# Decimal expansion of rational numbers: methods and examples (2023)

• Last modification on 01/24/2023 To expand rational numbers to decimals:A number of 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. The decimal form of a rational number can be either terminating or non-terminating.

The non-terminating decimal form of a rational number can be either a repeating decimal or a nonrecurring decimal. There are some special rules for converting the rational number to its decimal form. In this article we will learn about the decimal expansion of rational numbers.

## Decimal expansion of rational numbers: definition

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:

1. Each of the numbers $$\frac{3}{-2}, \frac{-4}{15}, \frac{-8}{5}, \frac{2}{19}$$ are rational numbers with a minus sign in the numerator or denominator.
2. Zero is a rational number because we can write $$0=\frac{0}{1}$$, which is the quotient of two integers with a nonzero denominator.
3. Every natural number is a rational number. We can write $$1=\frac{1}{1}, 2=\frac{2}{1}, 3=\frac{3}{1}$$ and so on. In general, if $$n$$ is a natural number, then we can write $$n=\frac{n}{1}$$, which is a rational number.
4. Every integer is a rational number. If $$m$$ is an integer, then we can write it as $$\frac{m}{1}$$, which is a rational number.
5. Every fraction is a rational number. Let $$\frac{a}{b}$$ be a fraction. Then $$a$$ and $$b$$ are integers and $$b \neq 0$$.

### How to Expand Rational Numbers to Decimals

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.

(Video) On the decimal expansion of rational numbers.

1. exit decimals
2. nonterminal decimals

1.final decimals
Decimal numbers with a finite number of decimal places are called final decimal numbers. Its number of decimal places is finite. These decimals are called exact decimals. We can represent these decimal numbers in the form $$\frac{p}{q}$$, where $$q \neq 0$$.
For example, $$2.3, 4.43$$ are the final decimal places.
$$2.3$$ is represented as $$\frac{23}{10}$$ if $$p=23$$ and $$q=10$$ and the number of decimal places $$=1$$.
$$4.433$$ is represented as $$\frac{4433}{1000}$$ if $$p=4433$$ and $$q=1000$$ and the number of decimal places $$=3$$.

2.nonterminal decimals
Decimal numbers with an infinite number of decimal places are called endless decimal numbers.
For example, $$0.3333…$$, $$4.43333…$$, $$5.34672310…$$ are examples of decimal numbers with no ending.
We can classify unterminated decimal numbers into two types, e.g. B. Recurring decimal numbers and nonrecurring decimal numbers.

A.repeating decimals
Decimal numbers with an infinite number of digits after the decimal point, and the digits repeat at equal intervals after the decimal point, are called repeating decimal numbers.
For example, $$0.111\ldots, 4.444444…, 5.232323…, 21.123123….$$, etc. are the repeating decimal places.
We can represent repeating decimal numbers in the form $$\frac{p}{q}$$, where $$q \neq 0$$ or we can represent these decimal numbers as rational numbers.

B.single decimals
Decimal numbers with an infinite number of digits after the decimal point and digits that do not repeat at equal intervals after the decimal point are known as nonrecurring decimal numbers. For example, $$0.1223589…, 4.4782451…., 5.67245….$$, etc. They are non-repeating decimal numbers.

We cannot represent single decimal numbers in the form $$\frac{p}{q}$$.
Numbers that cannot be represented in the form $$\frac{P}{q}$$, where $$q \neq 0$$ are known as irrational numbers. Therefore, we can say that the decimal numbers that do not repeat and do not end are irrational numbers.

Read more aboutNature of decimal expansions of rational numbersand about the decimal expansion of 1/2 and 1/3 in this Embibe article.

### Methods to expand rational numbers to decimals

We have already discussed that integers, natural numbers, and integers are also rational numbers, since they can be represented in the form $$\frac{p}{q}$$. Decimal numbers expressed as rational numbers can be repeating decimal numbers with or without a suffix. Let's 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.

(Video) Number System - Decimal Expansion of Rational Numbers

The decimal expansion of $$\frac{1}{3}$$ is $$0.333333\ldots$$.
Here the remainder at each step is $$1$$ 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. Remainder becomes 0 after a certain level or starts looping.
ii. The number of entries in the repeating remainder sequence is less than the divisor.
In $$\frac{1}{3}$$ only one number, namely $$1$$, is repeated as a remainder. In $$\frac{1}{7}$$ there are six entries $$3 , 2,6,4,5,1$$ in the repeating remainder sequence, and $$7$$ is the divisor.
iii. If the remainders are repeated, we get a repeating block of numbers in the quotient. For $$\frac{1}{3}, 3$$ repetitions in the quotient and for $$\frac{1}{7}$$ we obtain the repetition block as $$142857$$.

