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: