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Prove that 3+2 is irrational.

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CBSE/Class 10/Mathematics/UNIT I: Number systems/Real numbers/NCERT Solutions/Exercise 1.3

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To prove : 3+2√5 is rational number. Ans: 3 is rational number. 2 is rational number. 2√5 is irrational number (product of rational and irrational is an irrational number) 3+2√5 is an irrational number (sum of rational and irrational number is irrational) Or Let P = 3+2√5 is... read more

To prove : 3+2√5 is rational number.

Ans: 3 is rational number.

2 is rational number.

2√5 is irrational number (product of rational and irrational is an irrational number)

3+2√5 is an irrational number (sum of rational and irrational number is irrational)

Or 

Let P = 3+2√5 is rational number ( we are assuming )

P= 3+2√5

P-3 =2√5

(P-3)÷2=√5

L.H.S. we assume P is a rational number .

P-3 is also rational number substraction of rational number always rational number .

(P-3)÷2 rational number (P-3) divide by rational number 2 hence (P-3)÷2 is rational number .

L.H.S. is rational While R.H.S. √5 is irrational L.H.S ≠R.H.S.

Our assumption is wrong so we can say 3+2√5 is an irrational number.

 

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Experienced Mathematics Teacher

Root 5 is an irrational number. Therefore 2 root 5 is an irrational number as product of an rational and irrational number is also an irrational number. Again 3+ 2 root 5 is an irrational number as sum of a rational and an irrational number is also an irrational number. Hence proved.
Comments

Rationalize it and you can find it's irrational
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Tutor

Let 3+2V5= Rational no. => 3+2V5=p/q =>V5=(p/q-3)/2 R.H.S. is a rational no. but V5 is not a rational no. => Wrong assumption. => it is an irrational no.
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Since 3 is the whole number and 3=3/1 which is rational but √5 is the irrational number.so 2√5 is also a irrational number .if we add any number to irrational number will be irrational. Explanation : 3+2*2.236067977.....=7.472135954....
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Tutor

Let 3+2√5 is a rational number =a/b,then 2√5=a/b-3=(a-3b)/b and √5=(a-3b)/2b. This shows √5 is a rational number.But √5 is an irrational number. So 3+2√5 is irrational
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