**Number Systems**

**Page: 8**

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

**(i) Every irrational number is a real number.**

**Solution:**

**True**

Irrational Numbers – A number is said to be irrational, if it **cannot** be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Every irrational number is a real number, however, every real numbers are not irrational numbers.

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

**Solution:**

**False**

The statement is false since as per the rule, a negative number cannot be expressed as square roots.

E.g., √9 =3 is a natural number.

But √2 = 1.414 is not a natural number.

Similarly, we know that there are negative numbers on the number line but when we take the root of a negative number it becomes a complex number and not a natural number.

E.g., √-7 = 7i, where i = √-1

The statement that every point on the number line is of the form √m, where m is a natural number is false.

**(iii) Every real number is an irrational number.**

**Solution:**

**False**

The statement is false, the real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.

Real numbers – The collection of both rational and irrational numbers are known as real numbers.

i.e., Real numbers = √2, √5, , 0.102…

Irrational Numbers – A number is said to be irrational, if it **cannot** be written in the p/q, where p and q are integers and q ≠ 0.

i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….

Every irrational number is a real number, however, every real number is not irrational.

**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, the square roots of all positive integers are not irrational.

**For example,**

√4 = 2 is rational.

√9 = 3 is rational.

Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).

**3. Show how **√5** can be represented on the number line.**

**Solution:**

**Step 1:**Let line AB be of 2 unit on a number line.**Step 2:**At B, draw a perpendicular line BC of length 1 unit.**Step 3:**Join CA**Step 4:**Now, ABC is a right angled triangle. Applying Pythagoras theorem,

AB^{2}+BC^{2} = CA^{2}

2^{2}+1^{2} = CA^{2} = 5

⇒ CA = √5 . Thus, CA is a line of length √5 unit.

**Step 4:** Taking CA as a radius and A as a center draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0 because it is a radius of the circle whose center was A.

Thus, √5 is represented on the number line as shown in the figure.

**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 OP _{1} of unit length (see Fig. 1.9). Now draw a line segment P_{2}P_{3} perpendicular to OP_{2}. Then draw a line segment P_{3}P_{4} perpendicular to OP_{3}. Continuing in Fig. 1.9 :**

**Constructing this manner, you can get the line segment P _{n-1}Pn by square root spiral 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},….,Pn,… ., and joined them to create a beautiful spiral depicting √2, √3, √4, …**

**Solution:**

**Step 1:**Mark a point O on the paper. Here, O will be the center of the square root spiral.**Step 2:**From O, draw a straight line, OA, of 1cm horizontally.**Step 3:**From A, draw a perpendicular line, AB, of 1 cm.**Step 4:**Join OB. Here, OB will be of √2**Step 5:**Now, from B, draw a perpendicular line of 1 cm and mark the end point C.**Step 6:**Join OC. Here, OC will be of √3**Step 7:**Repeat the steps to draw √4, √5, √6….