## 2017-09-05

### Betascript Mathematical Notation

Betascript got a mathematical notation. I tried to shed off traditional math notation as well as I could from a lifetime of indoctrination. Things that specifically got to go were fractional notation, representing equations as, well, equations, zero, and all the shenanigans with logarithms. The latter was heavily inspired by the Triangle of Power, a very useful notation that helps (YMMV) unify exponentiation, taking roots, and logarithms. So here goes.

The number system is duodecimal (base 12, dozenal) and the digits are slightly uncomfortably simple. 12 would have been small enough of a set to have completely different glyphs, but let's say Beta culture was developed by the kind of people that develop number systems.

The system is bijectional, meaning every numeral corresponds with exactly one number, unlike our system that has has redundancies like 1 = 01 = 001 etc. This is a rather roundabout way of saying that the system doesn't use zero, like Ancient Klingon . This should clear out any confusion, things to pay attention to are underlined:

Decimals are expressed by basically using the equivalent of scientific notation to shift the 'point'

Sums and products can be expressed in a variety of ways, with binary operators or surrounding with different kinds of brackets, or both. Simply stringing symbols together is considered summation.

There is no separate symbols for the reverses, ie. subtraction and division, nor for equalness. The relation is expressed just by writing the result next to the operation. (This probably leads to problems, but let's call this notation a draft...)

The same principle of stating a relation is used to combine exponentiation, taking a root, and logarithms. The default form is <base>⊥<exponent> over <result>, and some additional notation allows you to write the result on the same line, left or right, optionally enclosed in brackets.

If the need arises to refer to one of the variables (ie. as in 'selecting' one of the three types of operation), one can use a dot to denote what is considered the result. The same can be used to denote division and subtraction.

The symbol for e is a square, and it forms a ligature with the exponentiation sign.

Minus one, the ubiquitous unsung constant, has its own symbol. Used with different operations it yields useful things like negative numbers, reciprocals, and another useful constant, namely zero. Ligatures get formed, and the imaginary unit gets a further simplified symbol.

Trigonometry tries to make sense with the circle constant and functions that hint at their meaning. Zero degrees is y-axis, not our x. This might be a bad idea. Again using the idea of expressing a relation, the reverse operations are expressed by omitting the argument of the forward version.

Finally, some familiar formulas rendered in Betascript. The last example adds a simple notation for summing series and a sign for infinity. As I'm a beginner in this notation, there might be mistakes or ambiguities.

I can't say anything about the practicality of this system in actually doing mathematics, but creating it certainly was interesting. Some things like the role of -1 fell nicely into place. I have a hunch that similar clicking might happen if one were to go further with this, like with learning any new way of looking at things, like the aforementioned Triangle of Power, reverse Polish notation, or Haskell.

The logical next steps from here would be a) to test the system on actually doing maths, and b) extending it to calculus. My understanding of the latter is, however, probably too shallow to see clearly enough to be able to create anything interestingly different or logical.