Although we only noticed this pattern in the previous examples, it holds for all rational numbers of the form $$\frac{p}{q}$$. When dividing $$p$$ by $$q$$ two main things happen, the remainder becomes zero or a repeating sequence of remainders is created.

### Finishing the decimal expansion of rational numbers

Final decimal expansion means that the decimal number ends after a certain number of decimal places. In the example above for $$\frac{7}{8}$$, we found that the remainder becomes zero after just a few steps, and we got the decimal expansion of $$\frac{7}{8}$$. (0.875\) using the long division method. There is a different method than the long division method to find the final decimal places.

A rational number is complete 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 $$5$$ produces a terminating decimal.

(Video) Decimal Representation of Rational Numbers | Don't Memorise

Consider the same example $$\frac{7}{8}$$.
Now the denominator is $$8$$, that is, $$2^{3}$$. To get the power of the denominator $$10$$, we need to multiply the denominator and numerator by $$5^{3}$$.
Also $$\frac{7 \times 5^{3}}{2^{3} \times 5^{3}}=\frac{875}{1000}=0.875$$.

### Non-terminating decimal expansion of rational numbers

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 $$5$$ produces an endless repeating decimal. In other words, we have a repeating block of numbers in the quotient part. We can say that the expansion is uninterrupted and recurring.

In $$\frac{1}{3} 3$$ the quotient is repeated after the comma. Therefore $$\frac{1}{3}=0 . \overline{3}$$. Similarly, in $$\frac{1}{7}$$, since the number block $$142857$$ is repeated as a quotient, we can say $$\frac{1}{7}=0 . \overline { 142857 }$$. The bar over the digits indicates that the number block is repeated.

So we can see that the decimal expansion of rational numbers has two possibilities, either they end or they don't repeat.

### Worked examples of expanding rational numbers to decimals

Let's see some examples of decimal expansion of rational numbers:

Q.1. Select the decimal expansion of the rational numbers from the following.$$1.444444…, 1.4, 5.67432145 \ldots$$
Answer:The decimal numbers that can be expressed as rational numbers are repeating decimal numbers with or without a suffix. Decimal numbers that do not repeat and do not terminate and that cannot be expressed in the form $$\frac{p}{q}$$ are called irrational numbers.
$$1.444444\ldots$$ is a repeating non-terminating decimal number. Therefore, it can be expressed as a rational number. $$1.4$$ is a final decimal number. $$1.4$$ is the decimal expansion form of a rational number. But $$5.67432145\ldots$$. is a single, non-terminating decimal number. Therefore, it cannot be expressed as a rational number. Therefore, the decimal expansion of the rational numbers is $$1.4, 1.44444...$$.

F.2. Find the decimal form of a rational number$$\frac{3}{4}$$.
Answer:A rational number is complete 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 $$5$$ produces a terminating decimal. Well, in $$\frac{3}{4}$$ the denominator is $$4$$, so $$2^{2}$$. To get the power of the denominator $$10$$, we need to multiply the denominator and numerator by $$5^{2}$$.
Also $$\frac{3 \times 5^{2}}{2^{2} \times 5^{2}}=\frac{75}{100}=0.75$$
Therefore, the decimal expansion form is $$0.75$$.

F.3. Find the decimal form of a rational number$$\frac{5}{{16}}$$
Answer:A rational number is complete 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 $$5$$ produces a terminating decimal. Well, in $$\frac{5}{16}$$ the denominator is $$16$$, so $$2^{4}$$. To get the power of the denominator $$10$$, we need to multiply the denominator and numerator by $$5^{4}$$.
Also $$\frac{5 \times 5^{4}}{2^{4} \times 5^{4}}=\frac{3125}{10000}=0.3125$$
Therefore, the decimal expansion form is $$0.3125$$.

F.4. Find the decimal form of a rational number$$\frac{10}{3}$$.
Answer:A rational number is a final 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 $$5$$ produces an endless repeating decimal. Well, in $$\frac{10}{3}$$ the denominator is $$3$$, not the factor of $$2$$ and $$5$$. Therefore, to find the decimal form, we need to use the long division method.
For example, consider $$\frac{{10}}{3}$$.

(Video) How to Convert Fractions to Decimals

We have $$\frac{10}{3}=3.333 \ldots$$
In $$3.333 \ldots .$$ the period is $$3$$ and the periodicity is $$1$$.
Therefore, the decimal expansion form is $$3. \overline{3}$$.

