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Great Proofs ## √2 is irrational

This proof, a standard bearer for all reductio ad absurdum proofs, is one of the early mathematics proofs - said to be from around 500 B.C. from the followers of Pythagoras.

A 'rational' number is a fraction a/b, where a and b are integers. Another elegant proof:
In 1952, Robert Gauntt, then a freshman at Purdue University, gave the following proof.
a2 = 2b2 cannot have a non-zero solution in integers because the last non-zero digit of a square, written in base 3, must be 1, whereas the last non-zero digit of twice a square is 2.
(See multiplication in any base!)

See HERE for other proofs and generalizations.

Remark
Robert Gauntt used this result: The last non-zero digit of a square, written in base 3, must be 1.
A more general statement is the following:
The last digit of the square of any positive integer written in base 3 can not be 2.
A proof of this is immediate on noting that any positive integer n can be written as n = 3q + r, where q and r are integers and r is one of the integers 0,1, or 2. Then n2 = 9q2 + 6rq + r2. If r is 0 or 1, then r2 = r = 0 or 1. If r = 2, then n2 = 3q'+ 1. This means, in base 3, n2 ends in a 0 or 1.

Similarly one can prove the following.
The last digit of the square of any positive integer in base 5 can not be 2 or 3.
The last digit of the square of any positive integer in base 10 can not be 2,3,7,or 8.

Gauntt used the result for base 3 noting further that the last non-zero digit of the square must be 1.

REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Oxford University Press, 1978
 American Mathematical Monthly, Vol 63 (1956), P. 247

(G.R.T.)

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