Nat Notation

Nat notation is a calligraphic way of writing numbers that describes numbers in terms of their prime factors. It's partially based off of Jan Misali's base-neutral number system for naming number systems. It's called Nat notation because it's used to represent the natural numbers, it's "natural" (i.e. doesn't rely on a specific base), and because I'm Nat and I made it.

The Rules

The rules of basic Nat notation:

  1. Two is a circle.
  2. Powers are stacked vertically.
  3. Multiples are concatenated horizontally.
  4. Numbers are expressed as combinations of their prime factorization.
  5. Prime numbers are expressed as the previous number plus one (represented by a flourish or swash).

Here are the numbers from one to twelve expressed in basic Nat notation:

Numbers from 1 to 12 in the notation described above. 1 is a swash, 2 is a circle. 3 = 2+1, which is a circle with a flourish. 4 is 2^2, a circle on a circle, and so on

Extended notation

If we restrict ourselves to only using addition, prime numbers can end up becoming pretty unwieldy to represent. Thus, we amend the fifth rule:

  1. Prime numbers are expressed as the previous number plus 1 or the next number minus 1, whichever is "simplest".

Definition for "simplest" may vary but I usually take it to mean "the option with the smallest maximum number that appears in its prime factorization".

With this expanded definition, the written forms of 7 and 11 are slightly different:

Comparison of the basic and extended versions of 7 and 11. In basic Nat notation, 7 is expressed as 23 + 1, but in extended, it's 2^3 - 1. In basic notation, 11 is represented as 25 + 1, but in extended notation it's 2^2*3 - 1.

This notation makes it easier to express certain prime numbers, such as the Mersenne primes.

Other encodings

Nat notation only requires:

  1. A way to express 2
  2. A way to express multiplication
  3. A way to express exponentiation
  4. A way to express incrementing (and decrementing) by 1

Any system where those requirements are met is a valid Nat notation. You can use whatever encoding you want to create Nat notation numbers.

For example, here is the largest known prime number, a Mersenne prime, expressed in a stylized format:

Nat notation representation of the largest known Mersenne prime, 2^82589933 − 1, in a stylized format akin to chinese or japanese calligraphy, drawn to look like a rearing dragon or snake

Purpose and Examples

Nat notation is pretty hard to calculate and pretty unwieldy to use for ordinary counting. I see it more of how we use Roman numerals nowadays - a fancy way to write numbers for special occasions. It's fun for mathematically or culturally significant large numbers. Here are some works that I've done:

52 factorial written in nat notation the factors forming a ring
52!, the number of possible ways to shuffle a deck of cards.
78 factorial written in nat notation with the factors forming a ring
78!, the number of possible ways to shuffle a tarot deck.
The order of the Monster group (a number around 8 times 10 to the power of 53) drawn in Nat notation as a ring of its prime factors, with the 2 factor in the center, making it look like a biblically-accurate angel
The order of the Monster group.
The second sublime number in Nat notation with its factors arranged in the shape of a fern.
The second of two known sublime numbers, the other of which is 6.
The smallest Euler brick (described below) with its side lengths, face diagonals, and volume represented in Nat notation in the vague shape of a prism.
Representation of the smallest Euler brick, with side lengths (44, 117, 240), face diagonals (125, 244, 267), and volume 1235520.

Requests and making your own

I encourage people to use Nat notation to make their own artwork! For calculating prime factors of large numbers, I recommend using Wolfram Alpha (I may get around to making my own Nat notation converter at some point). Feel free to reach out to me on Twitter with your own art, or if you a number you'd like me to write in Nat notation with my calligraphy!