Wednesday, October 06, 2010

Superparticular Craziness

For my classical arithmetic and geometry class I'm studying Nicomachus's Intro to arithmetic.  It's pretty funky stuff.  Today I studied superparticular numbers - really mind bending.  Not sure of their significance but cool!

Check this out


If 
x = the superparticular number
n = the number being compared to x, or the subsuperparticular number
y = the fraction of n that is a factor of x 

Then 
x= n + yn


For example when y = 1/2 the equation would be 
x = n + ½ n or 6 = 4 + ½ (4).

If y is 1/2 the equation will work with the following pairs of numbers

3:2
6:4
9:6
12:8
15:10

If y is 1/3 the equation works with these numbers

4:3
8:6
12:9
16:12
20:15

If y is 1/4 it works with these
5:4
10:8
15:12
20:16
25:20

If w is 1/5 it works with
6:5
12:10
18:15
24:20
30:25

Look at these number pairs arranged this way

(n is 1/2)            3:2         6:4         9:6        12:8        15:10
(n is 1/3)            4:3         8:6         12:9      16:12      20:15
(n is 1/4)            5:4        10:8        15:12    20:16      25:20
(n is 1/5)            6:5        12:10      18:15    24:20      30:25
For the # below        +1          +2            +3         +4           +5           

2 comments:

Juli said...

Congratulations. My mind is officially broken.

Jennifer said...

Mine too Juli. I'm drowning in the second half of the assignment.