Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Class 9 : Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

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Rational numbers:– These are real numbers which can be expressed in the form of Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes , where p and q are integers and Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes.

Ex. Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes –3, 0, 10, 4.33, 7.123123123......... 

  1. All natural numbers, whole numbers & integer are rational numbers.
  2.  Every terminating decimal is a rational number.
  3. Every recurring decimal is a rational number.
  4. A non-terminating repeating decimal is called a recurring decimal.
  5.  Between any two rational numbers, there is an infinite number of rational numbers.
  6. This property is known as the density of rational numbers. 
  7. Every rational number can be represented either as a terminating decimal or as a non-terminating repeating (recurring) decimals.
  8.  Types of rational numbers:– (a) Terminating decimal numbers and (b) Non-terminating repeating (recurring) decimal numbers.


Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Irrational numbers:– A number is called irrational number if it cannot be written in the form Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes, where p & q are integers and Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes. All Non-terminating & Non-repeating decimal numbers are Irrational numbers. 

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

(vi) Real numbers :– The totality of rational numbers and irrational numbers is called the set of real numbers.
i.e. rational numbers and irrational numbers taken together are called real numbers.
Every real number is either a rational number or an irrational number.

FINDING RATIONAL NUMBERS BETWEEN TWO NUMBERS
(A) 1st method : Find a rational number between x and y then,   Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes  is a rational number lying between x and y.

(B) 2nd method : Find n rational number between x and y (when x and y is non fraction number)
then we use formula.
Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

(C) 3rd method : Find n rational number between x and y (when x and y is fraction Number) then we use formula,
Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes
then n rational number lying between x and y are (x + d), (x + 2d), (x + 3d) .....(x + nd)

Remark : x = First Rational Number, y = Second Rational Number, n = No. of Rational Number.

Ex. Find 3 rational numbers between 2 and 5.

Sol. Let, a = first rational number.

b = second rational number.

n = number of rational number.

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Hence, three rational numbers between 2 and 5 are : Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes Ans.

 RATIONAL NUMBER IN DECIMAL REPRESENTATION:
Every rational number can be expressed as terminating decimal or non-terminating decimal.
(i) Terminating Decimal : The word "terminate" means "end". A decimal that ends is a terminating decimal.
OR
A terminating decimal doesn't keep going. A terminating decimal will have a finite number of digits after the decimal point.

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Ex. Express Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes in the decimal form by long division method.

Sol. We have,

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

∴ Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

(ii) Non terminating & Repeating (Recurring decimal) :–
A decimal in which a digit or a set of finite number of digits repeats periodically is called Non-terminating repeating (recurring) decimals.

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes
Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes
Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes


Ex. Find the decimal representation of Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Sol. By long division, we have 

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

COMPETITION WINDOW

NATURE OF THE DECIMAL EXPANSION OF RATIONAL NUMBERS

Theorem-1 : Let x be a rational number whose decimal expansion terminates. Then we can express x in the form Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes, where p and q are co-primes, and the prime factorisation of q is of the form 2m × 5n, where m,n are non-negative integers.

Theorem-2 : Let x = Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes be a rational number, such that the prime factorisation of q is of the form 2m × 5n, where m,n are non-negative integers . Then, x has a decimal expansion which terminates.

Theorem-3 : Let x = Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes be a rational number, such that the prime factorisation of q is not of the form 2× 5n, where m,n are non-negative integers . Then, x has a decimal expansion which is non terminating repeating.

Ex. (i)Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes  

we observe that the prime factorisation of the denominators of these rational numbers are of the form 2m × 5n, where m,n are non-negative integers. Hence, Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes has terminating decimal expansion.

(ii)  Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

we observe that the prime factorisation of the denominator of these rational numbers are not of the form 2m × 5n, where m,n are non-negative integers. Hence 17/6  has non-terminating and repeating decimal expansion. 

(iii) Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

So, the denominator 8 of 17/8 is of the form 2m × 5n, where m,n are non-negative integers. Hence 17/8 has terminating decimal expansion.

 (iv) Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Clearly, 455 is not of the form 2m × 5n. So, the decimal expansion of 64/455 is non-terminating repeating.

REPRESENTATION OF RATIONAL NUMBERS ON A NUMBER LINE
We have learnt how to represent integers on the number line. Draw any line. Take a point O on it. Call it 0(zero). Set of equal distances on the right as well as on the left of O.
Such a distance is known as a unit length. Clearly, the points A, B, C, D represent the integers 1, 2, 3, 4 respectively and the point A', B', C' D' represent the integers –1, –2, –3, –4 respectively

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Thus, we may represent any integer by a point on the number line. Clearly, every positive integer lies to the right of O and every negative integer lies to the left of O.
Similarly we can represent rational numbers.

Ex. Represent 1/2 and -1/2  on the number line.
Sol. Draw a line. Take a point O on it. Let it represent 0. Set off unit length OA and OA' to the right as well as to the left of O.
The, A represents the integer 1 and A' represents the integer –1.

Rational and Irrational Numbers - Number Systems, Class 9, Mathematics | EduRev Notes

Now, divide OA into two equal parts. Let OP be the first part out of these two parts.
Then, the point P represents the rational number 1/2.

Again, divide OA' into two equal parts. Let OP' be the first part out of these 2 parts. Then the point P' represents the rational number -1/2.

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