Counting the Hard Way
rupertmckay.com
rupertmckay.com
It has the nice property that it's a) binary, and b) each number contains a '11' substring at the end, but nowhere else. This makes it slightly less efficient than the usual base 2 binary, but handles variable length integers nicely; you always know where one number ends and the next begins, so you don't have to worry about overflow or specifying the length of a number.
The trick with this base is recognising that powers of 2i "rotate" through the Gaussian plane."
Powers:
-1: -(1/2)i, 0: 1, 1: 2i, 2: -4, 3: -8i, 4: 16, 5: 32i, 6: -64 etc.
PDS: Now this is highly interesting to me! A system which collapses a 2D number system, that is, non-imaginary + imaginary numbers (AKA, the "Complex (2D) Plane") -- back into what is basically a number line -- or a single-dimensional linear system (choose whatever terminology or linguistics you prefer...)
But the point is -- it reduces dimensionality from N to N-1 dimensions -- while still preserving imaginary numbers...
To accomplish this weird trick, Knuth spaces out imaginary numbers (and negative numbers) at intervals of 4 -- basically you cycle through properties (positive real, positive imaginary, negative real, negative imaginary) with every cycle of 4.
Could this be accomplished with other bases and other number systems having more than one dimension?
Based on Knuth's example, it would seem to be very possible!
I can only wonder what interesting mathematical properties future mathematicians might discover by using such structures...
They might turn out to be useful for a lot of things!
I'd be highly curious if there is any Mathematician who has applied such a structure to one or more specfic problems, and if it helped them solve a specific problem in an elegant or novel way...
What would also be interesting would be a list of mathematical problems (like a wiki page) whose computation could be aided by using such a structure, and how specifically to perform calculations using the structure, etc., etc.
Yes, all sets ℤⁿ (n > 0) have cardinality ℵ₀ and therefore a bijection can be defined for any set ℤⁿ↔ℤ.
The "computer science" way to think of this is, you can always e.g. interleave the bits of any number of integers, to encode them as a single unique integer. For two dimensions this is the Z-order curve [1].
Brilliant!
Each rational number has the form a / b for a pair of integers (a, b). This representation could be non-unique, e.g. 1/2 = -3/-6 are two different representations of the same rational number as fractions of integers. For this representation to be unique let's require the denominator b to be positive and a, b to have no common divisor. Let's consider (a, b) a point with integer coordinates on the lattice Z x Z, where Z is the set of integers (Z = { ..., -2, -1, 0, 1, 2, ... }).
We can count each of these points if we begin by labeling the origin (0, 0) as "0" then (1, 0) as "1" then (0, 1) as "2",(-1, 0) as "3", etc. We can order these integer lattice points first by their increasing magnitude |a| + |b| from the origin and then by the angle they make relative to the origin and count them all one by one.
Informally this shows that the set of rational numbers is not larger (in the sense of cardinality) as N_{>=0}, the set of non-negative integers. It's not smaller either as the set of non-negative integers N_{>=0} are a subset of rational numbers.
The cross products of countable set are countable and a bijection between N and NxN which you can then extend as been known for ages.
The whole thing falls apart as soon as you use continuous set.
In other words, infinity*infinity=infinity whilst 2^infinity>infinity
The proofs are pretty easy, but that doesn't make the result less surprising to people who don't know it yet.
That is, even within the same base, you can have different ways of doing things. "Balanced ternary" is the one that seems like it had the most potential outside of binary, in a computer. That ship has firmly sailed, at this point.
Base 10: 1,2,5,10
Base 12: 1,2,3,4,6,12
Base 60: 1,2,3,4,5,6,10,12,15,20,30,60
https://en.m.wikipedia.org/wiki/Superior_highly_composite_nu...
https://en.wikipedia.org/wiki/Multiplication_table#Modern_ti...
Unfortunately you have to memorize your times tables. There's no other way. You can't build number sense without having operations on small numbers in your meat L1 cache.
Base 10 strikes me as good tradeoff