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**REPRESENTATION OF REAL NUMBERS ON NUMBER LINE**

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

Represent âˆšx on number line.

Step 1:Draw a line and mark a point A on it.Step 2:Mark a point B on the line drawn such that AB = x cm.Step 3:Mark a point C on AB produced such that BC = 1 cm.Step 4:Find mid-point of AC (x+1). Let the mid-point be O.Step 5:Taking 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 6:Taking B as the centre and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents âˆšx.

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

Step 1:Draw a line and mark a point A on it.Step 2:Mark a point B on the line drawn such that AB = 9.3cm.Step 3:Mark a point C on AB produced such that BC = 1 cm.Step 4:Find mid-point of AC (9.3+1). Let the mid-point be O.Step 5:Taking 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 6:Taking B as the center and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents âˆš(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 â€˜closedâ€™ 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.

âˆš5 + (- âˆš5) = 0

âˆš3 - âˆš3 = 0

(âˆš2)(âˆš2) = 2

(âˆš7)/(âˆš7) = 1

All are rational numbers.

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

For example, âˆš6 is an irrational number so when we add or subtract any rational number to an irrational number the result will be irrational number only.

**Note: The numbers with same radicals are called like terms. Only like terms can be added or subtracted. In like terms only numbers before the radicals is added or subtracted and the radical will remain as it is.**

**Example: **

3âˆš2 and 5âˆš2 â†’ Like terms.

-âˆš7 and 3âˆš7 â†’ Like terms.

2âˆš3 and âˆš5 â†’ Unlike terms.

2âˆš11 and 5âˆš6 â†’ Unlike terms.**Operations on real numbers ****(1) ****Add the 2âˆš2 + 5âˆš3 and âˆš2 - 3âˆš3****Solution: **

2âˆš2 + 5âˆš3 + âˆš2 + (- 3âˆš3)

= __2âˆš2 __+ __5âˆš3 __+ __âˆš2__ __- 3âˆš3 __

= 3 âˆš2 + 2âˆš3**(2)**** Add the -6âˆš3 + 3âˆš2 and -2âˆš2 â€“ 4 âˆš3****Solution:**

-6âˆš3 + 3âˆš2 + (-2âˆš2 â€“ 4 âˆš3)

= __- 6âˆš3__ __+ 3 âˆš2____ - 2âˆš2__ __- 4âˆš3 __

= -10âˆš3 + 1âˆš2**(3) ****Multiply (2âˆš3) and (-3âˆš5)****Solution:**

2 âˆš3 x (-3âˆš5)

= (2 x -3) âˆš3 x âˆš5

= - 6 âˆš(3 x 5)

= -6 âˆš15**(4)**** Multiply 3âˆš2 (2 + 4âˆš3)****Solution:**

3âˆš2 (2 + 4âˆš3)

= 3âˆš2 x 2 + 3âˆš2 x 4âˆš3 [ use a distributive law]

= (3 x 2)âˆš2 + (3 x 4)(âˆš2 x âˆš3)

= 6âˆš2 + 12âˆš6**(5) ****Divide âˆš15 by âˆš3****Solution:**

âˆš15 / âˆš3

= (âˆš3 x âˆš5) / âˆš3

= âˆš5

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