SOLUTION: prove that root of three is irrational

Question 708870: prove that root of three is irrational

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Assume

Square both sides:

Then we can say:

is odd, then

is odd because the product of any two odd numbers is odd. Similarly,

must then also be odd. Consequently,

is odd, and since by definition an odd number has no even factor,

must be odd.

If

is even, then

, and

must be even. But if

and

are both even, then

is not irreducible. Consequently,

and

must both be odd.

If

is the

th odd number,

. Likewise,

can be represented as

Go back to

and substitute:

A little algebra (verification left as an exercise for the student) gives us:

Since the LHS is clearly odd and the RHS is clearly even, this equation has no solutions over the integers. Therefore, reductio ad absurdum, the square root of 3 cannot be expressed as a rational number, hence

is irrational.

John

Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it

The Out Campaign: Scarlet Letter of Atheism

RELATED QUESTIONS
prove that square root of 3 is irrational (answered by nabla)

Prove that square root of 5 is... (answered by Fombitz)

Prove that cube root of 7 is an IRRATIONAL... (answered by stanbon,cparks1000000,ikleyn)

if the product of root a and root b is irrational prove that thier sum is also... (answered by Edwin McCravy)

prove that: 7-2 root 3 is an irrational... (answered by solver91311)

1.prove that the square root of 5 is an irrational number? (answered by psbhowmick)

prove cube root 6 is... (answered by solver91311)

Prove that 0.258 is irrational (answered by ikleyn,Theo)

prove that... (answered by Edwin McCravy)