ex 1.1 class 9

Number System:

Real Numbers

In a Number System Real numbers are the numbers which include both Rational and Irrational numbers.

Natural Numbers:- 

In a Number System all counting numbers \(1,2,3,4…\) are called Natural numbers. If N is the set of natural numbers, then N = {1, 2, 3, 4,…}

Whole Numbers:

In a Number System all the natural numbers including \(0\) form the set of whole numbers. It is denoted by W. Thus W = {0,1,2,3,4,…}

Irrational Numbers: 

An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means not Rational.

An Irrational Number can NOT be written in the form \({n \over d}\), where \( n \) and \( d \) are integers and \( d \) is non-zero. 

Examples

\(\pi   \)                      \( \sqrt{3}\)                                \(\sqrt{5}\)

Composite Numbers :- 

A natural number “a“ is said to be Composite if it has at least one more divisor except 1 and ‘a’ itself.

• For example \(4,6,8,9,10,12,14,…\) are all composite numbers.

Integers:-

• In a Number System all the natural numbers, their negatives and zero form the set of integers.
• It is denoted by Z. Thus \(Z = {0,\pm 1, ±2,±3, ±4,…}\).
• Integers are also denoted by I.

Rational numbers:-

• A number in the form of  \({p  \over q}\) where p and q are integers; and i,e are co-primes is called a rational number.
• Rational numbers are denoted by Q (Q comes from word Quotient).
• Note:-The word ‘Rational’ is derived from the word ‘ratio’.

Twin Primes:- 

• A pair of prime numbers is said to be Twin Primes if they differ by 2. 
• For example \({ (3,5); (5,7); (11,13); (17,19) }\) are Twin Primes while as  \({ (3,5),(7,11); (13,17) }\) are not Twin Primes.

Co-prime:- 

Two numbers are said to be co-prime or relatively prime if their Highest Common Factor or Greatest Common Divisor is 1. 

ex 1.1 class 9

Question 1.

Is zero a rational number? Can you write it in the form ^p/_q, where p and q are integers and q ≠ 0?

Solution 1:

Consider the definition of a rational number.

A rational number is the one that can be written in the form of  ^p/_q , where p and q are integers and q 

≠0.

• Zero can be written as \(\displaystyle{{0 \over 1},{0 \over 2},{0 \over 3},{0 \over 4},{0 \over 5}...}\)

• Zero can be written as well \(\displaystyle{{0 \over -1},{0 \over -2},{0 \over -3},{0 \over -4},{0 \over -5}...}\)

So, we arrive at the conclusion that 0 can be written in the form of  ^p/_q , where p and q are integers ( q can be positive or negative integers).

Therefore, zero is a rational number.

Question  2.

Find six rational numbers between 3 and 4. 

Solution 2:

We know that there are infinite rational numbers between any two numbers.

As we have to find 6 rational numbers between 3 and 4 So multiply and divide by 7 (or any number greater than 6)

We get, \(3 = {\displaystyle {3 × 7 \over 7}} = {\displaystyle21 \over 7} \)

                \(4 = \displaystyle {4 × 7 \over 7} = \displaystyle {28 \over 7} \)

Thus the 6 rational numbers are \(\displaystyle{ {22 \over 7},{23 \over 7},{24 \over 7},{25 \over 7},{26 \over 7},{27 \over 7} }\).

Question 3.

Find five rational numbers between ³/₅ and ⁴/₅.

Solution 3:

We know that there are infinite rational numbers between any two numbers.

As we have to find 5 rational numbers between \( \displaystyle{ 3 \over 5}\) and \( \displaystyle{4 \over 5}\)

So, multiply and divide by 6 (Or any number greater than 5) 

\(\displaystyle{ {3 \over 5} = {3 \over 5} × {6 \over 6} = {18 \over 30}}\)

\(\displaystyle{ {4 \over 5} = {4 \over 5} × {6 \over 6} = {24\over 30}}\)

Thus the 5 rational numbers are \(\displaystyle{{19 \over 30}, {20 \over 30} ,{21 \over 30},{22 \over 30},{23 \over 30}}\)

Question 4.

State whether the following statements are true or false. Give reasons for your answers.

(i)Every natural number is a whole number.

(ii)Every integer is a whole number.

(iii)Every rational number is a whole number. 

Solution 4:

(i)

Consider the whole numbers and natural numbers separately.

We know that whole number series is 0,1, 2, 3, 4,5..... .

We know that natural number series is 1, 2, 3, 4,5..... .

So, we can conclude that every natural number lie in the whole number series.

(ii)

Consider the integers and whole numbers separately.

We know that integers are those numbers that can be written in the form of ^p/_q , where q = 1.

Now, considering the series of integers, we have – 4,–3,–2,–1, 0, 1, 2,3, 4..... .

We know that whole number are 0, 1, 2, 3, 4, 5..... .

We can conclude that whole number series lie in the series of integers. But every integer does not appear in the whole number series.

Therefore, we conclude that every integer is not a whole number.

But, clearly every whole number is an integer.

(iii)

Consider the rational numbers and whole numbers separately.

We know that rational numbers are the numbers that can be written in the form \( \displaystyle{p \over q}\) ,where q ≠ 0.

We know that whole numbers are 0, 1, 2, 3, 4, 5..... .

We know that every whole number can be written in the form of  \( \displaystyle{p \over q}\)as follows

\(\displaystyle{ ({0 \over 1},{1 \over 1},{2 \over 1},{3 \over 1},{4 \over 1})...}\).

We conclude that every whole number is a rational number.

But, every rational number  \(\displaystyle{ ({1 \over 2},{1 \over 3},{1 \over 4},{1 \over 5},{1 \over 6})...}\)is not a whole number. 

Therefore, we conclude that every rational number is not a whole number.

But, clearly every whole number is a rational number.