9th Class Mathematics Real Numbers Rationalization

Rationalization

Category : 9th Class

*         Rationalization

 

The process of making denominator of a irrational number to a rational by multiplying with a suitable number is called rationalization. This process is adopted when the denominator of a given number is irrational. The number by which we multiply the denominator or convert it into rational is called rationalizing factor.  

 

 

Rationalize the denominator of \[\frac{6}{\sqrt{7}+\sqrt{2}}\].

Solution:

We have:

\[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6\times (\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})}=\frac{6\times (\sqrt{7}-\sqrt{2})}{7-2}\]

Here, \[(\sqrt{7}-\sqrt{2})\] is rationalizing factor.

Therefore, \[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6}{5}(\sqrt{7}-\sqrt{2})\]  

 

*            Laws of Radicals Let

\[x>0\] be any real number if a and b rational number then

(i) \[({{x}^{a}}\times {{x}^{b}})={{x}^{a+b}}\]    

(ii) \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]

(iii) \[\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\]     

(iv) \[{{x}^{a}}\times {{y}^{a}}={{(xy)}^{a}}\]  

 

*            Euclid's Division Lemma

Let a and b be any two positive integer. Then, there exist unique integers q and r such that \[a=\text{bq}+\text{r},\text{ }0\le \text{r}<\text{b}\]

 

   

 

  • \[1\times 9+2=11\]                   \[9\times 9+7=88\]             \[9\times 9=81\]                                              \[6\times 7=42\]
  • \[12\times 9+3=111\]             \[98\times 9+6=888\]         \[99\times 99=980166\times 67=4422\]
  • \[123\times 9+4=1111\]        \[987\times 9+5=8888\]      \[999\times 999=998001\]                            \[666\times 667=444222\]

 

 

 

  • A particular point on the number line represents a particular rational number.
  • A rational number cannot be represented by two or more than two distinct points on a number line.
  • There are infinite real numbers between two distinct real numbers.
  • A and B be two rational number in which A < B then the n rational number between A and B are \[(A+x),(A+2x),........(A+nx)\] Where \[x=\frac{B-A}{n+1}\]
  • A non-terminating and non-repeating decimals are called irrational numbers.
  • The sum of two irrational numbers may or may not be irrational.
  • The difference of two rational numbers may or may not be irrational.
  • The quotient of two irrational number may not be irrational.
  • The quotient of two irrational number may or not be irrational.    

 

 

 

  Which one of the following statements is true for a rational number?

(a) It is in the form of \[\frac{p}{q}\], Where p and q are integers and\[p\ne 0\]

(b) The decimal representation of a rational number is either terminating or non-terminating and non-repeating decimals

(c) There are five rational number between two given rational numbers

(d) Rational numbers are either terminating or non-terminating and repeating decimals

(e) None of these  

 

Answer: (d)    

 

 

  Match the following:

(a) \[\frac{123}{128}\] (i) Non terminating and repeating decimal
(b) \[\frac{2318}{9900}\] (ii) Non terminating and non-repeating decimals.
(c) 0.01010010010001000... (iii) Rational number between two rational numbers\[x\]and y.
(d) \[\frac{1}{2}(x+y)\] (iv) Terminating decimals.

 

(a) a-iv, b-i, c-ii, d-iii                        

(b) a-ii, b-i, c-iv, d-iii

(c) a-iv, b-iii, c-ii, d-i                        

(d) a-ii, b-iv, c-iii, d-i

(e) None of these  

 

Answer: (a)  

Explanation:

(a) \[\frac{123}{128}\] Here, denominator is 128 whose prime factor \[~\text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\] have only 2 as a factor therefore, it is a terminating decimal, that is why (iv) is correct for (a).

(b) \[\frac{2318}{9900}\] the denominator is 9900 whose prime factor is \[\text{2}\times \text{3}\times \text{3}\times \text{5}\times \text{11}\]. Here, 3 and 11 as a factor which is other than 2 and 5, that is why \[\frac{2318}{9900}\] is non-terminating and repeating decimal, (b-i)

(c) 0.01010010010001000......... Here, number on the right of the decimal is not repeating periodically, that is why it is non-terminating and non - repeating decimal, (c-ii)

(d) For any two rational numbers \[x\] and y, the rational number which is between \[x\] and y is \[\frac{1}{2}(x+y).(d-iii)\].  

 

 

  Read the following statements.

(i) Rational number may or may not be an integer

(ii) Some rational number can be represented on a number line

(iii) On a number line only rational numbers can be represented

(iv) There are infinite number of rational number between two given rational numbers  

Which one of the following set of statements is correct?

(a) (i) and (ii)                                     

(b) (i) and (iv)

(c) (i), (iii) and (iv)                           

(d) (i), (ii) and (iv)

(e) None of these  

 

Answer: (b)      

 

 

Which one of the following is repeating decimals?

(a) \[\frac{22}{7}\]                                          

(b) \[\pi \]

(c) \[\frac{224}{135}\]                                   

(d) \[\frac{154}{448}\]

(e) None of these  

 

Answer: (d)    

 

 

  The length, breadth and height of a room are 5 m 25 cm, 3 m 25 cm and 1 m 25 cm respectively. The length of the longest rod which can measure the three dimensions of the room exactly will be:

(a) 50cm                                              

(b) 75cm

(c) 1m                                                  

(d) 25cm

(e) None of these  

 

Answer: (d)      

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