Dude, I am a 56 year old Ph.D. (Chemistry, from Purdue University) - I have always only learned the math I needed to "get by" in science. I have spent the last 5 hours watching you. You are amazing. I wish I had paid more attention in various math classes. Thank you! I cannot imagine why anyone would have a problem with any math with the internet available - especially with instruction by people like you. Thank you again!
Perhaps. ;) Or they might just be transcendental, though that might have its difficulties. Or they might be complex, which gets really unreal. Which just goes to show that, before you set out to argue with anyone, you'd better have their number! :D
Godel's theorems are also proved. Maths would be nowhere without 'By Definition' statements . By definition statements don't need to be proved because they can't be proved. Don't mistake examples or justifications as 'proof'.
That a square has 360 degrees is an assumption that only works in Euclidean (flat) geometry. It comes from Euclid's parallel postulate, which actually speaks on the terms of when straight lines must intersect (if the sum of the interior angles of two lines crossing a third is less than 180 degrees, the two lines must intersect on that side of the third), but is logically equivalent to "if two lines A and B cross a third line C at perpendicular angles, A and B are parallel (will never intersect)." You can apply this definition again, "if two lines C and D cross a third line A at perpendicular angles, C and D are parallel" and "if two lines C and D cross a third line B at perpendicular angles, C and D are parallel." Now you have constructed a rectangle (four sided regular polygon with four right angles). The problem, however, is that the parallel postulate has no evidentiary foundation; it cannot be proven to be an unbreakable rule. However, we built an entire system of geometry on it, to the point that if we discard it, our entire geometry changes. This may sound ridiculous, but we actually have real world situations where rectangles do not sum to 360 degrees (and thus triangles do not sum to 180 degrees). The two most common are spherical geometry (no lines can be parallel, triangles sum >180 degrees, like on the surface of a planet) and hyperbolic geometry (infinitely many unique lines parallel to line L can be drawn through point P, triangles sum
I understand that to present sophisticated ideas while teaching euclidean geometry is a terrible idea . But a teacher must be careful NOT TO MISLEAD the student into to thinking that euclidean geometry is iron cld, always true A proof in euclidean geometry is SEMI - iron clad! IF an ONLY IF the axioms of euclidean are accepted as true... then YES indeed euclidean theorems are true. But only I F-f-f-f euclidean axioms are true. Are they true? BUT (1) Euclidean geometry applied to outer space is a problem. Einstein has shown that the shortest line between two points is NOT straight but curved due to space-time, (2) Riemann geometry discards euclidean axiom 5 : that there are such a thing as parallel lines . Instead Riemann a geometry assumes the OPPOSITE of the axiom 5 ,. In Riemann geometry lines intersect at infinity
6:20 "we can demonstrate easily that given parallel lines, then alternate angles are equal." Exactly how do you demonstrate this easily? Isn't that an axiom?? Mathematicians have tried for 2000 years, in vain, to prove this The converse "if alternate interior angles are equal, then the lines are parallel" is a theorem.
the way i've seen it taught most often is through the study of transversals, which use the knowledge that supplementary angles sum to 180 degrees to demonstrate several other properties of angles
