Prove that √3 – √2 and √3 + √5 are irrational.
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Explanation:
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Class 10
>>Maths
>>Real Numbers
>>Revisiting Irrational Numbers
>>Prove that √(3) + √(5) is irrational.
Question
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Prove that
3
+
5
is irrational.
Medium
Solution
verified
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To prove :
3
+
5
is irrational.
Let us assume it to be a rational number.
Rational numbers are the ones that can be expressed in
q
p
form where p,q are integers and q isn't equal to zero.
3
+
5
=
q
p
3
=
q
p
−
5
squaring on both sides,
3=
q
2
p
2
−2.
5
(
q
p
)+5
⇒
q
(2
5
p)
=5−3+(
q
2
p
2
)
⇒
q
(2
5
p)
=
q
2
2q
2
−p
2
⇒
5
=
q
2
2q
2
−p
2
.
2p
q
⇒
5
=
2pq
(2q
2
−p
2
)
As p and q are integers RHS is also rational.
As RHS is rational LHS is also rational i.e
5
is rational.
But this contradicts the fact that
5
is irrational.
This contradiction arose because of our false assumption.
so,
3
+
5
irrational.
Solve any question of Real Numbers with:-
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Answer:
Let √3 इस an rational no.
√3= p/q < where p and q are integers and q is not is equal to 0>
on taking Sq. of both rhe side
3= p²/q²
3q²=p²
p² is divisible by 3 so , p is also divisible by 3
1-p² = 3q²
(3p)² =3q²
9p²= 3q²
3p²= q²
q²is divisible by 3 so , q is also divisible by 3
we know that in a rational no. p and q have no common factor , Hence, √3 is an irrational no.
######### Let √2 is an rational no.
√2= p/q < where p and q are integers and q is not is equal to 0>
on taking Sq. of both rhe side
2= p²/q²
2q²=p²
p² is divisible by 2 so , p is also divisible by 2
1-p² = 2 q²
(2p)² =2q²
4p²= 2q²
2p²= q²
q²is divisible by 2 so , q is also divisible by 2
we know that in a rational no. p and q have no common factor , Hence, √2 is an irrational no.
irrational no - irrational no = irrational no
so, √3 – √2 is an irrational no
use the same method for√3 + √5