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How to Expand Rational Numbers in Decimals: A number in the form \(\frac{p}{q}\) or a number that can be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers, and \(q \neq 0\) is called a rational number. A rational number can be expressed as a decimal number. The decimal form of a rational number can be terminating or non-terminating.
The non-terminating decimal form of a rational number can be a recurring decimal or a non-recurring decimal. There are some specific rules to convert the rational number into its decimal form. In this article, we will know about the decimal expansion of rational numbers.
The numbers of the form \(\frac{p}{q}, w\) where \(p\) and \(q\) are integers, and \(q \neq 0\) are called rational numbers. Some examples of rational numbers are,
When the numerator of a rational number is divided by its denominator, we get the decimal expansion of the rational number. The decimal numbers thus obtained can be of two types.
1. 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 \(q \neq 0\).
For example, \(2.3, 4.43\) 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 \(=3\).
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, \(0.3333…\), \(4.43333…\), \(5.34672310…\), 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.
a. 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, \(0.111\ldots, 4.444444…, 5.232323…, 21.123123….\) etc., are the 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.
b. Non-recurring Decimals
The decimal numbers having infinite numbers of digits after the decimal point and the digits are not repeated at equal intervals after the decimal point are known as the non-recurring decimal numbers. For example, \(0.1223589…, 4.4782451…., 5.67245….\), etc., are non-recurring decimals.
We cannot represent non-recurring decimals in \(\frac{p}{q}\) form.
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.
Read more about Nature of the Decimal Expansions of Rational Numbers and about the Decimal expansion of 1/2 and 1/3 in this article on Embibe.
We discussed earlier that the whole numbers, natural numbers, integers are also rational numbers as these can be represented as \(\frac{p}{q}\) form. The decimal numbers that are expressed as rational numbers can be terminating or non-terminating recurring decimals. Let us take some examples of rational numbers and find their decimal expansion.
Example 1: Find the decimal expansion of \(\frac{1}{3}\)
Divide the numerator by the denominator.
The decimal expansion of \(\frac{1}{3}\) is \(0.333333\ldots\).
Here, the remainder is \(1\) in every step, and the divisor is \(3\).
Example 2: Find the decimal expansion of \(\frac{1}{7}\).
The decimal expansion of \(\frac{1}{7}\) is \(0.142857 \ldots\)
Remainders: \(3,2,6,4,5,1,3,2,6,4,5,1\), and so on the divisor is \(7\).
Example 3: Find the decimal expansion of \(\frac{7}{8}\).
The decimal expansion of \(\frac{7}{8}\) is \(0.875\). Remainders: \(6,4,0\)
The divisor is \(8\).
Now, we can see three things:
i. The remainder either become 0 after a particular stage or start repeating themselves.
ii. The number of entries in the repeating string of remainders is less than the divisor.
In \(\frac{1}{3}\), only one number, i.e., \(1\), repeats itself as remainder. In \(\frac{1}{7}\), there are six entries \(3,2,6,4,5,1\) in the repeating string of remainders, and \(7\) is the divisor.
iii. If the remainders repeat, then we get a repeating block of digits in the quotient. For \(\frac{1}{3}, 3\) repeats in the quotient, and for \(\frac{1}{7}\), we get the repeating block as \(142857\).
Although we have noticed this pattern using only the examples above, it is true for all rational numbers of the form of \(\frac{p}{q}\). On the division of \(p\) by \(q\), two main things happen, either the remainder becomes zero or gets a repeating string of remainder.
The terminating decimal expansion means the decimal number terminates after a particular number of digits after the decimal point. In the earlier example of \(\frac{7}{8}\), we found that the remainder becomes zero after some steps, and we got the decimal expansion of \(\frac{7}{8}\) is \(0.875\) using the long division method. There is another method other than the long division method to find the terminating decimal numbers.
A rational number is terminated if it can be represented as \(\frac{p}{2^{n} \times 5^{m}}\). The rational number whose denominator has no prime factors other than \(2\) and \(5\) gives a terminating decimal number.
Consider the same example \(\frac{7}{8}\).
Now, the denominator is \(8\) that means \(2^{3}\). To make the denominator \(10\)’s power, we need to multiply the denominator and the numerator by \(5^{3}\).
So, \(\frac{7 \times 5^{3}}{2^{3} \times 5^{3}}=\frac{875}{1000}=0.875\).
In the decimal expansion of some rational numbers, the remainder never becomes zero. The rational number whose denominator has prime factors other than \(2\) and \(5\) gives a non-terminating recurring decimal number. In other words, we have a repeating block of digits in the quotient part. We can say the expansion is non-terminating and recurring.
In \(\frac{1}{3} 3\) repeats in the quotient after the decimal point. Therefore, \(\frac{1}{3}=0 . \overline{3}\). Similarly, in \(\frac{1}{7}\) since the block of digits \(142857\) repeats as the quotient, we can say, \(\frac{1}{7}=0 . \overline{142857}\). The bar above the digits indicates that the block of digits repeats.
So, we can see that the decimal expansion of the rational numbers has two choices, either they are terminating, or they are non-terminating recurring.
