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**Representation of Real Numbers on number line**

For representation of real numbers on number line, use the following steps :

Represent √x on number line.

Step 1Draw a line and mark a point A on it.Step 2Mark a point B on the line drawn such that AB = x cm.Step 3Mark a point C on AB produced such that BC = 1 cm.Step 4Find mid-point of AC (x+1). Let the mid-point be O.Step 5Taking O as centre and OC = OA [(x+1)/2] as radius draw a semi-circle. Also draw a line passing through B perpendicular to OB. Let it cut the semi-circle at DStep 6Taking B as the centre and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents √x.

**Examples on representation of real numbers on number line : ****1) Represent √(9.3) on the number line. **

Step 1Draw a line and mark a point A on it.Step 2Mark a point B on the line drawn such that AB = 9.3cm.Step 3Mark a point C on AB produced such that BC = 1 cm.Step 4Find mid-point of AC (9.3+1). Let the mid-point be O.Step 5Taking O as center and OC = OA [10.3/2] as radius draw a semi-circle. Also draw a line passing through B perpendicular to OB. Let it cut the semi-circle at DStep 6Taking B as the center and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents √(9.3).

**2) Visualize 3.765 on the number line using successive magnification. **

Step 1:Since the given number lies between 3 and 4, look at the portion of the number line between 3 and 4.Step 2:Divide the portion between 3 and 4 into 10 equal parts and mark each point of the division as shown in Fig. 1.Step 3:The 7th mark and 8th mark of this sub-division corresponds to 3.7 and 3.8 respectively and 3.765 lies between them (3.7 < 3.765 < 3.8)Step 4:Again divide the portion between 3.7 and 3.8 into 10 equal parts. Now 3.765 lies between its 6th and 7th mark (3.76 < 3.765 < 3.77) .Step 5:Divide the portion between 3.76 and 3.77 again into ten equal parts. Therefore, the 5th mark of this sub-division mark represents 3.765 as shown in fig.3

**Operations on real numbers**The rational numbers satisfy the commutative, associative and distributive laws for addition and multiplication Moreover, if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number (that is, rational numbers are ‘closed’ with respect to addition, subtraction, multiplication and division). It turns out that irrational numbers also satisfy the commutative, associative and distributive laws for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational.

√5 + (- √5) = 0

√3 - √3= 0

(√2)(√2)= 2

(√7)/( √7)= 1

All are rational numbers.

When we add and multiply a rational number with an irrational number.

For example, √6 is an irrational number so when we add or subtract any rational number to an irrational number the result will be irrational number only.

Example :

-√7 and 3√7 -------------> Like terms.

2√3 and √5 --------------> Unlike terms.

2√11 and 5√6 ------------> Unlike terms.

2√2 + 5√3 + √2 + (- 3√3)

=

= 3 √2 + 2√3

-6√3 + 3√2 + (-2√2 – 4 √3)

=

= -10√3 + 1√2

2 √3 x (-3√5)

= (2 x -3) √3 x √5

= - 6 √(3x5)

= -6 √15

3√2 ( 2 + 4√3)

= 3√2 x 2 + 3√2 x 4√3 [ use a distributive law]

= (3x2)√2 + (3x4)( √2 x √3)

= 6√2 + 12√6

√15 / √3

= (√3 x √5) / √3

= √5

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