Cool Number Systems

Just a compilation of a bunch of neat number systems I've found or made.

The basics

The systems showcased will follow a few basic rules:

• Every system will have a guaranteed notation for every positive integer from one to infinity
• No number has more than 1 way one way to write a given integer
• Systems will use Arabic numerals with any additional characters being filled in by Latin characters (and then theoretically infinite hypothetical characters)

Every system showcased will have numbers shown in a table with decimal (base ten) digits on the left, and the system's digits on the right

Many systems showcased will use positional notation or variations of the idea.

System 1

Base 1 (Unary)

Unary is the simplest number system, it's representing a number with the same amount of characters.

The closest thing that we currently use is tally marks, with the only difference grouping into 5s for countability.

The biggest downside of unary is of course that it can't represent large numbers readably.

System 10

Base 2 (Binary)

Other than decimal, this is the most well known system on this list, being used in computer science.

Being based around the number 2, the system makes itself very universal and applicable to a lot of problems.

This is my personal favorite base, see this fantastic video for a great argument for human usability

System 3

Base 5 (Quinary)

A system based around 5's should be somewhat more familliar to decimal users.

Interestingly, because it's an odd numbered base, 10 is an odd number, while 20 is even, the places switch back and forth accordingly.

System 10000

Base √2

This is the first "cursed base" as I'll call it, with the base number being a square root.

Like many sytems on this list, this is mostly just a mathmatical side effect rather than a human usable base.

powers of √2 flip-flops between rational and irrational: (1, √2, 2, 2√2, 4, etc...) so only every other number is used to represent integers.

It ends up just looking like spaced out binary with zeros filling in the gaps.

System 101

Base -2 (Negabinary)

I quite enjoy how unique this one ends up looking, with numbers having to cancel out.

Powers of -2 flip-flop between positive and negative: (1, -2, 4, -8, 16, etc...)

In order to write something like 8, it has to be -8 + 16.

System 1-0

Base 3 (Balanced Ternary)

While this one is very similar to base 3, it has the cool property of incorporating negatives.

In essence the system is just base 3 with the 2 replaced with -1.

It counts up very similar to a normal base, but with digit places halfway to the power of 3.

The system has a cool property where negative numbers are just positives when you swap 1 and - .

System 1000

Prime number multiplying system

This one is a system of my creation, its built around prime multiplication.

Its a neat fact of primes that you can form any integer by multiplying them.

This system takes full advantage of this to have each digit place represent a given prime.

Its actually pretty useful for visualizing the make up of numbers, but is terrible for addition.

System 110

Factorial (factoradic)

This one is interesting because it gets a new symbol every digit place.

It goes perfectly with the joke that bigger numbers should get the biggest number symbols.

Interesting while the only rule is that each place is worth the place factorial, it also ends up with each digit place being its own number base.

System 3.1

Base 1/6

This system is unique in the aspect that its just one over base 6.

It ends up as base 6 with all the digits flipped around to be past the decimal point.

This is because the fractional base cancels out the negative power.

System 10202

Base 2i (Base √-4)

Since imaginary numbers are mathmatically just the square root of a negative number, the base gains the properties of both the square root base, and the negative base.

It has the same count down thing as the negative and the same spacing as the root.

System ~101.0110

Base π (pinary)

This is another cursed one because it ends up as having all integers being an approximation.

Pi is more than 3 so we need a 3 to get to the point that we can use the second digit place.

As terrible as it is, its really just a classic positional notation base.

System 41

Base 3 Tetration system (Tetrary 3)

This one uses the same idea as positional notation, but uses tetration instead of exponentiation.

Base 3 was a nice size (relatively) and is as such: ⁰3, ¹3, ²3, ³3 or 1, 3, 27, 7625597484987.

It starts out using reasonably sized numbers, but very quickly gets out of hand in symbol count.

Hey, thanks for reading. Nothing here is too serious, just some fun number systems I've found. Hope this page was interesting, have a good one.