Mathematics of Boxy Numbers

June 22, 2008 at 3:48 pm | Posted in Uncategorized | Comments Off on Mathematics of Boxy Numbers

I’ve found some interesting mathematical concepts through researching my ideas presented in my previous blog posts on a scheme for representing numbers using Basic Primes, and drawing that representation as Boxy Numbers.

There is a function named pi, (written as the greek symbol), but pi isn’t used in reference to circles’ diameters and circumferences, but is instead (as far as I can tell) P. I. standing for prime index. pi(x) is equal to the number of primes equal to or less than x.

By altering my representation slightly so that the number one is represented as a single box:

So that two is the first prime, so two is a box with the representation of one inside it (which is how I previously drew three).

Then the representation of any number n is drawn by producing n‘s prime factors, f0 to fk and each factor fi is drawn as a box with the representation of pi(fi) inside it.

To make the representation slightly more aesthetic, you can fill in all the gaps with the representation of one. This is because everything is multiplied, and you can of course multiply by one as many times as you like without altering the product. Multiplication is also commutative, so the boxes may be rearranged to give a pleasant looking number, so long as the number of boxes within any box stays the same.

Multiplication is simply a case of drawing both sets of boxes for the numbers being multiplied.

Division and rational numbers are probably best drawn with a numerator over a divisor separated by a line. Common factors are easily identified and removed.

By contrast, addition and subtraction are intensely difficult.

Here’s a walk-through of representing your birthday as a bunch of boxes:

You could draw the number for the day, month and year individually, but you’ll probably end up with more a more interesting drawing if you start with one large number n = YYYYMMDD. So for my birthday, 8th September 1972, n=19720908

The first step is to produce the prime factors. There are lots of ways of doing this. The easiest way if you are reading this online is to use an online tool. The one from Maths Is Fun says my prime factors are 6763 x 3 x 3 x 3 x 3 x 3 x 3 x 2 x 2.

So I need nine boxes, containing the representation of pi of each factor.

Though boxes can be arranged in any way you like, in order to demonstrate the technique I’ll stay consistent, showing these vertically, as otherwise by the end of the post I’ll be running out of room. Web pages have a fixed width, but can be infinitely long:

 pi(6763)
 pi(3)
 pi(3)
 pi(3)
 pi(3)
 pi(3)
 pi(3)
 pi(2)
 pi(2)

There’s an online calculator of pi(n) too here.

It reports that there are:

871 primes less than or equal to 6763

2 primes less than or equal to 3

1 prime less than or equal to 2

So our boxes need to be:

 871
 2
 2
 2
 2
 2
 2
 1
 1

We repeat the process for each of these numbers, drawing a box for each prime factor, and writing pi(factor) in each box.

The prime factors of 871 are 67 x 13, so we need two boxes.

2 is prime, so each of the 2s need a single box.

1 has been defined as a box with nothing in it.

 pi(67)
 pi(13)

 pi(2)

 pi(2)

 pi(2)

 pi(2)

 pi(2)

 pi(2)

pi(67) = 19

pi(13) = 6

pi(2) we’ve previously seen = 1

So our boxes need to be:

 19
 6

 1

 1

 1

 1

 1

 1

Producing prime factors again, 19 is prime, so a single box is needed. 6s prime factors are 3 x 2, so two boxes needed

We’ll replace 1s as before.

So we’ve got:

 pi(19)

 pi(3)
 pi(2)

Evaluating the pi functions gives:

 8

 2
 1

Factorising gives:

 pi(2)
 pi(2)
 pi(2)

 pi(2)

Evaluating the pi functions gives:

 1
 1
 1

 1

Replacing the last 1s with their representation of an empty box gives the final representation:

Rearranging this into a pleasing arrangement gives