Astralica
Since Astralica is based on Logic and Computer Science as well,
Astralica has its own set of mathematical symbols,
some of them are easy one by one substitutions,
and some of them require a bit of explanation because some symbols work in a slightly different
from how we normally would use them.
And here are the list of them :
You may notice that some of them would share the same symbol
because in some way, they are connected.
Power, roots, logarithm, and natural logarithm
share the same symbol.
Factorial, nPr, nCr
share the same symbol because they are almost the same.
You might think
addition, subtraction, multiplication, division, and modulation
are simple substitutions, but it also can be used in a story-like behavior.
Soon we will disect each group
and understand how are the symbols made,
and how to use them.
Basic Operations
Addition, Subtraction, Multiplication, Division, and Modulation.
These 5 operations are the 5 basic operators,
they modify the quantity of an environment.
So here are the concept of each of them.
if you look at the symbols, they all share the "┐" part， which represents the box,
when you add things into it, it goes in from the top,
that's why the "+" symbol has the "⌠" part to show something is going inside from the top.
when you minus things from it, it drops from the bottom,
that's why the "-" symbol has the "⌡" part to show something is dropping outside from the bottom.
multiplication is the slight superior version of addition,
if you compare the add symbol and the multiply symbol, you can find that the multiply symbol has an extra stroke, that stroke is used to say "this is the superior version of it".
A similar concept goes for division,
by comparing the minus symbol and divide symbol, we can see that the divide symbol has one more stroke than the minus symbol.
Modulation is a modification of division,
that is why the symbol is derived from the divide symbol and an extra stroke is given to say "this is the modified version of division."
as you can see, there are 2 ways to use the symbols,
either we can use it as a string, or we can use "Story Script".
From the 1st example,
we can see that 3 is being added to the box with 2 sitting inside the box, and the result is 5.
From the 2nd example,
we can see that 4 is taken away from the box which has 7 inside of it before, and the result is 3.
From the 3rd example,
we can combine the number with the same operation and separate the numbers with a "math comma",
which is something like a lighting bolt.
From the 4th example,
we can combine both addition and subtraction together as well.
here we can see a similar pattern exists.
still, Astralica follows the "MultiplyDivision first, then AdditionSubtraction" rule, that's why sometimes we need parenthesis.
we also can use the "Story Script" for multiplication and division.
we can interpret multiplication as : it will scale the box upwards by how many times.
we can also interpret division as : it will slice the box downwards into several pieces.
you can notice that when we are writing negative numbers, we don't use the minus symbol to represent the negativity, instead, we have its specific symbol for that, and since we have the symbol for negativity, we also have the symbol to represent the positivity of a number, but most of the time, we don't use it.
Power/Root/Log
here we have a slightly complicated situation,
let's say we have a magical triangle,
the left corner will be a slot for the base,
the slot on the right would be the slot for a power,
and the slot on the top would be the slot for a product.
the main idea for this symbol is it acts like a function that takes 2 inputs,
and the output is the missing corner.
base on the different output, the triangle will act as different operations or functions
product
base
power
we can also choose to use the string script, let's say we use "Δ" to represent the triangle, and "," to represent the "math comma", and "_" to represent "empty space"
the main syntax would be: Δ(base, power, product)
it will be something like this :
Note that we can literally use the symbol as the triangle.
Factorial, nPr, nCr
here a similar concept occurs,
it is just the symbol for Factorial,nPr, nCr is different.
since these 3 operations are related to multiplication and division,
the symbol is derived from the multiplication symbol
the top left slot is the "starting number",
the top right slot is the "range number".
so take 8P5 for example,
a way to interpret this is "starting on 8, we need 5 terms"
so we would end up with:
8P5 = 8 x 7 x 6 x 5 x 4
but if the order doesn't matter (which is Combination instead of Permutation)
the bottom slot is the "non order number",
means "divide by the factorial of non order number"
usually the "non order number" and the "range number" are the same,
but they can be different
Pos & Neg
usually, when we want to express the positivity or the negativity of a quantity,
we reuse the plus symbol and the minus symbol,
which sometimes may cause confusion
and sometimes it costs more parenthesis.
Here in Astralica,
"positive" and "add" is different,
"negative" and "minus" is different,
that's why when we want to mention something is positive or negative,
we can use the positive symbol and negative symbol.
Equalities & Inequalities
The main idea for this part is that "either the scale is balanced or tilted"
so for "=", it is a balanced scale.
the approximation sign is a "kinda balanced" situation
for the "not equal" sign, the scale is not even balanced,
such that it even give up measuring (the horizontal stroke represents the level is on the floor under it's base)
for the "defined as" sign, the scale is so balanced, such that we need to emphasize its balance-ness
for the rest of them, just think about "which one is heavier ?" and it should be a piece of cake.
Geometry Components
when it comes to shapes, each shapes can be broken down into the following parts:
Component of 0 dimension : a point or a dot
Component of 1 dimension : a line
Component of 2 dimensions : a face, or a surface, or an area
Component of 3 dimensions : a mesh, or a volume
by theory, we can keep on going, but since we are just 3D creatures, let's stop here.
the L shape at the bottom left of the symbol means "it is a noun or a function"
Component Relationships
Since we have relationships between numbers and quantities,
we also have relationships between shapes
on the top left corner of each symbol here indicates that it is a Geometry Relation Symbol,
they are used to describe a certain relationship between 2 or more shapes.
the Perpendicular sign is just 2 perpendicular lines
the Parallel sign is 2 parallel lines with point symmetry tails
the Similar sign is 2 similar triangles
Specific Components
Here we have symbols made specially for these two shapes : Circle and Triangles
These symbols are used to mention a part of a Circle or a function relative with Triangles
to describe which shape we are referring to, we attach the shape to the L shape.
for the Circle:
for Circumference, it is the perimeter of the circle, or the outside of the circle, so here we marked the outside
for Diameter, it is the line that pass through the center of the circle enclosed by the perimeter, the symbol marks the diameter.
for radius, it is the half of diameter, and so the symbol marks it
Pi is a special number, it is related between the Circumference and the Diameter, so the both of them are marked
for line 01,
we state the Geometry Component, then the name of the component.
for line 04,
sine is a trigonometry function, so the name will be stated, then the input of the function will be enclosed by brackets
Set Theory - Set Relation
In Astralica Math, we also talk about Sets as well.
Here are the four main symbols to describe
if an element or a set belongs to a set or not,
note that the symbol can be flipped
if an element is not contained inside the set, meaning iy is outside the set,
therefore the stroke is outside the circle, otherwise, it is inside the circle (set)
a set is bigger than an element, thats why it has 2 strokes which is more than 1 stroke (an element)
Set Theory - Set Operation
Since Set Theory and Logic can be combined, both topic shares the same symbols.
Union & OR
if an element is either in any of the sets you mentioned, that element is considered,
if a situation requires any of the events you mentioned, that situation is considered.
the symbol is derived from the shape of 2 sets from a Venn diagram.
Intersect & AND
if an element is in all the sets you mentioned, that element is considered,
if a situation fulfills all the events you mentioned, that situation is considered.
the symbol is derived from the shape of the intersection of 2 sets on a Venn diagram.
Difference
All the elements in a set but not in another set mentioned are considered,
it can be considered as a filter.
the symbol is derived from
"the set is mentioned by the part which is not needed are cut off"
Compliment & NOT
all the elements not in the set mentioned are all considered.
this is can be considered as a NOT gate
the symbol is derived from "the element outside the set".
Set Theory - Special Sets
an Empty set contains nothing,
we call it "Set Zero" in Astralica,
in every system, there is only one Universal Set,
we call it "Set One"
the set which contains all the subsets of a set is known as the "Power set"
the cardinality of a powerset is 2^n where "n" is te cardinality of that set.
since it has something to do with the number 2, we call it "Set Two"
Serieses & Calculus
the way we use Sigma and Pi is the same as how we normally we would use it.
the same goes for differentiation and intergration.
just incase, in differentiation, we usually say "dy/dx", the letter "d" is used to represent "small changes", in Astralica we have a specific symbol for it
Proving & Reasoning
in Astralica, the "Sapling" Symbol represents "an event".
the root of an event is a reason, which also means "because".
the outcome of an event is a result, which also means "therefore".
In Math proving, sometimes we will use the symbol "for all" and "there exsists",
since the both symbols are just upside down letters,
Astralica is the same over here.
