Chapter 1 - Number Systems, Solved Examples, Class 9, Maths PDF Download

INTRODUCTION

"God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (1823-1891), the reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers played in the evolution of human thought.

N : The set of natural numbers,
W : The set of whole numbers,
Z : The set of Integers, R
Q : The set of rationals,
R : The set of Real Numbers.

In our day to life we deal with different types of numbers which can be broady classified as follows.

(i) Natural Numbers (N) : The counting numbers 1, 2, 3, ..... are known as natural numbers.

The collection of natural number is denoted by 'N'
N = {1, 2, 3, 4...∞}

(i) The set N is infinite i.e. it has unlimited members.
(ii) N has the smallest element namely '1'.
(iii) N has no largest element. i.e., give me any natural number, we can find the bigger number from the given number.
(iv) N does not contain '0' as a member. i.e. '0' is not a member of the set N.
(v) If we go on adding 1 to each natural number ; we get next natural number.

(ii) Whole numbers (W) : The number '0' together with the natural numbers 1, 2, 3, .... are known as whole numbers.
The collection of whole number is denoted by 'W'
W = {0, 1, 2, 3, 4...∞}

(i) The set of whole number is infinite (unlimited elements).
(ii) This set has the smallest members as '0'. i.e. '0' the smallest whole number. i.e., set W contain '0' as a member.
(iii) The set of whole numbers has no largest member.
(iv) Every natural number is a whole number but every whol number is not natural number.
(v) Non-zero smallest whole number is '1'.

(iii) Integers (I or Z) : All natural numbers, 0 and negative of natural numbers are called integers.
The collection of integers is denoted by Z or I.
Integers (I or Z) : I or Z = {-∞, ..... -3, -2, -1, 0, +1, +2, +3 .... +∞}
Positive integers : {1, 2, 3...},

Negative integers : {.... -4, -3, -2, -1}

(i) This set Z is infinite.
(ii) It has neither the greatest nor the least element.
(iii) Every natural number is an integer.
(iv) Every whole number is an integer.
(iv) The set of non-negative integer = {0, 1, 2, 3, 4,....}
(v) The set of non-positive integer = {......–4, – 3, – 2, –1, 0}
(iv) Rational numbers :– These are real numbers which can be expressed in the form of p/q , where p and q are integers 

(i) All natural numbers, whole numbers & integer are rational numbers.

(ii) Every terminating decimal is a rational number.

(iii) Every recurring decimal is a rational number.

(iv) A non- terminating repeating decimal is called a recurring decimal.

(v) Between any two rational numbers there are an infinite number of rational numbers.

This property is known as the density of rational numbers.

(vi) Every rational number can be represented either as a terminating decimal or as a non-terminating repeating (recurring) decimals.
(vii) Types of rational numbers :–
(a) Terminating decimal numbers and
(b) Non-terminating repeating (recurring) decimal numbers
(v) Irrational numbers :– A number is called irrational number, if it can not be written in the form p/q , where p & q are integers and q  0. All Non-terminating & Non-repeating decimal numbers are Irrational numbers.

(vi) Real numbers :– The totality of rational numbers and irrational numbers is called the set of real number 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, 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.

(C) 3rd method : Find n rational number between x and y (when x and y is fraction Number)
then we use formula

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 1. Find 3 rational number between 2 and 5.

Sol. Let, a = first rational number.

b = second rational number

n = number of rational number
Here a = 2, b = 5

A rational number between 2 and 5 

Second rational number between 2 and 

Third rational number between 

Hence, three rational numbers between 2 and 5 are :

Ex 2. Find 4 rational numbers between 4 and 5.

Sol. a = 4, b = 5, n = 4

Hence, reational number between 4 and 5 are : 

Ex. 3 Find three rational number between (6/7), (7/5)

Sol. a =  (6/7)
b = (7/5)

n = 3

∴3 rational numbers between 6/7 and 7/5 are : (a + d), (a + 2d), (a + 3d) ........

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 decial doesn't keep going. A terminating decimal will have a finite number of digits after the decimal point.

Ex 4. Express 7/8 in the decimal form by long division method.

Sol. We have,

∴  7/8= 0.875

Ex 5. Convert 35/16  into decimal form by long division method.

Sol. We have,

∴ 35/16 = 2.1875

(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.

Ex 6. Find the decimal representation of (8/3)

Sol. By long division, we have

Ex 7. Express 2/11  as a decimal fraction.
Sol. By long division, we have

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 p/q , where p and q are co-primes, and the prime factorisation of q is of the form 2^m × 5ⁿ, where m,n are non-negative integers.

Theorem-2 : Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2^m × 5ⁿ, where m,n are non-negative integers . Then, x has a decimal expansion which terminates.

Theorem-3 : Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2^m × 5ⁿ, where m,n are non-negative integers . Then, x has a decimal expansion which is non - terminating repeating.

Ex 8. 
we observe that the prime factorisation of the denominators of these rational numbers are of the form 2^m × 5ⁿ, where m,n are non-negative integers. Hence,  has terminating decimal expansion.

 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 has non-terminating and repeating decimal expansion.

So, the denominator 8 of is of the form 2^m × 5ⁿ, where m,n are non-negative integers.

Hence  has terminating decimal expansion.

Clearly, 455 is not of the form 2^m × 5ⁿ. So, the decimal expansion of 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

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 9. Represent  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.

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

Ex 10. Represent 4/7 on number line.

Sol. Divide the line segment between 0 and 1 into 7 equal parts (because 4/7 lies between 0 and 1)