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Your question is so confusing that I almost trashed it and am not sure yet what you want to know but I have a possible idea : consider a right triangle each of whose legs have length 1. By the Pythagorean theorem, the hypotenuse has length equal to the square root of 2. The square root of 2 is irrational- one can prove it is not equal to any fraction of integers, yet it is obviously is a number of some kind. Thus the number system had to be extended to include numbers of this kind.

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Q: Why was Pythagorean Theorem become necessary to incorporate in the number system?
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