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What is the most motivating way in introduction to Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is there any resources for this?
I have for gotten the details, but I believe to remember, that on a documentation about Stonehenge they said that the constructors must have known the theorem, or the triples. But anyway, I'm sure that they can be found in ancient constructions prior to Pythagoras.What is the most motivating way in introduction to Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is there any resources for this?
You could refer to what square numbers mean - the number of dots in a square array.What is the most motivating way in introduction to Pythagorean triples to undergraduate students? I am looking for an approach that will have an impact. Good interesting or real life examples will help. Is there any resources for this?
Is this a proof that all triples are given by the algorithm I gave? (Don't attribute it to me - it is well known and can easily be verified by computing ##a^2,b^2,c^2##).the proof of mathman's algorithm is short and sweet by geometry. note that the problem is equivalent to finding all pairs of rational numbers (x,y) such that x^2+y^2 = 1, i.e. to finding allpoints of the unit circle with both coordinates rational. By projecting along a line from the north pole (0,1), rational points on the circle correspond one - one with rational points on the real axis. then just compute where the line joining the points (0,1) and (n/m,0) meets the unit circle, and you get all points with coordinates (2mn/(m^2+n^2), (m^2-n^2)/(m^2+n^2)), i.e. you get all pythagorean triples as (2mn), (m^2-n^2), (m^2+n^2). (I changed a sign which switched the order of the first two points.)