**NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.1****NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.2****NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.3****NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.4****NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.5**

**NCERT Solutions for Class 9 Maths Chapter 1 Number Systems Ex 1.2**

**Ex 1.2 Class 9 Maths Question 1.
**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 √m, where m is a natural number.

(iii) Every real number is an irrational number.

**Solution:
**(i) True

Because the collection of all rational numbers and all irrational numbers is called a set of real numbers.

(ii) False

Because negative numbers cannot be the square root of any natural number.

(iii) False

Because real numbers have rational and irrational both types of numbers. For example, 5, ½, 12, 2/3, etc. are real numbers but they are not irrational.

**Ex 1.2 Class 9 Maths Question 2.
**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, if we take a positive integer, say 16, its
square root is 4, which is a rational number. Again, the square root of 25 is 5
which is a rational number.

**Ex 1.2 Class 9 Maths Question
3.****
**Show how √5 can be represented on
the number line.

**Solution:**

Method 1:

Draw a number line and mark a point
O on it. Take a point A on it such that OA = 1 unit. Draw BA ⊥ OA as BA = 1 unit. Join OB = √2 units.

Now draw BB_{1} ⊥ OB such that BB_{1} =1 unit. Join OB_{1} =
√3 units.

Next, draw B_{1}B_{2 }⊥ OB_{1 }such that B_{1}B_{2} = 1
unit. Join OB_{2} =
√4 = 2 units.

Again, draw B_{2}B_{3} ⊥ OB_{2} such that B_{2}B_{3} =
1 unit. Join OB_{3} = √5 units.

Take O as centre and OB

_{3}as radius, draw an arc which cuts the number line at D. Point D represents √5 on the number line and OD = √5 units.

**Method 2: **We have √5 = √(4 + 1) = √(2^{2} + 1^{2})

Draw a number line and mark a point
O on it. Mark … , -2, -1, 0, 1, 2, … as shown in the figure below. Take a point
Q such that OQ = 2 units. Draw PQ ⊥ OQ. With point Q as centre and radius as 1 unit, cut an arc
at P. Join OP.

Now, O as centre and OP as radius,
draw an arc which cuts the number line at R. Point R represents √5 and OR = √5.

We can verify the result using
Pythagoras theorem,

OP^{2} = OQ^{2} +
PQ^{2}

OP^{2} = 2^{2} + 1^{2}

OP^{2} = 4 + 1

OP^{2} = 5

OP = √5

OR = OP = √5 units

**Ex 1.2 Class 9
Maths Question 4.**

A classroom activity (constructing the ‘square root spiral’).

**Solution:**

Take a large sheet of paper and construct the
‘square root spiral’ in the following fashion. Start with a point O and draw a
line segment OP_{1} of unit length. Draw a line segment P_{1}P_{2} perpendicular
to OP, of unit length (see figure).

_{2}P

_{3}perpendicular to OP

_{2}. Then draw a line segment P

_{3}P

_{4}perpendicular to OP

_{3}. Continuing in this manner, you can get the line segment P

_{n-1}P

_{n}by drawing a line segment of unit length perpendicular to OP

_{n-1}. In this manner, you will have created the points P

_{2}, P

_{3}, …., P

_{n }…, and joined them to create a beautiful spiral depicting √2, √3, √4, …