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Topic: Reductio Ad Absurdum
Replies: 4   Last Post: Jan 19, 1998 3:01 PM

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Alan Lipp

Posts: 41
Registered: 12/6/04
Re: Reductio Ad Absurdum
Posted: Jan 16, 1998 12:01 PM
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On Fri, 16 Jan 1998, William wrote:

> Dear Sir,
> please help me prove that the square root of 2 is an irrational
> number using "reductio ad absurdum". Please explain the method.
> Thanks a million



This is not the usual proof that you can find in many math books.
But I like it.

If the square root of two were not irrational then it could be written
as a fraction a/b where a and b are both whole numbers. Since
fractions can be reduced we may as well assume that a and b have
no common factor. If a and b have no common factor then a^2 and
b^2 cannot have a common factor either.

Square both sides to get a^2 / b^2 = 2 and multiply through by b^2
to get a^2 = 2 ( b^2)

In decimal notation the only possible last digits of a and b are
0,1,2,3,4,5,6,7,8, or 9. The last digits of a^2 and b^2 can only be
0, 1, 4, 9, 6, or 5.

And 2(b^2) must end in 0, 2, or 8.

Since a^2 = 2(b^2) the only possibility is that both end in 0. So a^2
ends in zero and b^2 ends in either 0 or 5. In either case 5 is a
common factor of a^2 and b^2 . . . a contradiction.


Alan Lipp
Williston Northampton School
Easthampton, MA

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