will multiply the decimal by \(100\) .
Now, \(100 x=213.636363 \ldots…(2)\)
Now, subtracting \((1)\) from \((2)\) we have,
\(100 x-x=213.636363 \ldots-2.13636363 \ldots \).
\(\Rightarrow 99 x=211.5\)
\(\Rightarrow x=\frac{211.5}{99}\)
To remove the decimal point in the numerator, multiply both numerator and denominator by \(10\).
\(\Rightarrow x=\frac{2115}{99 \times 10}=\frac{2115}{990}\)
We can reduce this to standard form.
\(x = \frac{{2115 \div 5}}{{990 \div 5}} = \frac{{423}}{{198}}\)
\( = \frac{{423 \div 3}}{{198 \div 3}} = \frac{{141}}{{66}}\)
\(=\frac{141 \div 3}{66 \div 3}=\frac{47}{22}\)
\(\therefore 2.13636363 \ldots=2.1 \overline{36}=\frac{47}{22}\)

Solved Example Questions on Recurring Decimals

Find some recurring decimals examples with solutions below:

Q.1: Select the recurring decimals from the following.
\(22.4666666 \ldots ., 1.444444 \ldots, 1.4,5.67432145 \ldots\)
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 \(2 . \overline{6}\) into a fraction.
Ans: Given, \(2 . \overline{6}\)
\(2.\overline{6}=2.6666666 \ldots \ldots\)
We need to convert the decimal in \(\frac{p}{q}\) form.
Let us say \(x=2.6666666 \ldots…(1)\)
There is only one digit that is repeating. So, we will multiply it by \(10\).
Now, \(10 x=26.66666 \ldots…(2)\)
Now, subtracting \((1)\) from \((2)\) we have,
\(10 x-x=26.66666 \ldots .-2.666666\)
\(\Rightarrow 9 x=24\)
\(\Rightarrow x=\frac{24}{9}\) or, \(x=\frac{8}{3}\)

Q.3: Convert \(0 . \overline{5}\) into fraction.
Ans: Given, \(0 . \overline{5}\)
\(0 . \overline{5}=0.5555555 \ldots \ldots\)
We need to convert the decimal in \(\frac{p}{q}\) form.
Let us say \(x=0.5555555 \ldots \ldots \ldots…(1)\)
There is only one digit that is repeating. So, we will multiply it by 10.
Now, \(10 x=5.555555 \ldots \ldots…(2)\)
Now, subtracting \((1)\) from \((2)\) we have,
\(10 x=5+0.55555 \ldots\)
\(\Rightarrow 10 x=5+x\)
\(\Rightarrow 10 x-x=5\)
\(\Rightarrow 9 x=5\)
\(\Rightarrow x=\frac{5}{9}\)

Q.4: Convert \(3.2 \overline{45}\) into a rational number.
Ans: Given, \(3.2 \overline{45}\)
\(3.2 \overline{45}=3.2454545 \ldots \ldots\)
We need to convert the decimal in \(\frac{p}{q}\) form.
Let us say \(x=3.2454545 \ldots..\)
There are two digits that are repeating. So, we will multiply it by \(100\).
Now, \(100\,x = 324.54545….\)
\( \Rightarrow 100\,x – x = 324.54545….. – 3.2454545……\)
\( \Rightarrow 99\,x = 321.3\)
\(\Rightarrow x=\frac{3213}{99 \times 10}=\frac{3213}{990}=\frac{1071}{330}=\frac{357}{110}\)
\(\therefore 3.2 \overline{45}=\frac{357}{110}\)

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

Summary

This article covered the definition of decimal, types of decimals, and recurring decimals. We have discussed that all non-terminating decimals are not recurring decimals. We have learned the conversion of recurring decimals to fractions.

FAQs on Equivalent Decimals

Frequently asked questions related to decimals is listed as follows:

Q.1: What is recurring and non-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 as the recurring decimal numbers. Non-recurring decimals can not be 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 point, the digits/digit are repeating in an equal interval.

Q.3: Explain recurring decimal with an example.
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.
\(0.111…, 4.444444…, 5.232323…, 21.123123…\) are examples of recurring decimals.

Q.4. Is \(0.4\) a terminating decimal?
Ans: \(0.4\) can be represented as \(\frac{4}{10}\), and there is a finite number after the decimal point that is \(4\). Therefore, \(0.4\) is a terminating decimal number.

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

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