Hiking 18/6/2009 - More Pictures

Hiking 18/6/2009

Do you know that Greeks find it very hard to write down equations and number problems?

Because the Greeks had only very clumsy ways of writing down numbers,they didn't like algebra.

instead,Greeks mathematicians were more focused on geometry,and used geometric methods to solve problems that you might use algebra for.

They were also very interested in proving that certain mathematical ideas were true.
So they spent a lot of time using geometry to prove that things are always true,even though people like Egyptians and Babylonians already knew that they were true most of the time anyway.

The Greeks in general,were very interested in rationality,in things making sense and hanging together.They wanted to tie up the loose ends.They liked music,because music followed strict rules to produce beauty.So did architecture and so did mathematics.

Interesting and Little-Known Algebra and Geometry Facts

Here are a few helpful and neat little facts that evade most students and teachers of algebra and geometry:

a) Though everyone can factor a^2 - b^2, a^3 - b^3, a^3 + b^3 and a^4 - b^4, most folks do not know that:
a^4 + b^4 = (a^2 + ab(sqrt(2)) + b^2) (a^2 - ab(sqrt(2)) + b^2
b) An approximation exists for the factorial function (for large n) which seems hardly related but works:
Stirling's Formula ---> n! ~~ e^(-n) n^n sqrt(2 (pi) n)
c) The area of any regular polygon of n sides, each of length x, is given by:
Area = (1/4)nx^2 (cot(180°/n))
d) The radii of circumscribed (R) and inscribed (r) circles within such regular polygons are given by:
R = (x/2) csc (180°/n) and r = (x/2) cot (180°/n)
e) The radius of a circle inscribed within any triangle of sides a, b, and c with semi-perimeter s is given by:
r = (sqrt (s (s-a) (s-b) (s-c)) / s
e) The radius of a circle circumscribed about any triangle of sides a, b, and c with semi-perimeter s is given by:
R = abc / 4 (sqrt (s (s-a) (s-b) (s-c))
f) The perimeter P and area A of polygons (of n sides) inscribed in a circle of radius r is given by:
P = 2nr sin(pi/n) and A = (1/2) nr^2 sin (2pi/n)
g) The perimeter P and area A of polygons (of n sides) circumscribed about a circle of radius r is given by:
P = 2nr tan (pi/n) and A = nr^2 tan (pi/n)
h) Factoring the seemingly prime expression a^4 + 4b^4 becomes (and there are a family of these)
a^4 + 4b^4 = (a^2 + 2b^2)^2 - (2ab)^2 (thanks to N. Hobson)