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
P2=Q2 *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 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 P2=Q2 *2.
there is no common measuring unit

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