Decimal Expansions of Real Numbers: In Mathematics, numbers are the main building blocks. The complete number system is divided into real numbers and imaginary numbers. Imaginary numbers are numbers that cannot be represented on the number line. On the contrary, real numbers are the number that can be represented on the number line.

The numbers which can be written in the form of  \(\frac{a}{b},b \ne 0\) where \(a, b\) are integers are known as rational numbers. We get the decimal form of real numbers if we simplify the real numbers further. 

Decimal Expansions of Real Numbers: Definition

The decimal expansion of real numbers is nothing but simplifying the real numbers further and writing in the decimal form, whose values are numerically equal to the real number. We can write the decimal expansion of real numbers by using the long division method. 

In the long division method, divide the numerator of the real number by the denominator of the real number and observe the quotient obtained, which is known as the decimal value of the given real numbers.

Example: The decimal expansion of the real number \(\frac{1}{2}\) is given by \(0.5.\)

What are Real Numbers?

The complete number system is divided into real numbers and imaginary numbers. Imaginary numbers are numbers that cannot be represented on the number line. On the contrary, real numbers are numbers that can be represented on the number line.

Simply, we can say that any numbers that can be found in our real world are known as real numbers. In mathematics, real numbers are the combination of both rational numbers and irrational numbers.

Numbers which can be written in the form of \(\frac{a}{b},b \ne 0,\) where \(a, b\) are integers are known as rational numbers. The numbers which cannot be written in the form of \(\frac{a}{b}\) are known as irrational numbers. Irrational numbers are real numbers other than rational numbers.

Example:
\(\sqrt {11} ,\sqrt 2 ,\pi , \ldots .\)etc.

Real Numbers and Their Decimal Expansions

Any numbers that can be found in our real world are known as real numbers. The decimal expansion of real numbers is nothing but simplifying the real numbers further and writing in the decimal form, whose values are numerically equal to the real number. 

We can write the decimal expansion of real numbers by using the long division method. In the long division method, divide the numerator of the real number by the denominator of the real number and observe the quotient obtained, which is known as the decimal value of the given real numbers.

Any real numbers can be expressed in the form of \(\frac{p}{q},\) in which \(q≠0,\) dividing the value of \(p\) by \(q,\) we have two types of decimal expansions. They are

1. Terminating decimals
2. Non-terminating decimals

In the long division method, while dividing the numerator by the denominator, if we get the remainder as zero, then the quotient obtained is known as terminating decimal. While the remainder obtained is not zero and continues, the division is known as non-terminating decimals.

Non-terminating numbers are of two types: non-terminating recurring (repeating) and non-terminating non-recurring (repeating) decimals. All terminating numbers and non-terminating recurring numbers are known as rational numbers. And, the non-terminating non-recurring numbers are known as irrational numbers.

Decimal Representation of Terminating Numbers

A terminating decimal number is a number obtained in the decimal form that has a finite number of terms after the decimal point. The decimal expansion of the rational number with a fixed number of digits after the decimal point is known as the terminating decimal number.

All rational numbers are terminating decimal numbers. The decimal expansion of the rational number has a finite number of terms after the decimal point.

Example: The decimal expansion of a rational number \(\frac{1}{4} = 0.25\) is terminating decimal as it has only two digits after the decimal point.

Decimal Representation of Non-Terminating Numbers:

Rational numbers in which the decimal expansion has an infinite or not finite number of digits after the decimal point are known as non-terminating decimal numbers. There are two types of non-terminating decimals

1. Non-terminating recurring decimal
2. Non-terminating non-recurring decimal

If one or more numbers or the patterns of the numbers are repeating after the decimal point are known as non-terminating recurring decimals. And, these are known as rational numbers.

The decimal form of any numbers, which are non-terminating and non-recurring, are known as irrational numbers.

Example: The decimal expansion of the real number \(\frac{4}{9}\) is \(0.4444444……\)
Here, digit \(4\) is repeating after the decimal point.

Decimal Expansion of Irrational Numbers

The numbers which cannot be written in the form of \(\frac{a}{b}\) are known as irrational numbers. Irrational numbers are real numbers other than rational numbers. 

Example: \(\sqrt 2 ,\sqrt 3 \) are irrational numbers.

Irrational numbers have an infinite number of terms or digits after the decimal point. One or more numbers or the patterns of the numbers are not repeating after the decimal point in the decimal expansion of the irrational numbers. 

Thus, the decimal expansion of the irrational numbers is non-terminating and non-recurring decimals.

Example: Pi \((\pi ) = \frac{{22}}{7}\) is an irrational number, and the decimal expansion is \(3.145……,\) which is non-terminating and non-recurring.

Theorems of Decimal Expansion of Real Numbers

The decimal expansion of real numbers is nothing but simplifying the real numbers further and writing in the decimal form, whose values are numerically equal to the real number. We can write the decimal expansion of real numbers by using the long division method. 

Finding the factors of the denominator of any real number in the form of \(\frac{p}{q}\) by prime factorization, we can decide the decimal expansion of the real numbers.

Theorem-1: Let any real number be expressed in the form of \(\frac{p}{q}\) and the prime factorization of the denominator \(q\) expressed in the form of \({2^m} \times {5^n},\) where \(m, n\) are non-negative integers, then we can say that the decimal expansion of the real number is terminating.

