<![CDATA[This study included the geometric, algebraic, and arithmetic measurement of one-dimensional length for an important ancient theorem in mathematical history, named Pythagoras’ theorem, for a unit square. In this paper, we also tried to disclose a rigorous evaluation for the Pythagorean hypotenuses, which were founded incomplete square numbers by us through classical construction instead of considering flexible arithmetic approximation (root extraction) and incommensurable abstract algebraic point of view. Every conscious math reader knows that the characteristic of a theorem is that it produces true results, and the characteristic of a formula is that it may give approximate mathematical results. The much-discussed Pythagorean relation for the side of a right triangle does not satisfy the aspirations of mathematicians to measure the hypotenuse of a unit square, which remains elusive. Therefore, in some cases, this formula exhibits limitations in providing complete results in order to maintain the properties of a theorem.]]>
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