Chapter 01 Number Systems Exercise-02
EXERCISE 1.2
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 $\sqrt{m}$, where $m$ is a natural number.
(iii) Every real number is an irrational number.
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Solution
(i) True; since the collection of real numbers is made up of rational and irrational numbers.
(ii) False; as negative numbers cannot be expressed as the square root of any other number.
(iii) False; as real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.
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.
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Solution
If numbers such as $\sqrt{4}=2, \sqrt{9}=3$ are considered,
3. Show how $\sqrt{5}$ can be represented on the number line.
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Solution
We know that,
$ \sqrt{4}=2 $
$ \sqrt{5}=\sqrt{(2)^{2}+(1)^{2}} $
Show howAnd,
Mark a point ’ $A$ ’ representing 2 on number line. Now, construct $A B$ of unit length perpendicular to $OA$. Then, taking $O$ as centre and $OB$ as radius, draw an arc intersecting number line at $C$.
C is representing $\sqrt{5}$.
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 $\mathrm{O}$ and draw a line segment $\mathrm{OP_1}$ of unit length. Draw a line segment $\mathrm{P_1} \mathrm{P_2}$ perpendicular to $\mathrm{OP_1}$ of unit length (see Fig. 1.9). Now draw a line segment $\mathrm{P_2} \mathrm{P_3}$ perpendicular to $\mathrm{OP_2}$. Then draw a line segment $\mathrm{P_3} \mathrm{P_4}$ perpendicular to $\mathrm{OP_3}$. Continuing in this manner, you can get the line segment $P_{n-1} P_{n}$ by
Fig. 1.9 : Constructing square root spiral drawing a line segment of unit length perpendicular to $\mathrm{OP_\mathrm{n}-1}$. In this manner, you will have created the points $\mathrm{P_2}, \mathrm{P_3}, \ldots ., \mathrm{P}_{\mathrm{n}}, \ldots$, and joined them to create a beautiful spiral depicting $\sqrt{2}, \sqrt{3}, \sqrt{4}, \ldots$
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Solution
Construct the “square root spiral”.
Step 1. On a large sheet of paper, mark a point $\mathrm{O}$ and draw a line segment $\mathrm{OP}_1$ of unit length.
Step 2. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length and join $OP_2$.
Here, $\mathrm{OP}_2$ is of length $\sqrt{2}$ units. As from Pythagoras theorem: $\left(O P_2\right)^2=\left(P_1 P_2\right)^2+\left(O P_1\right)^2$
Step 3. Draw a line segment $\mathrm{P}_2 \mathrm{P}_3$ perpendicular to $\mathrm{OP}_2$ of unit length and join $\mathrm{OP}_3$.
Here, $\mathrm{OP}_3$ is of length $\sqrt{3}$ units. As from Pythagoras theorem: $\left(\mathrm{OP}_3\right)^2=\left(\mathrm{P}_2 \mathrm{P}_3\right)^2+\left(O P_2\right)^2$ Step 4. Repeat the steps to get the points $\mathrm{P}_4, \mathrm{P}_5, \mathrm{P}_6, \ldots$ and join them with $\mathrm{O}$ to obtain $\sqrt{4}, \sqrt{5}, \sqrt{6}, \ldots$
Hence, the ‘square root spiral’ is constructed.
