We have actually talked about the Pythagorean Triple for a little bit, and I found it really interesting. Here are two links about the Pythagorean Triple, one of which is from Wikipedia, and the other is from Wolfram Math World. I think the Math World one is probably a bit too much for our discussion, but I like its idea of showing the Pythagorean Triple in graphs.
Here is a link about the Babylonian Pythagoras (Pythagoras's Theorem in Babylonian mathematics)
Oddly enough, the smallest Pythagorean Triple (3,4,5) when cubed and summed add up to 216, which is a perfect cube, giving the equation 3^3+4^3+5^3=6^3. This has been called Plato's Number http://en.wikipedia.org/wiki/Plato's_number
- Please edit your entry to use $\TeX$ for the mathematics. Ethan.
For the Pythagorean Triples:
Let x, y, z be three real numbers.
If x is an odd number starting from 3, then the sum of y and z is the square of x with a difference of 1.
(If x is 1, then either one of y or z has to be 1 and the other one has to be the square root of 2, which is not a real number.)
If x is an even number starting from 4, then the sum of y and z is half of the square of x with a difference of 2.
(If x is 2, then either one of y or z has to be 2 and the other one has to be 2 times the square root of 2, which is not a real number)
Also, if x, y, z is a Pythagorean Triple, and a, b, c is another Pythagorean Triple, and a is equal to the double of x, then b is also equal to the double of y and c is equal to the double of z.
I came up with this during class, that is why I posted this as soon as the class ended on Tuesday.
- Nice! Ethan
As i was searching the internet for some good ways to learn the Pythagorean triple formula i stumbled upon a silly video using the triples. It seems silly but when you use food in examples. I'm hooked.
I also found this other video which i remember my high school geometry teacher showing us. In the Wizard of Oz when the scare crow gets a brain he rattles off the Pythagorean Theorem wrong. He says "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side"
"Pythagorean Triples" just remind me a interesting experiment..I think someone may saw before, but I just wanna share it because I think it's really good to get idea for Pythagorean Triples..
supposed we have three box A,B, and C. For each Box, the top and bottom are square, x, y,and z and every box has height h.First we full box A and B with water, and then, we pour the water from A,B to C. if box C is completely full, 100% math up, then we can get V(A)+V(B)=V(C), which follows that h*x^2+h*y^2=h*z^2. Finally, we get x^2+y^2=z^2..