Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev

Mathematics (Maths) Class 9

Class 9 : Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev

The document Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev is a part of the Class 9 Course Mathematics (Maths) Class 9.
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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 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 D
Step 6: Taking B as the centre and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents √x. 

Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev 
Examples on representation of real numbers on number line: 
(1) Represent √(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 D
Step 6: Taking B as the center and BD as radius draw an arc cutting OC produced at E.Point E so obtained represents √(9.3). 

Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev 
(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

Representation of Real Numbers on number line & Operations on real numbers Notes | EduRev


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.

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: 
3√2 and 5√2 → Like terms.
-√7 and 3√7 Like terms.
2√3 and √5 Unlike terms.
2√11 and 5√6 Unlike terms.


Operations on real numbers 
(1) Add the 2√2 + 5√3 and √2 - 3√3
Solution: 
2√2 + 5√3 + √2 + (- 3√3)
2√2 5√3 √2 - 3√3 
= 3 √2 + 2√3

(2) Add the -6√3 + 3√2 and -2√2 – 4 √3
Solution: 
-6√3 + 3√2 + (-2√2 – 4 √3)
- 6√3 + 3 √2 - 2√2 - 4√3 
= -10√3 + 1√2

(3) Multiply (2√3) and (-3√5)
Solution: 
2 √3 x (-3√5)
= (2 x -3) √3 x √5
= - 6 √(3 x 5)
= -6 √15


(4) Multiply 3√2 (2 + 4√3)
Solution: 
3√2 (2 + 4√3)
= 3√2 x 2 + 3√2 x 4√3     [ use a distributive law]
= (3 x 2)√2 + (3 x 4)(√2 x √3)
= 6√2 + 12√6

(5) Divide √15 by √3
Solution: 
√15 / √3
= (√3 x √5) / √3
= √5

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