Neat little proof

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3 total messages Started by Sol Lerner Tue, 04 Aug 1992 12:04

#213

Author: Sol Lerner
Date: Tue, 04 Aug 1992 12:04

15 lines
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I am trying to remember a neat little proof that sqrt(2) is irrational.  I 
think it went something like this:

Assume sqrt(2) is rational, then sqrt(2) = p/q where q is the LCD.

Then 2 = p squared / q squared
...SOME MAGIC OCCURS...
q can not be the LCD therefore sqrt(2) is irrational.

Can anyone fill in the "...SOME MAGIC OCCURS..." line?

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Re: Neat little proof

#216

Author: Kelly D. Crawford
Date: Fri, 07 Aug 1992 23:45

30 lines
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In article <873@ceylon.gte.com>, lerner@hilbert.gte.com (Sol Lerner) writes:
> I am trying to remember a neat little proof that sqrt(2) is irrational.  I 
> think it went something like this:
> 
> Assume sqrt(2) is rational, then sqrt(2) = p/q where q is the LCD.
> 
> Then 2 = p squared / q squared
> ...SOME MAGIC OCCURS...
> q can not be the LCD therefore sqrt(2) is irrational.
> 
> Can anyone fill in the "...SOME MAGIC OCCURS..." line?
>

Here it is from G. H. Hardy, "A Course of Pure Mathematics", Cambridge
University Press, 1952 (note: x**2 means x squared)

Suppose, if possible, that p/q is a positive fraction, in its lowest terms,
such that (p/q)**2 = 2 or p**2 = 2q**2.  It is easy to see that this involves
(2q-p)**2 = 2(p-q)**2; and so (2q-p)/(p-q) is another fraction having the same
property.  But clearly q < p < 2q, and so p-q < q.  Hence there is another
fraction equal to p/q and having a smaller denominator, which contradicts
the assumption that p/q is in its lowest terms.

Kelly
-- 
Kelly D. Crawford             | Amoco Production Research | These are my own
kcrawford@trc.amoco.com       | 4502 East 41st Street     | opinions, not
phone: (918) 660-4043         | Tulsa, OK  74102          | Amoco's...
          >>> *** This line intentionally left blank *** <<<

Re: Neat little proof

#217

Author: Vineet Nene
Date: Sat, 08 Aug 1992 01:15

6 lines
682 bytes

Proof that Sqrt(2) is irrational :

Assume Sqrt(2) = p/q where p and q do not have any common factor.
=> p^{2}/q^{2} = 2 => p^{2} and q^{2} have a common factor.
=> p and q must also have a common factor.
=> Contradiction.

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