Two Proofs of the Irrationality of the Square Root of 2
<p>The Pythagorean theorem states that the square constructed on the hypotenuse of a right triangle (side <em>c</em> of the triangle in the following image) equals the sum of the squares constructed on the legs (sides <em>a</em> and <em>b </em>in the image). From this identity, it follows that <em>c</em> is equal to the square root of <em>a² +</em> <em>b²</em>.</p>
<p><img alt="" src="https://miro.medium.com/v2/resize:fit:700/1*dYxuhGiie5d0N7vxK4Liow.png" style="height:685px; width:700px" /></p>
<p>Given any unit of measurement, if the two legs of a right triangle are both 1 unit long, according to the Pythagorean theorem, the square constructed on the hypotenuse <em>c </em>must be equal to 1² + 1², that is, 2. It follows that <em>c</em> is equal in this case to the <strong>square root of 2</strong>.</p>
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