Class 9 Chapter 1 Number Systems | Notes, Problems And Solutions

CLASS 9 Mathematics Chapter 1 Solutions | NUMBER SYSTEMS

Notes

NATURAL NUMBERS:

Numbers which are used for counting and ordering are called natural numbers. It is denoted by N.

WHOLE NUMBERS:

Zero and natural numbers are collectively known as whole numbers which is denoted by the symbol W.

INTEGERS:

Natural numbers, zero and negative of natural numbers are known as integers and it is denoted by the symbol Z.

RATIONAL NUMBERS:

A number ‘r’ is called a rational number, if it can be written in the form p/q, where p and q are integers and q not equal to zero. The collection of rational numbers is denoted by Q.

IRRATIONAL NUMBERS:

A number ‘s’ is called irrational, if it cannot be written in the form p/q, where p and q are integers and q not equal to zero.

REAL NUMBERS:

The collection of rational numbers and irrational numbers are called real numbers which is represented by R.

Example questions:

1. Are the following statements true or false? Give reasons for your answers.

a) Every whole number is a natural number.
b) Every integer is a rational number.
c) Every rational number is an integer.

Solution:

a) False, because zero is a whole number but not a natural number.
b) True, because every integer m can be expressed in the form m/1, and so it is a rational number.
c) False, because ¾ is not an integer.

2. Is zero a rational number? Can you write it in the form p/q, where P and q are integers and q not equal to zero.

Solution:

Yes, we can write 0 in the form 0/1, where 0 and 1 are integers

And 1 is not equal to zero.

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

a) Every natural number is a whole number.
b) Every integer is a whole number.
c) Every rational number is a whole number.

Solution:

a) True, since the collection of whole numbers contains all the natural numbers.
b) False, for example -3 is not a whole number.
c) False, for example 1/3 is a rational number but not a whole number.

4. Find five rational numbers between 1 and 2.

Solution:

Since we want five numbers, we write 1 and 2 as rational numbers with denominator 5+1. We can write 1 = 6/6 and 2 = 12/6.
Then 7/6, 8/6, 9/6, 10/6 and 11/6 are all rational numbers between 1 and 2.So the five rational numbers are
7/6, 4/3,3/2,5/3 and 11/6.

5. Find six rational numbers between 3 and 4.

Solution:

We can write 3 = 21/7 and 4 = 28/7

Then the six rational numbers are 22/7, 23/7, 24/7, 25/7, 26/7 and 27/7.

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

Solution:

We can write 3/5 = 30/50 and 4/5 = 40/50
Therefore, five rational numbers are 31/50, 32/50,33/50,34/50
and 35/50.

7. State whether the following statements are true or false. Justify your answers.

a). Every irrational number is a real number.

b). Every point on the number line is of the form square root of m
where m is a natural number.
c). Every real number is an irrational number.

Solution:

a). True, since collection of real numbers is made up of rational and irrational numbers.
b). False, no negative number can be the square root of any natural number.
c). False, for example 2 is real but not irrational.

8. Are the square roots of all positive integers irrational? If not give an example of the square root of a number that is a rational number.

Solution:

No, For example, square root of 4 is 2, which is a rational number.
Square root of 9 is 3, which is a rational number.

Real Numbers and their Decimal Expansions

1. Write the following in decimal form and say what kind of decimal expansion each has:

a) 36/100
b) 1/11
c) 4 1/8
d) 3/13
e) 2/11
f) 329/400

Solution:

a). 0.36, terminating.
b). 0.09090909……, non terminating repeating.
c). 4 1/8 = 4 X 8 +1/8 = 33/8 =4.125, terminating.
d). 0.230769230769…….,non terminating repeating.
e). 0.18181818……,non terminating repeating.
f). 0.8225, terminating.

2. Express the following in the form p/q, where p and q are integers and q is not equal to zero.

a). 0.66666…………
b). 0.477777…………..
c). 0.001001001…………..

Solution:

a). Since we do not know what 0.66666……….. is, let us call it x and so x = 0.66666……………
Here only one number 6 is repeating.
So we can multiply by 10
10x = 6.6666…………
= 6 + 0.6666………
= 6 + x
Solving for x, we get 9x = 6
i.e, x = 6/9 = 2/3

b). Since we do not know what 0.47777…………. is, let us call it x and so x = 0.4777……….
Here only one number 7 is repeating.
So we can multiply by 10
10 x = 4.777……….
= 4.3 + 0.4777……
= 4.3 + x
Solving for x, we get 9x = 4.3
i.e, x = 4.3/9 = 43/90

c). Since we do not know what 0.001001001……………..is, let us call it x and so x = 0.001001………
Here three numbers 001 are repeating. So we can multiply by 1000.

1000x = 1.001001………….
= 1 + 0.001001……………..
= 1+ x
999x = 1
X = 1/999

3. Express 0.9999…………… in the form p/q .Are you surprised by your answer?

Solution:

Since we do not know what 0.9999……….. is, let us call it x and so
X = 0.9999………….

Here only one number 9 is repeating.
So we can multiply by 10.
10 x = 9.999…………
= 9 + 0.999………….
= 9 + x
9x = 9
X = 1

4. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.

Solution:

We can observe that while dividing 1 by 17 we will get 16 number of digits in the repeating block of decimal expansion which will continue after carrying 16 continuous divisions.

Therefore, 1/17 = 0.0588235294117647……………………

5 Write three numbers whose decimal expansions are non terminating non recurring.

Solution:

All irrational numbers are non terminating and non recurring.

0.01001000100001……..
0.202002000200002…………….
0.003000300003…………

6. Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Solution:

Let us convert 5/7 and 9/11 into decimal form, we get 5/7 = 0.714285………….. and 9/11 = 0.818181………….

Three irrational numbers that lie between 0.714285………… and 0.818181…………. are

0.75075007500075000075……………
0.767076700767000767………..
0.808008000800008…………

7. Classify the following numbers as rational or irrational:

a) Square root of 23
b) Square root of 225
c) 0.3796
d) 7.478478……………
e) 1.101001000100001……………

Solution:

a) Irrational, the value of square root of 23 is 4.795831………….
b) Rational, the value of square root of 225 is 15.
c) Rational, it is terminating.
d) Rational, it is a non terminating recurring decimal which can be converted into p/q form.
e). Irrational, it is a non terminating non recurring decimal. Thus it cannot be converted into p/q form.