# NCERT Solutions for Class 9 Maths Unit 1

## Number Systems Class 9

### Unit 1 Number Systems Exercise 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 Solutions

The Natural Numbers
The natural (or counting) numbers are 1, 2, 3, 4, 5, etc. There are infinitely many natural numbers. The set of natural numbers is sometimes written N for short.
The whole numbers are the natural numbers together with 0.
The Integers
The integers are the set of real numbers consisting of the natural numbers, their additive inverses and zero. The set of integers is sometimes written J or Z for short. The sum, product, and difference of any two integers is also an integer.
But this is not true for division... just try 1 ÷ 2.

The Rational Numbers
The rational numbers are those numbers which can be expressed as a ratio between two integers. For example, the fractions 1/3 and –1111/8 are both rational numbers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z/1.

### Exercise 1.1 : Solutions of Questions on Page Number : 5

Q1 :

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

Yes. Zero is a rational number as it can be represented asetc.

Q2 :

Find six rational numbers between 3 and 4.

There are infinite rational numbers in between 3 and 4.

3 and 4 can be represented asrespectively.

Therefore, rational numbers between 3 and 4 are

Q3 :

Find five rational numbers between.

Q4 :

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.

### Exercise 1.2 : Solutions of Questions on Page Number : 8

Q1 :

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

(i) Every irrational number is a real number.

(ii) Every point on the number line is of the form, where m is a natural number.

(iii) Every real number is an irrational number.

Q2 :

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.

Q3 :

Show howcan be represented on the number line.

### Exercise 1.3 : Solutions of Questions on Page Number : 14

Q1 :

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

(i) (ii) (iii)

(iv) (v) (vi)

Q2 :

You know that. Can you predict what the decimal expansion of are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of carefully.]

Q3 :

Express the following in the form, where p and q are integers and q ≠ 0.

(i) (ii) (iii)

Q4 :

Q5 :

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

Q6 :

Look at several examples of rational numbers in the form (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Q7 :

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

Q8 :

Find three different irrational numbers between the rational numbers and.

Q9 :

Classify the following numbers as rational or irrational:

(i) (ii) (iii) 0.3796

(iv) 7.478478 (v) 1.101001000100001…

### Exercise 1.4 : Solutions of Questions on Page Number : 18

Q1 :

Visualise 3.765 on the number line using successive magnification.

Q2 :

Visualise on the number line, up to 4 decimal places.

### Exercise 1.5 : Solutions of Questions on Page Number : 24

Q1 :

1Classify the following numbers as rational or irrational:

(i) (ii) (iii)

(iv) (v) 2π

Q2 :

Simplify each of the following expressions:

(i) (ii)

(iii) (iv)

Q3 :

Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

Q4 :

Represent on the number line.

Q5 :

Rationalise the denominators of the following:

(i) (ii)

(iii) (iv)

Q1 :

Find:

(i) (ii) (iii)

Q2 :

Q2. Find:

(i) (ii) (iii)

(iv)

Q3 :

Simplify:

(i) (ii) (iii)

(iv)