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

Class 9: Real Numbers & Their Decimal Expansion Notes | Study Mathematics (Maths) Class 9 - Class 9

The document Real Numbers & Their Decimal Expansion Notes | Study Mathematics (Maths) Class 9 - 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

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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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 Notes | Study Mathematics (Maths) Class 9 - 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
73 videos|351 docs|110 tests

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

study material

,

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

,

Objective type Questions

,

past year papers

,

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

,

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

,

Viva Questions

,

Previous Year Questions with Solutions

,

MCQs

,

Important questions

,

shortcuts and tricks

,

Summary

,

ppt

,

video lectures

,

Extra Questions

,

Free

,

Exam

,

Sample Paper

,

pdf

,

mock tests for examination

,

practice quizzes

,

Semester Notes

;