Class 9 Exam  >  Class 9 Notes  >  Mathematics (Maths) Class 9  >  Real Numbers & Their Decimal Expansion

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9 PDF Download

Before going into a representation of the decimal expansion of rational numbers, let us understand what rational numbers are. Any number that can be represented in the form of p/q, such that p and q are integers and q ≠ 0 are known as rational numbers. So when these numbers have been simplified further, they result in decimals. Let us learn how to expand such decimals here.

Examples: 6,−8.1, 4/5 etc. are all examples of rational numbers.
Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

How to Expand Rational Numbers in Decimals?

The real numbers which are recurring or terminating in nature are generally rational numbers.
Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

For example, consider the number 33.33333……. It is a rational number as it can be represented in the form of 100/3. It can be seen that the decimal part .333…… is the non-terminating repeating part, i .e. it is a recurring decimal number.
Also the terminating decimals such as 0.375, 0.6 etc. which satisfy the condition of being rational (0.375 = 3/23 0.6 = 3/5)
Consider any decimal number. For e.g. 0.567. It can be written as 567/1000 or 567/103. Similarly, the numbers 0.6689,0.032 and .45 can be written as 6689/104, 32/103 and 45/102 respectively in fractional form.
Thus, it can be seen that any decimal number can be represented as a fraction which has denominator in powers of 10. We know that prime factors of 10 are 2 and 5, it can be concluded that any decimal rational number can be easily represented in the form of p/q, such that p and q are integers and the prime factorization of q is of the form 2x 5y, where x and y are non-negative integers.
This statement gives rise to a very important theorem.

Theorems

Theorem 1: If m be any rational number whose decimal expansion is terminating in nature, then m can be expressed in form of p/q, where p and q are co-primes and the prime factorization of q is of the form 2x 5y, where x and y are non-negative integers.
The converse of this theorem is also true and it can be stated as follows:

Theorem 2: If m is a rational number, which can be represented as the ratio of two integers i.e. p/q, and the prime factorization of q takes the form 2x 5y, where x and y are non-negative integers then, then it can be said that m has a decimal expansion which is terminating.

Consider the following examples:

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

Moving on, to decimal expansion of rational numbers which are recurring, the following theorem can be stated:

Theorem 3: If m is a rational number, which can be represented as the ratio of two integers i.e. p/q, and the prime factorization of q does not takes the form 2x 5y, where x and y are non-negative integers. Then, it can be said that m has a decimal expansion which is non-terminating repeating (recurring).

Consider the following examples:

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9
Rational Number to decimal Examples

Case 1: Remainder equal to zero

Example: Find the decimal expansion of 3/6.
Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9
Here, the quotient is 0.5 and the remainder is 0. Rational number 3/6 results in a terminating decimal.

Case 2: Remainder not equal to zero
Example: Express 5/13 in decimal form.
Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

Here, the quotient is 0.384615384 and the remainder is not zero. Notice that the number…384 after the decimal is repeating. Hence, 5/13 gives us a non-terminating recurring decimal expansion. And this can be written as
5/13 = Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9
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.

The document Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9 is a part of the Class 9 Course Mathematics (Maths) Class 9.
All you need of Class 9 at this link: Class 9
Are you preparing for Class 9 Exam? Then you should check out the best video lectures, notes, free mock test series, crash course and much more provided by EduRev. You also get your detailed analysis and report cards along with 24x7 doubt solving for you to excel in Class 9 exam. So join EduRev now and revolutionise the way you learn!
Sign up for Free Download App for Free
45 videos|409 docs|53 tests

Up next

FAQs on Real Numbers & Their Decimal Expansion - Mathematics (Maths) Class 9

1. What are real numbers and why are they important in mathematics?
2. How are decimal numbers related to real numbers?
Ans. Decimal numbers are a way of representing real numbers in base-10 notation. A decimal number consists of a whole number part and a fractional part separated by a decimal point. The fractional part can be expressed using place values such as tenths, hundredths, etc. So, every decimal number can be seen as a real number, but not every real number can be expressed as a decimal.
3. Can all real numbers be expressed as decimal expansions?
Ans. No, not all real numbers can be expressed as decimal expansions. Rational numbers, which are a subset of real numbers, can be expressed as terminating decimal expansions or repeating decimal expansions. However, irrational numbers, which are also real numbers, cannot be expressed as finite or repeating decimals. Examples of irrational numbers are √2, π, and e.
4. How do we represent irrational numbers in decimal form?
Ans. Irrational numbers are represented in decimal form as non-repeating and non-terminating decimals. Since they cannot be expressed exactly, we often round them to a certain number of decimal places for practical purposes. For example, the square root of 2 (√2) is approximately 1.41421356 when rounded to eight decimal places.
5. Are there any patterns in the decimal expansions of real numbers?
Ans. The decimal expansions of some real numbers exhibit patterns, while others do not. Rational numbers have repeating or terminating decimal expansions, which means their decimal digits follow a fixed pattern. For example, 1/3 = 0.333... with the digit 3 repeating indefinitely. On the other hand, irrational numbers have non-repeating and non-terminating decimal expansions, with their digits appearing in a seemingly random manner, without any discernible pattern.
45 videos|409 docs|53 tests
Download as PDF

Up next

Explore Courses for Class 9 exam
Related Searches

ppt

,

Summary

,

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

,

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

,

video lectures

,

Important questions

,

mock tests for examination

,

Exam

,

Viva Questions

,

Extra Questions

,

past year papers

,

Previous Year Questions with Solutions

,

Sample Paper

,

shortcuts and tricks

,

Objective type Questions

,

study material

,

pdf

,

Semester Notes

,

practice quizzes

,

Real Numbers & Their Decimal Expansion | Mathematics (Maths) Class 9

,

Free

,

MCQs

;