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Langara College - Department of Mathematics and Statistics
Internet Resources for the Calculus Student - Topics in Precalculus
The Irrationality of Root M for M not a Perfect Square
The irrationality of root2 is a special case of a more general fact.
Any square root that is not an integer is in fact irrational.
An easy way to see this is to show that the square of a rational number cannot be an integer unless the number itself is.
If the number has a finite decimal expansion this is clear because the
last nonzero digit of the square will come from squaring the last
nonzero digit of the number and so will be nonzero.
But not every rational number has a finite decimal expansion so this is not enough.
To deal with fractions that correspond to repeating decimals, we might
try using a similar argument in a different base, but will need to be
careful about that nonzero last digit. . . . Here goes:
Write our candidate square root as R=a+b/c for integers a,b,c with b<c and b and c having no common factors.
(In 'base c' notation this would be R=a.b)
Expanding gives R^2=(a+b/c)^2=a^2+2ab/c+b^2/c^2
(Here the first term is a whole number, the second term contributes in
the first 'c-imal' place and the last term contributes in the second
'c-imal' place. So the result certainly won't be an integer - unless
after "carrying" nothing is left to the right of the 'c-imal point'. So
what we need to show is that the last 'c-imal digit' is in fact
non-zero. That 'digit' would be the remainder on dividing b^2 by c, so
that is what we look at next.)
In fact, if b and c have no common factors, then c cannot divide
exactly into b^2, because if it did then each of its prime factors
would have to do so also and if a prime number divides into b^2 it must
also divide into b.
Of course the 'c-imal' stuff is not really necessary, it's just a cute
way of expressing the fact that the expansion of R^2 as a fraction has
denominator c^2 and is not reducible to lower terms (so is definitely
not an integer).
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Here is a shorter but perhaps less direct proof. (It can be found in
'The Book of Numbers' by Conway and Guy and is included along with several others on the 'Cut-the-Knot' website maintained by Alex Bogomolny):-
If RootN=B/A, then B/A=NA/B and equating fractional parts gives a/A=b/B (with a<A and b<B).
So unless B/A is an integer we have B/A=b/a, which gives an
expression for RootN in lower terms than we started with - which would
be a contradiction if we had started in lowest terms.