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I don't know why but whenever I'm bored I find myself coming up with systems of numerals. I like to set a few base shapes and combine these to make larger numbers. I like the challenge of representing large numbers in this way without it getting too messy. I posted one system like this five years ago (holy balls!) here but that was very limited, I think this one is much better.

I wanted to base the system on prime numbers -- this makes sense if I'm constructing other numbers from a base set -- and I also wanted an algebraic feel to it.


(Image created in Inkscape: very nice program that makes precise vector graphics easy.)

The Three Basic Methods of Operation
1. Multiplication - Unconnected symbols placed side-by-side are multiplied together.
2. Exponentiation - When one symbol is above another the bottom symbol is raised to the power of the top symbol.
3. Addition - Symbols that are combined are added together and treated as a single symbol. Symbols can be combined in two ways: either by joining the symbols together or placing one inside the other. See below for detailed rules on combining symbols.

Four is shown as the sum of two lines but can also be made using the other operations: "||" (2 x 2) or "=" (2 squared). Most numbers can be written in a variety of ways, but I have tried to only show the most efficient forms.

Rules for Combining Symbols
Some of these rules may seem odd but they are in place to maintain readability.
* Only unambiguous connections are valid: the direction of the first symbol approaching the join must be different (by at least 90 degrees) to the direction of the second symbol leaving the join. For example: five can be constructed by placing an upward curve at the top of a vertical line or by crossing the top of the curve with a horizontal line. joining the curve to the bottom of the line would be invalid as the curve leads into the line (or vice versa). Note that this rule only applies when using lines, there is no ambiguity when combining curves -- such as ~ (conjoined upward and downward curves representing 32) where one curve leads into the other -- so these combinations are valid.
* The maximum width of any combined symbol is two symbols, with vertical lines treated as half-symbols.
* Vessels are symbols that can accommodate other symbols or expressions, these include curves and combined symbols where the extremities form the boundary of an enclosed space. For example, the M-shaped form of 7 can hold a symbol or expression between its legs, however, the curve of the M cannot hold a symbol or expression because the vertical lines are outside the curve (the curve is not the extremity of the symbol). Whereas the curve in the vesseled form of 5 is valid as the horizontal line is within the curve.

It's looking pretty good so far, there are only two numbers that I think look a little messy (22 and 26). I could have introduced the second curve earlier (eg for 19) but I wanted to save it for 29 as it has a neighbouring prime only two away; that means I can simply add a line to the curve to make an h symbol for 31.

I like the way curves act as natural brackets, that was a nice coincidence. I'm not sure how high this can go but I'd hope it can get to at least 100 without introducing another curve.

looks pretty simple when you look on it closely.
maybe someone will make an encryption system based on it. who knows
it might be a good one too.

That's really cool. I like how there are multiple ways to write the same number.

If this were to be used formally, would you use the additive, multiplicative, or exponential form of the number first? e.g., "4" would be represented as "||", "˥", or "=" respectively. (Those are approximations for the characters, of course.)

IMO, it would be cool to see this system used in games for "alien messages," or some such.

EDIT: How would you represent zero?

Cheers,
Aaron

related: http://en.wikipedia.org/wiki/Penrose_graphical_notation
No, I don't understand it at all. xD

I have my own, 100%, totally unique, not-derived-from-anything, completely-my-creation way of representing numbers:


Seriously, I do get where you're coming from Obese. It's kinda like Kanji for numbers.

Tolkien for Lord Of The Rings used to make up languages, and symbols. Led Zeppelin had a few symbols, they looked like Tolkien's.



Well you should program the shapes to build themselves, then you can test them for any number.

haha, like a unit test. Anyway, pretty neat concept.

I've added some detailed rules and made the diagram look better.
I realised I was missing two tricks that would help create more numbers: Closing the curves with a horizontal lines I can now add 5 or 31 to anything. Shapes like the M-shaped 7 can also be used as vessels for addition (see 11). I'm finding out how important it is to try and make vessels out of prime numbers.


That would be pretty nice, seems quite a challenge though.
It would need to find the prime factors or ones that produce a number that could be added to a vessel to reach the target number. It would then choose the narrowest and most economic (fewest strokes) solution.

I wanted to come back to this to explore how arithmetic could work.
I decided that I'd use the dot symbol, which represents one, to also represent addition. This has a nice poetic ring to it if you think of "oneness": addition could be described as combining two numbers into a single number. Hopefully it wont cause confusion using the same symbol, but one is never used in other numbers since it's not a prime so it should be okay.
I also decided to use a colon (or 1^1 as valued in the prime-base system) as an equals sign (since "=" means 4!) for similar reasons (why would I ever need 1^1?).

I don't really like the idea of representing non-primes with joined symbols (such as the alternative forms of four and six), that goes against the principle of prime-based numerals, but it's very convenient and I feel that if this were a real system used by a real culture they would have used these informal joined forms of non-primes.



Going through the arithmetic example:
The first step is to break both numbers down into terms of two and three. This is simple for 14 since the 7 just needs to be broken apart and each part multiplied by 2, but for 11 you're required to know the arithmetical form written in terms of two and three. Then we count up the twos and threes and find we have five of each. Since the multiples of each are equal we can add the three and two together to make five, and this turns out to be five squared (25).

It's a cumbersome process but I think it has a charm to it. It's quite easy to see relationships between numbers in this way.

PS Penrose graphical notation looks very interesting and a similar (if way more complex) idea. Thanks for posting that.

This is the last time I'll bump this thread, I'll let it die if there's no interest after this.

I decided I didn't like using vessels (a symbol containing other symbols to be added to it) they felt like a really arbitrary and cheesy way of increasing the number of numbers that can be made. I simplified the curves to try and increase the possibilities. It had an interesting effect making the numerals look like a cross between Chinese and Arabic, while the old style looked like a mix of Egyptian hieroglyphs and Nordic runes.

I've attached the first twenty-four numbers, can you tell me why I coloured them in this way?:

Have you every seen "Imagining The Tenth Dimension" ?

http://www.youtube.com/watch?v=JkxieS-6WuA

Zeroth dimension is a point, first dimension is a line (your 1), second dimension is line with a branch off of it (your 4), third dimension is a fold (your 5). Just a cool coincidence I suppose.

although, that video is more or less gibberish http://www.youtube.com/watch?v=enqn2NLQ4As&t=20m41s

I didn't say it was a good video, I was just pointing out the similarities between his numbers and the symbols that the video shows representing each dimension.

@Neuro - I had to force myself to close that video. Interesting discussion.

I've added an important capability to the mathematics of this system. It involves the dot being used again for a special purpose: when placed above a number (acting as an exponent) it does not have its usual value of 1, but can you work out what its value is from this "proof"?

If you have not been following here's everything you need to know:
* The down-left-curve is equal to three
* A straight line (both when horizontal or vertical) is equal to two
* The larger symbol is a combination of the curve and line so, as one would expect, is equal to five
* ":" states equivalence
* Symbols above other symbols denote exponents (except ":" of course)
* Symbols beside each other represent a value equal to their product, as in algebra.