How to generate Pythagorean triples (example: 3,4,5), well one way at least.
Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy
we can write the difference of two squares
(x + y)2 – (x – y)2 = 4xy
and if we write x = A2 and y = B2 the right hand side is a square as well.
(A2 + B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2
which can be written as
(A2 + B2) 2 = (A2 – B2) 2 + (2AB) 2
the Pythagoras form.
Now just put in some integers for A and B
2 and 1 gives 3,4,5
Conjecture1: This process generates ALL the Pythagorean triples.
Conjecture2: Every odd number belongs to some Pythagorean triple.
My next post will be about finding the radius of the inscribed circle in a right angled triangle…..
The following is unreadable. Use the browser zoom or click the picture, or download the .doc file from https://howardat58.files.wordpress.com/2014/10/sums-of-powers.doc
As h approaches zero
I quietly despair.
It really is the limit.
Please don’t take me there.
The funny thing about the calculus is that it was brought into existence by Isaac Newton in 1666 or earlier, and was developed and used without the idea of limits for over 150 years. The first attempt to get rid of the troublesome infinitesimals was by Cauchy in 1821, where he introduced the chord slope (f(x + h) – f(x))/h. The whole business of finding a satisfactory definition of the derivative was finally achieved by Weierstrass in the mid 19th century.
So here we go with cubics, and the same approach can be used for any whole number power of x, even negative ones. You should try it.
Next time sin(x) and cos(x), so no more sin(h)/h stuff.
Here’s another one, from a grade 8 test, same source:
9. What is the decimal notation of 7×10-4? ( they mean 10 to the power 4)
Answer and EXPLANATION!!!!!!!!!!!!!!!!!!!!
9. D: Because the exponent of 10 is -4, the decimal which is located behind the 7 will move 4 spaces to the left, and any of the empty spaces will fill with 0’s. so 7x 10? -4=0.0007
I hope that the computer system used to present these tests can cope better with “strange” symbols. I looked as well behind the 7 and could not see a decimal.