Let us look at some of the examples of decimal expansion of rational numbers:
Q.1. Select the decimal expansion of rational numbers from the following. \(1.444444…, 1.4, 5.67432145 \ldots\)
Ans: The decimal numbers that can be expressed as rational numbers are either terminating or non-terminating recurring decimals. The non-terminating non-recurring decimals that cannot be expressed in \(\frac{p}{q}\) form are called irrational numbers.
\(1.444444\ldots\) is non terminating recurring decimal. So, it can be expressed as a rational number. \(1.4\) is a terminating decimal number. So, \(1.4\) is the decimal expansion form of a rational number. But \(5.67432145\ldots\). is a non-recurring and non-terminating decimal. So, it can not be expressed as a rational number. Therefore, the decimal expansion of rational numbers is \(1.4, 1.44444…\).
Q.2. Find the decimal form of a rational number \(\frac{3}{4}\).
Ans: A rational number is terminated if it can be represented as \(\frac{p}{2^{n} \times 5^{m}}\). The rational number whose denominator has no factors other than \(2\) and \(5\) gives a terminating decimal number. Now, in \(\frac{3}{4}\) the denominator is \(4\) that means \(2^{2}\). To make the denominator \(10\)’s power, we need to multiply the denominator and the numerator by \(5^{2}\).
So, \(\frac{3 \times 5^{2}}{2^{2} \times 5^{2}}=\frac{75}{100}=0.75\)
Hence, the decimal expansion form is \(0.75\).
Q.3. Find the decimal form of a rational number \(\frac{5}{{16}}\)
Ans: A rational number is terminated if it can be represented as \(\frac{p}{2^{n} \times 5^{m}}\). The rational number whose denominator has no factors other than \(2\) and \(5\) gives a terminating decimal number. Now, in \(\frac{5}{16}\) the denominator is \(16\) that means \(2^{4}\). To make the denominator \(10\)’s power, we need to multiply the denominator and the numerator by \(5^{4}\).
So, \(\frac{5 \times 5^{4}}{2^{4} \times 5^{4}}=\frac{3125}{10000}=0.3125\)
Hence, the decimal expansion form is \(0.3125\).
Q.4. Find the decimal form of a rational number \(\frac{10}{3}\).
Ans: A rational number is a terminating decimal if it can be represented as \(\frac{p}{2^{n} \times 5^{m}}\). The rational number whose denominator has prime factors other than \(2\) and \(5\) gives a non-terminating recurring decimal number. Now, in \(\frac{10}{3}\) the denominator is \(3\) that is not the factor of \(2\) and \(5\). Therefore, to find its decimal form, we need to apply the long division method.
For 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 \(1\).
Hence, the decimal expansion form is \(3. \overline{3}\).
Q.5. Find the decimal form of a rational number \(\frac{2}{25}\).
Ans: A rational number is terminated if it can be represented as \(\frac{p}{2^{n} \times 5^{m}}\). The rational number whose denominator has no factors other than \(2\) and \(5\) gives a terminating decimal number. Now, in \(\frac{2}{25}\) the denominator is \(25\) that means \(5^{2}\). To make the denominator \(10\)’s power, we need to multiply the denominator and the numerator by \(2^{2}\).
So, \(\frac{2 \times 2^{2}}{5^{2} \times 2^{2}}=\frac{8}{100}=0.08\)
Hence, the decimal expansion form is \(0.08\).
In this article, we have learned about rational numbers, types of rational decimal expansions, the methods to convert rational numbers to the decimal form. We learnt that by using the long division method, we can find the decimal form of any rational number. We have discussed every non-terminating decimals can not be expressed as a rational number.
Frequently asked questions related to decimals of a rational number is listed as follows:
Q.1: How do you find the decimal of a rational number?
Ans: Using the long division method, we can find the decimal form of any rational number. A unique approach is to find the terminating decimal expansion of a rational number is to check whether the denominator has no prime factors other than \(2\) and \(5\). The rational number whose denominator has prime factors other than \(2\) and \(5\) gives a non-terminating recurring decimal.
Q.2: What is the decimal expansion of an irrational number?
Ans: The decimal expansion of an irrational number is always non-terminating and non-recurring. The non-terminating non-recurring decimals that cannot be expressed in \(\frac{p}{q}\) form are called irrational numbers.
Q.3: What are the types of decimal expansion?
Ans: There are two different types of decimal expansion. They are,
1. Terminating decimals
2. Non-terminating decimals
a. Non-terminating recurring decimal
b. Non-terminating non-recurring decimals
Q.4: Can we represent all non-terminating decimal numbers as rational numbers?
Ans: No, we cannot. Only non-terminating recurring decimals can be expressed as the rational number.
Q.5: What type of decimal expansion does a rational number have?
Ans: The decimal numbers that are expressed as rational numbers are terminating or non-terminating recurring decimals. The non-terminating non-recurring decimals that cannot be expressed in \(\frac{p}{q}\) form are called irrational numbers.
We hope this detailed article on decimal expansion of rational numbers proves to be helpful. If you have any doubts or queries regarding this concept, feel to ask us in the comment section.
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