non euclidean geometry pythagorean theorem

If you skipped 10th grade plane geometry because you just loved algebra, your path would take you right to the complex numbers. No, there are different axiomatisations possible so they are not even able to track all back to on and the same set.

Yes you can proof the Pythagorean Theorem in any axiomatisation, if anything if you could not prove it it would not be Euclidean geometry.ĭo they all track back to the same axioms? But for being the same theory they have to proof the same theorems. Theories (and Euclidean geometry is a theory) are defined by their theorems (everything that follows from the axioms and rules of inference) not by their axioms, so many different axiomatisations can give the same Theory. PS: Make sure you use the axioms for Euclidean geometry, you need to add the parallel axiom or an axiom that (together with the other axioms) can proof it. See for example: (there are many more and even within these examples there are different options) The axioms of his geometry were only found in late 19, beginning 20 century so it is reasonably fresh. (not just in relation to this proof but to any proof at all)Įuclid was a bit lacks with his Postulates and Common Notions,

What do you mean by equivalent in relation to proofs? Your Question is much more complicated than it looks.

Non euclidean geometry pythagorean theorem