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  • (i)Insert a rational number and an irrational number between the following. 2 and 3

(i)Insert a rational number and an irrational number between the following. 2 and 3
(ii)Insert a rational number and an irrational number between the following.0.1 and 0.1
(iii)Insert a rational number and an irrational number between the following.\frac{1}{3}\, and\, \frac{1}{2}
(iv)Insert a rational number and an irrational number between the following.\frac{-2}{5}\, and\, \frac{1}{2}
(v)Insert a rational number and an irrational number between the following.0.15 and 0.16
(vi)Insert a rational number and an irrational number between the following.\sqrt{2} and \sqrt{3}
(vii)Insert a rational number and an irrational number between the following.2.357 and 3.121
(viii)Insert a rational number and an irrational number between the following..0001 and .001
(ix)Insert a rational number and an irrational number between the following.3.623623 and 0.484848
(x) Insert a rational number and an irrational number between the following.6.375289 and 6.375738

 

 

 

 

 

Answers (1)

(i) Answer.  Rational number: \frac{5}{2}

Irrational number: 2.040040004 ……….
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.

Between 2 and 3
Rational number: 2.5 = \frac{25}{10}= \frac{5}{2}

and irrational number : 2.040040004

(ii) Answer. Rational number: \frac{19}{1000} 

Irrational number 0.0105000500005 ……..
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0 and 0.1:
0.1 can be written as 0.10
Rational number: 0.019 = \frac{19}{1000}
and irrational number 0.0105000500005

(iii)Answer. Rational number \frac{21}{60}
Irrational number : 0.414114111 ……
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
LCM of 3 and 2 is 6.
We can write \frac{1}{3} as \frac{1\times 20}{3\times 20}= \frac{20}{60} 

and \frac{1}{2} as \frac{1\times 30}{3\times 30}= \frac{30}{60}
Also, \frac{1}{3} = 0.333333….
And  \frac{1}{2}= 0\cdot 5
So, rational number between \frac{1}{3}  and \frac{1}{2} is \frac{21}{60}
and irrational number : 0.414114111 ……

(iv)Answer. Rational number: 0
Irrational number: 0.151551555 ……. 
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.

\frac{-2}{5}= -0\cdot 4 and \frac{1}{2}= -0\cdot 5
Rational number between -0.4 and 0.5 is 0
And irrational number: 0.151551555 …….

(v) Answer. Rational number: \frac{151}{1000}
Irrational number: 0.151551555 ……….
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.15 and 0.16
Rational number : 0.151 = \frac{151}{1000}
and irrational number 0.151551555

(vi) Answer.  Rational number: \frac{3}{2}
Irrational number: 1.585585558 ………
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature.
Between \sqrt{2}\, and\, \sqrt{3}

\sqrt{2}= 1\cdot 414213562373
\sqrt{3}= 1\cdot 732050807568
Rational number: 1.5 = \frac{3}{2}
and irrational number: 1.585585558

(vii) Answer.  Rational number: 3
Irrational number: 3.101101110………
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 2.357 and 3.121
Rational number: 3
Irrational number: 3.101101110………
(viii) Answer.  Rational number: \frac{2}{10000}
Irrational number: 0.000113133133 ……….
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.0001 and 0.001
Rational number: 0.0002 = \frac{2}{10000}
Irrational number: 0.000113133133

(ix) Answer. Rational number: 1
Irrational number: 1.909009000 ……
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 3.623623 and 0.484848
A rational number between 3.623623 and 0.484848 is 1.
An irrational number between 3.623623 and 0.484848 is 1.909009000 ……

(x) Answer. A rational number is \frac{63753}{10000}
An irrational number is 6.375414114111……..
Solution.        
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 6.375289 and 6.375738:
A rational number is 6.3753 = \frac{63753}{10000}
An irrational number is 6.375414114111……..

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