How to prove that 2+?3 are irrational numbers.

Answers (1)

To prove that sqrt2+sqrt3 is an irrational number
Let us assume them to rational number first
sqrt2+sqrt3=fracab,where a and b are integers
sqrt2=fracab-sqrt3
On squaring both sides,we get
(sqrt2)^2=(fracab-sqrt3)^2
2=fraca^2b^2+3-2*sqrt3(fracab)
On further solving,we get
fraca^2b^2+3-2=2*sqrt3(fracab)
fraca^2b^2+1=2*sqrt3(fracab)
(fraca^2+b^2b^2)*fracb2a=sqrt3
(fraca^2+b^22ab)=sqrt3
Since a,b are integers ,(fraca^2+b^22ab) is a rational number
Now sqrt3 is a rational numbere.
It cotradicts to our assumption thatsqrt3 is an irrational number.
Hence our assumption is wrong and sqrt2+sqrt3 is an irrational number.

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