Aleak: Outcast Number
~ I am still waiting to find where this leads… ~
TL;DR — We are used-to Base 10 numbers, zero through nine. And folks have seen plenty of Binary numbers, all zeroes or ones. Here is a DIFFERENT way to express number, which also uses ONLY zeroes and ones. It has a different ordering, which is a goldmine for Mathematicians. I am only a grim, lonesome surveyor; many men after me will be digging.
Minimal Change
A few Principles guide the creation of Aleak:
“We Begin with the value Zero expressed as 0. For any block of data, that would be all-zeroes, such as 0000 0000 for an eight-bit register.”
“Smaller change is on the right, large change is on the left of a register; same as Binary.”
“We want to FLIP ONLY ONE BIT when we ADD ONE to a value. This minimizes the amount of transistor-activity required to store an incrementor.”
Together, these rules determine Aleak. Look at how counting proceeds!
Counting in Aleak
0: 0000
1: 0001 ← “flip first bit”
2: 0011 ← “flip second bit”
3: 0010 ← “flip first bit, again”
4: 0110 ← “flip third bit”
5: 0111 ← “flip first bit”
6: 0101
7: 0100 ← “flip first bit”
8: 1100
9: 1101 ← “flip first bit”
10: 1111
11: 1110 ← “flip first bit”
12: 1010
13: 1011 ← “flip first bit”
14: 1001
15: 1000 ← “flip first bit”
16: 0000 ← “return register to Zero”
Aleak selects which bit-location to flip using that same saw-toothed fractal as Binary, in a different way! As a result, the smallest Aleak term oscillates between 0 & 1 every TWO increments. The second-smallest term oscillates every FOUR terms. There is a simple cellular automata which converts a Binary value into Aleak, and another one to convert Aleak into Binary.
…But Why, Though?
You aren’t going to like my answer. Mathematicians have to live with this fact, so I hope you can stomach it: “I have NO CLUE. This just seems weird and cool in a totally new way. And almost every time Mathematicians find something like that, then somebody else uses it to do cool stuff after the Mathematician is long-dead.”
Aleak is my Time Capsule to the Future of Mathematics.
My hunch is that there are operators on Aleak, using Boolean or cellular automata, which are a shorter path to the correct answer when compared to computing with your numbers expressed-in-Binary. Take your Binary values, convert them into Aleak, then perform the Aleak operators; the answer is converted back to Binary. I haven’t even figured-out basic arithmetic in Aleak, however. It’s like looking into a fog — wild and wispy new terrain! I love Maths. :)
