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Noah11
@Noah11
September 2021
2
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prove that:
√3 is Irrational!
Neat answer please!
Answers & Comments
MasterYaduvanshi
Let to be √3 be Rational number
Let it's simplest form be a/b
then,
Here,
Note:-
a and b are Integer having no common factor other than 1.
Again,
√3=a/b
Now,
Squaring the both sides means (L.H.S and RHS)
we get,
(√3)²=(a/b)²
(3)=a²/b²
Here, using crossmultiplication...
3b²=a²
Now,.
3 Divide a²
Again,
3 Divide a
Note Reason:- 3 is a prime Number and 3 Divide by a²
Now,
Let to be a=3c
then,
(3b²)=9c²
Note:- 9 divide by 3
b²=3c²
Again,
3 Divide by b²
3 Divide by a
Now,
3 is a common factor of a and b...
Thus,
√3 is an irrational number..
Note :- Conclusion- √3 is an irrational number.
1 votes
Thanks 15
DaIncredible
Great :o
MasterYaduvanshi
Thanks brother. !!!!!
DaIncredible
:p
MasterYaduvanshi
:-)
Prakhar2908
Gr8
MasterYaduvanshi
Thanks jee
Prakhar2908
: )
MasterYaduvanshi
:-)
MasterYaduvanshi
Thanks!!
vikram991
Here is your answer OK dude
We can provide a contradictional proof for it..
Firstly let us assume
Assumption:let √3 be a rational ….then as every rational can be represented in the form p/q where q≠0
Let √3=p/q where p,q have no common factor.
Now squaring on both sides we get 3=p^2/q^2
i.e 3*q^2=p^2
Which means 3 divides p^2 which implies 3 divides p
Hence we can write p=3*k
This gives 3*q^2=9*k^2
q^2=3*k^2
Which means 3 divides q^2 which implies 3 divides q.
3 divides p and q which means 3 is common factor for p and q.
And this is contradiction for our assumption that p and q have no common factor…
Hence we can say our assumption that √3 is rational is wrong…
And therefore √3 is an irrational…
hope it help you
0 votes
Thanks 4
nitya1984
hi
DaIncredible
No chats ✌
Noah11
@nitya1984,don't tend to start a conversation here
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Answers & Comments
Let to be √3 be Rational number
Let it's simplest form be a/b
then,
Here,
Note:-
a and b are Integer having no common factor other than 1.
Again,
√3=a/b
Now,
Squaring the both sides means (L.H.S and RHS)
we get,
(√3)²=(a/b)²
(3)=a²/b²
Here, using crossmultiplication...
3b²=a²
Now,.
3 Divide a²
Again,
3 Divide a
Note Reason:- 3 is a prime Number and 3 Divide by a²
Now,
Let to be a=3c
then,
(3b²)=9c²
Note:- 9 divide by 3
b²=3c²
Again,
3 Divide by b²
3 Divide by a
Now,
3 is a common factor of a and b...
Thus,
√3 is an irrational number..
Note :- Conclusion- √3 is an irrational number.
We can provide a contradictional proof for it..
Firstly let us assume
Assumption:let √3 be a rational ….then as every rational can be represented in the form p/q where q≠0
Let √3=p/q where p,q have no common factor.
Now squaring on both sides we get 3=p^2/q^2
i.e 3*q^2=p^2
Which means 3 divides p^2 which implies 3 divides p
Hence we can write p=3*k
This gives 3*q^2=9*k^2
q^2=3*k^2
Which means 3 divides q^2 which implies 3 divides q.
3 divides p and q which means 3 is common factor for p and q.
And this is contradiction for our assumption that p and q have no common factor…
Hence we can say our assumption that √3 is rational is wrong…
And therefore √3 is an irrational…
hope it help you