F.5. Find the decimal form of a rational number$$\frac{2}{25}$$.
Answer:A rational number is complete 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 $$5$$ produces a terminating decimal. Well, in $$\frac{2}{25}$$ the denominator is $$25$$, so $$5^{2}$$. To get the power of the denominator $$10$$, we need to multiply the denominator and numerator by $$2^{2}$$.
Also $$\frac{2 \times 2^{2}}{5^{2} \times 2^{2}}=\frac{8}{100}=0.08$$
Therefore, the decimal expansion form is $$0.08$$.

### SSummary of expanding rational numbers to decimals

In this article, we learned about rational numbers, types of rational decimal expansions, and methods for converting rational numbers to decimal form. We learned that we can use the long division method to find the decimal form of any rational number. We have discussed that all nonterminal decimal numbers cannot be expressed as a rational number.

## Frequently Asked Questions (FAQ) - Decimal expansion of rational numbers

The most frequently asked questions about the decimals of a rational number are listed below:

Q.1: How do you find the decimal place of a rational number?
Using the long division method, we can find the decimal form of any rational number. A unique approach is to find the final decimal expansion of a rational number by checking that the denominator has no prime factors other than $$2$$ and $$5$$. The rational number whose denominator has prime factors other than $$2$$ and $$5$$ produces a non-terminating repeating decimal.

Q.2: What is the decimal expansion of an irrational number?
Answer:The decimal expansion of an irrational number does not always terminate or repeat. Decimal numbers that do not repeat and do not terminate and that cannot be expressed in the form $$\frac{p}{q}$$ are called irrational numbers.

Q.3: What types of decimal expansion are there?
There are two different types of decimal expansion. They are,
1. Leave the decimals
2. Non-terminal decimals
A. Nonterminal repeating decimals
B. Nonrecurring and Nonterminal Decimals

Q.4: Can we represent all nonterminal decimal numbers as rational numbers?
No, we can not. Only non-terminating repeated decimal numbers can be expressed as a rational number.

Q.5: What kind of decimal expansion does a rational number have?
Decimal numbers expressed as rational numbers are repeating decimal numbers with or without a suffix. Decimal numbers that do not repeat and do not terminate and that cannot be expressed in the form $$\frac{p}{q}$$ are called irrational numbers.

We hope you find this detailed article on the decimal expansion of rational numbers useful. If you have any doubts or questions about this concept, feel free to ask us in the comment section.

## FAQs

### How do you solve the decimal expansion of a rational number? ›

The decimal expansion of a rational number is obtained by dividing the numerator by the denominator of the rational number.

What type of decimal expansions the rational numbers have give examples? ›

Rational number 3/6 results in a terminating decimal. Example: Express 5/13 in decimal form. A rational number gives either terminating or non-terminating recurring decimal expansion. Thus, we can say that a number whose decimal expansion is terminating or non-terminating recurring is rational.

What is decimal expansion and examples? ›

Decimal expansion is the form of a number that has a decimal point, either actual or implied. Examples of numbers with actual decimal points are 10.2 and 0.0084. An example of a number with implied decimal points is the whole number 17, which could actually be written as 17.000000000000.

What are examples of rational numbers in decimal form? ›

Rational numbers can be represented as decimals. The different types of rational numbers are Integers like -1, 0, 5, etc., fractions like 2/5, 1/3, etc., terminating decimals like 0.12, 0.625, 1.325, etc., and non-terminating decimals with repeating patterns (after the decimal point) such as 0.666..., 1.151515..., etc.

What is the decimal expansion of rational number 14587 by 1250? ›

Hence, given rational number will terminate after 4 decimal places.

What is the decimal expansion of the rational number 43 2 4 * 5 3? ›

Solution : (43)/(2^(4) xx 5^(3)) = (43 xx 5)/(2^(4) xx 5^(4)) = (215)/(10^(4)) = 0.0215 <br> therefore It will terminate after 4 places of decimals. Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

What is terminating and non-terminating decimal expansion with examples? ›

Terminating and Non-Terminating Decimals

A terminating decimal is a decimal, that has an end digit. It is a decimal, which has a finite number of digits(or terms). Example: 0.15, 0.86, etc. Non-terminating decimals are the one that does not have an end term.

How to identify decimal expansion of rational and irrational numbers? ›

The decimal expansion of rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. The decimal expansion of an irrational number is non-terminating non-recurring.

