Paradoxes

Have you heard about Zeno’s paradoxes (Achilles and the Tortoise and The Arrow Paradox). Check them out. Well I have thought of one now. It is “You have half a circle then you add half of that and then half of that and keep on going.” Here is a demonstration.

Stages 1-3Stages 4-6No matter how many times you add another half it will still never reach one. When you think of it being fractions the pattern is 1/2, 3/4, 7/8, 15/16, 31/32, 63/64 and so on. The denominator is two to the power of whatever stage and the numerator is the denominator minus one. To show it in algebra, Number of Stage=S, Numerator=N and Denominator=D. The equations are D=2^S and N=D-1. ^ means to the power of. In decimals it is 0.5, 0.75, 0.875, 0.9375, 0.96875, 0.984375. You would think Infinity times a number equals Infinity but in this case it equals one.

pi (π), circles and spheres

pi – π

pi is a number that goes on forever. So it has every number pattern possible somewhere inside it. It can be used to work out area and volume of circles and spheres.

Usually we can use pi with just 2 decimal places:

pi = 3.14

I have memorised it to 7 places as this is what a calculator shows:

pi = 3.1415927

My Dad memorised it to 21 decimal places when he was 10:

pi = 3.141592653589793238462

Circles

Circumference of a circle:

2πr (r is the radius which you can see from my picture below)

Diameter is 2 times the radius so the circumference can also be πd

Area of a circle:

πr2 (r2 means radius squared which means radius times itself)

(this picture I drew using Inkscape but I had to save it as a png as the svg file could not be seen).

Spheres

Surface area of a sphere:

4πr2

Volume of a sphere:

4/3πr3

(this picture was from wikipedia)