The square root of 2 is not rational.
Apply the Euclidean algorithm in an attempt to find a
common measuring unit between the hypotenuse and a side of an isosceles
right triangle.

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Suppose there is a common measuring unit for both sides.

Then P units/Q units = root 2 where P and Q are
counting numbers.

Thus P = Q root 2, and

P^{2}=Q^{2} *2.

Now count the number of 2's on each side of the equation.

On the left side there must be an even number of 2's
because P^{2 } must have twice as many 2's as P.

But on the right side there are an odd number of 2's.

This is impossible because there are the same number
of 2's on both sides of the equation P^{2}=Q^{2} *2.

*Thus *

**there is no common measuring unit**

**and "the square root of 2 is not a rational number".**