## Number Systems

## Chapter 1

## Exercise 1.5

**EX 1.5 QUESTION 1.**

**Classify the following numbers as rational or irrational:**

**(i) 2 –√5**

**(ii) (3 +√23)- √23**

**(iii) 2√7/7√7**

**(iv) 1/√2**

**(v) 2π**

**Solution:**

(i) 2 –√5

Irrational number.

(ii) (3 +√23)- √23

⇒ 3 +√23- √23 = 3

Rational number.

(iii) 2√7/7√7

⇒** = = **

Rational number.

(iv) 1/√2

Irrational number.

(v) 2π

Irrational number.

**EX 1.5 QUESTION 2.**

**Simplify each of the following expressions:**

**(i) (3+√3)(2+√2)**

**(ii) (3+√3)(2+√2 )**

**(iii) (√5+√2) ^{2}**

**(iv) (√5-√2)(√5+√2)**

**Solution:**

(i) (3+√3)(2+√2 )

(3×2)+(3×√2)+(√3×2)+(√3×√2)

= 6+3√2+2√3+√6

(ii) (3+√3)(2+√2 )

(3+√3)(2+√2 )

= 3^{2}-(√3)^{2}

= 9-3

= 6

(iii) (√5+√2)^{2}

(√5+√2)^{2 }=

√5^{2}+(2×√5×√2)+ √2^{2}

= 5+2×√10+2

= 7+2√10

(iv) (√5-√2)(√5+√2)

(√5-√2)(√5+√2)

= (√5^{2}-√2^{2})

= 5-2

= 3

**EX 1.5 QUESTION 3.**

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

**Solution:**

When we measure the length of a line with a scale or a tape, we get only an approximate rational number. Therefore, the value of c and d both are irrational.

c/d is an irrational number then π is irrational.

**EX 1.5 QUESTION 4.**

**Represent (√9.3) on the number line.**

**Solution:**

Draw a line segment AB = 9.3 units

Now produce AB to C such that BC = 1 unit.

Draw the perpendicular bisector of AC which intersect AC at O.

Draw a semicircle taking O as centre and AO as the radius. Draw BD ⊥ AC.

Draw an arc taking B as centre and BD as radius meeting AC produced at E such that BE = BD = √9.3 units.

**EX 1.5 QUESTION 6.**

**Rationalize the denominators of the following:**

**(i) 1/√7**

**(ii) 1/(√7-√6)**

**(iii) 1/(√5+√2)**

**(iv) 1/(√7-2)**

**Solution:**