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) We know that real number is either rational or irrational. So we can say that every irrational number
is a real number.
Hence, the given statement is true.
(ii) We know that real numbers can be represented on the number line. Thus, every point on the number
line is of the form m, where m is a natural number.
Hence, the given statement is true.
(iii) We know that, rational numbers and irrational numbers taken together are known as real numbers.
Hence, the given statement is false. Ans.
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. “The square roots of all positive integers are irrational.”, is not correct.
Since, square root of 9 i.e., 9 = 3, which is a rational number.
Hence, the given statement is not correct. Ans.
QUESTION 3. Show how 5 can be represented on the number line.
SOLUTION. We shall now show how to represent 5 on the number line.
Draw a number line l and mark a point O, representing zero
(0), on it. Let point A represents 2 as shown in the figure.
Now, construct a right-angled OAB, right-angled at A such
that OA = 2 units and AB = 1 unit (see figure).
By Pythagoras theorem, we have
OB = OA2 AB2 4 1 5
Draw an arc with centre O and radius OB to cut the number line at C. Clearly, OC = OB = 5.
Thus, the point C represents the irrational number 5. Ans.
QUESTION 4. Classroom activity (Constructing the ‘square root spiral’) : 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 OP1 of
unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see figure). Now draw a
line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3.
Continuing in this manner, you can get the line segment Pn – 1Pn by drawing a line segment of unit
length perpendicular to OPn – 1. In this manner, you will have created the point P2, P3, … Pn, … and
join them to create a beautiful spiral depicting 2 , 3 , 4 , ….
SOLUTION. Classroom activity – Do as directed.