Examine whether √2 is rational or irrational number . with solution also
Answers & Comments
Sprb0406
Assume root 2 as rational Root 2=p/q Squaring on both sides 2=p^2/q^2 2*q^2=p^2 Therefore,2 is a factor of p^2 2 is also factor of p. 2*m=p 2=p^2/q^2 2*q^2=p^2 2*q^2=(2m)^2 2*q^2=4m^2 q^2=4m^2/2 q^2=2m^2 q^2=2*m^2 Therefore, 2 is a factor of q^2 2 is also the factor of q But p and q are co-prime Therefore, our assumption is wrong Therefore, root 2 is irrational.
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Minisuresh
Let us assume that √2 is rational.that is √2 can be written as a/b.squaring on both side.2=asquare/bsquare. Asquare=2bsquare That is 2 divides asquare. By theorem 2 divides a So we can write a=2c for some integerc Squaring on both side asquare=4csquare Asquare can be written as bsquare So asquare= 4csquare 2bsquare=4csquare Bsquare=4/2 Bsquare=2csquare That is 2divides bsquare By theorem 2divides b Therefore a&b have at least 2as the common factor But this condredict that a and b have no other factor than one Therefore our assumption √2 is rational is wrong Therefore √2 is irrational
Answers & Comments
Root 2=p/q
Squaring on both sides
2=p^2/q^2
2*q^2=p^2
Therefore,2 is a factor of p^2
2 is also factor of p.
2*m=p
2=p^2/q^2
2*q^2=p^2
2*q^2=(2m)^2
2*q^2=4m^2
q^2=4m^2/2
q^2=2m^2
q^2=2*m^2
Therefore, 2 is a factor of q^2
2 is also the factor of q
But p and q are co-prime
Therefore, our assumption is wrong
Therefore, root 2 is irrational.
Asquare=2bsquare
That is 2 divides asquare.
By theorem 2 divides a
So we can write a=2c for some integerc
Squaring on both side asquare=4csquare
Asquare can be written as bsquare
So asquare= 4csquare
2bsquare=4csquare
Bsquare=4/2
Bsquare=2csquare
That is 2divides bsquare
By theorem 2divides b
Therefore a&b have at least 2as the common factor
But this condredict that a and b have no other factor than one
Therefore our assumption √2 is rational is wrong
Therefore √2 is irrational