What are some examples of expansion? ›

• Thermal expansion of railway tracks in summer.
• Thermal expansion of metallic joints in a bridge.
• The rise in the level of mercury in a thermometer.
• The gaps left between the walls in buildings.
• The design of the wings of a jet plane.

What are examples of decimal numbers? ›

Numbers such as 7, 28, 199 and 532.11 are all examples of decimal numbers. The decimal system is also referred to as the Hindu-Arabic system. Additionally, the term decimal is often used to refer to a fraction that is represented as a number in the decimal system, such as 19.368.

### What is a decimal expansion in math? ›

Decimal expansion may sound complicated, but it only means turning a fraction or a whole number into its decimal representation. This is helpful if you are working with a calculator or want to perform operations on numbers in a decimal format.

What are 10 examples of rational numbers? ›

Any fraction with non-zero denominators is a rational number. Some of the examples of rational numbers are 1/2, 1/5, 3/4, and so on. The number “0” is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc. are not rational, since they give us infinite values.

What are 3 examples of rational numbers? ›

Any number in the form of p/q where p and q are integers and q is not equal to 0 is a rational number. Examples of rational numbers are 1/2, -3/4, 0.3, or 3/10.

What are the two types of decimals that are rational numbers? ›

Decimal expansions for rational numbers can be either terminating or repeating decimals. Terminating Decimal: ↑ A decimal expansion that only has a finite number of non-zero decimal digits. For example, 3.125 is a terminating decimal.

What is the decimal expansion of 13 by 6250? ›

136250= 13×24105=208105= 0.00208.

What is the decimal expansion of 15 upon 1600? ›

Therefore, the decimal expansion of the given number is 0.009375.

What is the decimal expansion of 6 by 1250? ›

Therefore, Decimal representation of $\dfrac{6}{{1250}}$ is $0.0048$. Hence the representation will terminate after four decimal places.

How do you write 2.034 in expanded form? ›

2.034=2×1+0×110+3×1100+4×11000
1. Write the following decimal numbers in the expanded form: (i) 20.03 (ii) 2.03 (iii) 200.03. ...
2. Write the following decimal numbers in expanded from: 5.55 = 5+510+5100.
3. Question 1(iv) Write the following numbers in the expanded form: ...
4. Question 6 (iv) ...
5. Question 1(i)

Has the rational number 441 2² * 5⁷ * 7² a terminating or a non terminating decimal representation? ›

Hence, it has non-terminating decimal expansion.

What is the decimal expansion of 257 upon 5000? ›

The denominator of the given rational number is 5000. ∴ 257/5000 = (257 x 2)/(5000 x 2) = 514/10000 = 0.0514 is its decimal expansion.

### How do you find if a decimal expansion is terminating or non terminating? ›

Just divide the numerator by the denominator . If you end up with a remainder of 0 , then you have a terminating decimal. Otherwise, the remainders will begin to repeat after some point, and you have a repeating decimal.

Which of the following is an example of terminating decimal expansion? ›

The rational number has terminating decimal expansion.

How do you write 67.83 in expanded form? ›

What is 123.45 in expanded form? ›

123.45 = 12345 × 102. The same value can also be represented in normalized form with 1.2345 as the fractional coefficient, and +2 as the exponent (and 10 as the base): 123.45 = 1. 2345 × 10+2.

What are the 3 types of decimal expansion? ›

The three different classifications of decimal expansion are terminating decimals, non-terminating repeating decimals and non-terminating non-repeating decimals.

Which of the following decimal expansion are rational? ›

(i) All whole numbers are rational numbers. (ii) Negative integers are not rational numbers. (iii) Zero is not a rational number.

What are 5 examples of irrational numbers? ›

Example: √2, √3, √5, √11, √21, π(Pi) are all irrational.

What is decimal expansion of rationals? ›

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. The decimal numbers having finite numbers of digits after the decimal point are known as the terminating decimal numbers.

How do you find the decimal expansion? ›

The decimal expansion terminates or ends after finite numbers of steps. Such types of decimal expansion are called terminating decimals. It means that, after the decimal point, the numbers come to an end at a certain point. For example, ½ is a rational number and its decimal expansion is 0.5.

How do you find the decimal expansion of a rational number without actual division? ›

Hint: To find the decimal expansion of a number without actually dividing the numbers, we will multiply both the numerators and denominators with a constant k. We will first find the value of k and then we will multiply such that the denominator becomes in the form of ${10^n}$ where n is an integer.

### How do you find the expanding decimal of an irrational number? ›

The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational. S = 0.10110111011110… (non-terminating and non-recurring).

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