Example: \(\frac{3}{4} = 0.75\)

Theorem-2: The decimal expansion of any real number in the form of \(\frac{p}{q}\) is terminating in nature, then the denominator \(q\) expressed in the form of \({2^m} \times {5^n},\) where \(m, n\) are non-negative integers.

Example: \(\frac{3}{{40}} = \frac{3}{{{2^3} \times 5}}\)

Theorem-3: Let any real number can be expressed in the form of \(\frac{p}{q}\) and the prime factorization of the denominator \(q\) cannot be expressed in the form of \({2^m} \times {5^n},\) where \(m, n\) are non-negative integers, then we can say that the decimal expansion of the real number is non-terminating.

Example: \(\frac{1}{6} = 0.16666… = 0.1\underline 6 \)

Solved Examples – Decimal Expansion of Real Numbers

Q.1. Check the given pie chart below (shaded portion). And, observe the given pie chart represents the terminating decimal or not?

Ans: Given pie chart is

The fraction that represents the shaded portion is \(\frac{4}{6}.\)
The simplest form of the above fraction is \(\frac{2}{3}.\)
The prime factor of the denominator \(3\) are not in the form of \({2^m} \times {5^n}.\)
So, the decimal expansion of the given pie chart (shaded portion) is a non-terminating decimal.

Q.2. The length of the rectangle is 2.3 feet, and the breadth of the rectangle is 1.8 feet. Check whether the area of the rectangle is terminating or non-terminating decimal?
Ans: Given the length of the rectangle is \(2.3\) feet.
The breadth of the rectangle is \(1.8\) feet.
We know that area of the rectangle is given by length \( \times \) breadth.
Area\({\rm{ = 2}}{\rm{.3 \times 1}}{\rm{.8 = 4}}{\rm{.14}}\,{\rm{sq}}{\rm{.feet}}\)
The digits after the decimal point are finite.
Hence, the area of the rectangle obtained is terminating decimal.

Q.3. Express the real number \(\frac{1}{{13}}\) in the decimal form.
Ans: Given real number is \(\frac{1}{{13}}\)
Divide the numerator \(1\) by the denominator \(13\) as given below:

\(\therefore \,\,\frac{1}{{13}} = 0.0769230769…. = 0.\underline {076923} \)

Q.4. Write the decimal expansion of \(\frac{{13}}{{3125}},\) that have terminating decimals.
Ans: Given rational number is \(\frac{{13}}{{3125}}\)
Also, given that they have terminating decimal expansion.
Write the prime factors of the denominator \(3125\) of the given number.
\( \Rightarrow \frac{{13}}{{3125}} = \frac{{13}}{{5 \times 5 \times 5 \times 5 \times 5}} = \frac{{13}}{{{5^5}}}\)
Multiply and divide both sides of the number by \({{2^5}}\)
\( \Rightarrow \frac{{13}}{{{5^5}}} \times \frac{{{2^5}}}{{{2^5}}} = 13 \times \frac{{{2^5}}}{{{{10}^5}}}\)
\( \Rightarrow \frac{{13 \times 32}}{{{{10}^5}}} = \frac{{416}}{{{{10}^5}}} = 0.00416\)
The decimal expansion of \(\frac{{13}}{{3125}}\) is \(0.00416\)

Q.5. Check whether the number \(\sqrt 3 \) has terminating decimal or non-terminating decimal?
Ans: Given number is \(\sqrt 3 .\)
The number \(\sqrt 3 \) is an irrational number.
We know that the decimal expansion of irrational numbers is non-terminating and non-recurring.
Therefore, the number \(\sqrt 3 \) has a non-terminating non-recurring decimal.

Summary

In this article, we have studied the real number, which is the collection of rational numbers and irrational numbers. We have discussed the meaning of decimal expansion of real numbers. This article also gives the decimal expansion of rational numbers and irrational numbers. We have studied the decimal expansion of terminating and non-terminating recurring, non-terminating non-recurring numbers.

We have studied the theorems of the decimal expansion of real numbers and solved examples which helps us to understand the concept easily.

Learn the Properties of Real Numbers

Frequently Asked Questions (FAQs) – Real Numbers and Their Decimal Expansions

Q.1. Does every real number have a unique decimal expansion?
Ans: No. Every real number doesn’t have a unique decimal expansion. Irrational numbers are non-terminating non-recurring decimals, and they are unique. Some rational is non-terminating, and some are terminating.

Q.2. How many decimal expansions are there?
Ans: There are three decimal expansions are there.
1. Terminating decimal
2. Non-terminating recurring decimal
3. Non-terminating non-recurring decimal

Q.3. What are real numbers and their decimal expansions?
Ans: The decimal expansion of real numbers is nothing but simplifying the real numbers further and writing in the decimal form, whose values are numerically equal to the real number.

Q.4. How do you identify real numbers?
Ans: The complete number system is divided into real numbers and imaginary numbers. The imaginary numbers are the numbers that cannot be represented on the number line. On the contrary, real numbers are the number that can be represented on the number line.

Q.5. How do you find the terminating decimal expansion of the real numbers?
Ans: Let any real number can be expressed in the form of \(\frac{p}{q}\) and the prime factorization of the denominator \(q\) expressed in the form of \({2^m} \times {5^n},\) where \(m, n\) are non-negative integers, then we can say that the decimal expansion of the real number is terminating.

We hope this detailed article on real numbers and their decimal expansion helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!