Update

number.md

@@ -4,9 +4,11 @@ WIP
 
 { Sowwy I'm not a mathematician, please excuse if I'm wrong, lemme know if you spot something, thank u <3 ~drummyfish }
 
-Numbers are one of the most elementary [mathematical](math.md) objects, building stones serving most often as quantitative values (that is: telling count, size, length, order etc.), in higher math also used in much more [abstract](abstraction.md) ways which have only distant relationship to traditional counting. Examples of numbers are minus [one](one.md) half, [zero](zero.md), [pi](pi.md) or [i](i.md). Numbers constitute the basis and core of mathematics and as such they sit almost at the [lowest level](low_level.md) of it, i.e. most other things such as algebra, [functions](function.md) and [equations](equation.md) are built on top of numbers or require numbers to even be examined. In modern mathematics numbers themselves aren't on the absolute bottom of the foundations though, they are themselves built on top of [sets](set.md), as set theory is most commonly used as a basis of whole mathematics, however for many purposes this is just a formalism that's of practical interest only to some mathematicians -- on the other hand numbers just cannot be avoided anywhere, by a mathematician or just a common folk. The word *number* may be the first that comes to our mind when we say *mathematics*. The area of [number theory](number_theory.md) is particularly focused on examining numbers (though it's examining almost exclusively integer numbers because these seem to have the deepest pattern related e.g. to divisibility).
+Numbers (from Latin *numerus* coming from a Greek word meaning "to distribute") are one of the most elementary [mathematical](math.md) objects, building stones serving most often as quantitative values (that is: telling count, size, length, order etc.), in higher math also used in much more [abstract](abstraction.md) ways which have only distant relationship to traditional counting. Examples of numbers are minus [one](one.md) half, [zero](zero.md), [pi](pi.md) or [i](i.md). Numbers constitute the basis and core of mathematics and as such they sit almost at the [lowest level](low_level.md) of it, i.e. most other things such as algebra, [functions](function.md) and [equations](equation.md) are built on top of numbers or require numbers to even be examined. In modern mathematics numbers themselves aren't on the absolute bottom of the foundations though, they are themselves built on top of [sets](set.md), as set theory is most commonly used as a basis of whole mathematics, however for many purposes this is just a formalism that's of practical interest only to some mathematicians -- on the other hand numbers just cannot be avoided anywhere, by a mathematician or just a common folk. The word *number* may be the first that comes to our mind when we say *mathematics*. The area of [number theory](number_theory.md) is particularly focused on examining numbers (though it's examining almost exclusively integer numbers because these seem to have the deepest pattern related e.g. to divisibility).
 
-Let's not confuse numbers with digits or figures (numerals) -- a number is a purely abstract entity while digits serve as symbols for numbers so that we can write them down. One number may be written in many ways, using one of many [numeral systems](numeral_system.md) (Roman numerals, dots, Arabic numerals of different [bases](base.md) etc.), for example 4 stands for a number than can also be written as IV, four, 8/2, 16:4, 2^2, 4.00 or 0b100. There are also numbers which cannot exactly be captured within our traditional numeral systems, for some of them we have special symbols -- most famous example is of course [pi](pi.md) whose digits we cannot ever completely write down -- and there are even numbers for which we have no symbols at all, ones that are yet not well researched and are only described by equations to which they are the solution.
+Let's not confuse numbers with digits or figures (numerals) -- a number is a purely abstract entity while digits serve as symbols for numbers so that we can write them down. One number may be written in many ways, using one of many [numeral systems](numeral_system.md) (Roman numerals, tally marks, Arabic numerals of different [bases](base.md) etc.), for example 4 stands for a number than can also be written as IV, four, 8/2, 16:4, 2^2, 4.00 or 0b100. There are also numbers which cannot exactly be captured within our traditional numeral systems, for some of them we have special symbols -- most famous example is of course [pi](pi.md) whose digits we cannot ever completely write down -- and there are even numbers for which we have no symbols at all, ones that are yet not well researched and are only described by equations to which they are the solution. Sure enough, a number by itself isn't too interesting and probably doesn't even make sense, it's only in context, when it's placed in relationship with other numbers (by ordering them, defining operations and properties based on those operations) that patterns and useful attributes emerge.
+
+Humans first started to use positive natural numbers, i.e. 1, 2, 3 ..., so as to be able to trade, count enemies, days and so on -- since then they kept expanding the concept of a number with more [abstraction](abstraction.md) as they encountered more complex problems. First extension was to fractions, initially reciprocals of integers (like one half, one third, ...) and then general ones. Around 6th century BC Pythagoras showed that there even exist numbers that cannot be expressed as fractions ([irrational numbers](irrational_number.md), which in the beginning was a controversial discovery), expanding the set of known numbers further. A bit later negative numbers were discovered/invented, likely in China. Adoption of the number [zero](zero.md) also took some time, with it first just having a limited use as a mere placeholder digit. Since 16th century a highly abstract concept of [complex numbers](complex_number.md) started to appear, which was later (19th century) expanded further to [quaternions](quaternion.md). With more advancement in mathematics -- e.g. with the development of set theory -- more and more concepts of new kinds of numbers appeared and still appear to this day. Nowadays we have greatly abstract numbers, ones existing in many dimensions, capable of counting and measuring infinitely large and infinitely small entities, and it seems we still haven't nearly discovered everything there is to know about numbers.
 
 Basically **anything can be encoded as a number** which makes numbers a universal abstract "medium" -- we can exploit this in both mathematics and programming. Ways of encoding [information](information.md) in numbers may vary, for a mathematician it is natural to see any number as a multiset of its [prime](prime.md) factors (e.g. 12 = 2 * 2 * 3, the three numbers are inherently embedded within number 12) that may carry a message, a programmer will probably rather encode the message in [binary](binary.md) and then interpret the 1s and 0s as a number in direct representation, i.e. he will embed the information in the digits. You can probably come up with many more ways.
 
@@ -96,10 +98,11 @@ The following is a table demonstrating just one way of how you can play around w
 
 ## Numbers In Math
 
-There are different types of numbers, in mathematics we classify them into [sets](set.md). Though we can talk about finite sets of numbers perfectly well (e.g. [modulo](mod.md) arithmetic, [Boolean algebra](boolean_algebra.md) etc.), we are firstly considering [infinite](infinity.md) sets (curiously some of these infinite sets can still be considered "bigger" than other infinite sets, e.g. by certain logic there is more real numbers than rational numbers, i.e. "fractions"). Some of these sets are subsets of others, some overlap and so forth. Here are some notable number sets (note that a list can potentially not capture all relationships between the sets):
+There are different types of numbers, in mathematics we classify them into [sets](set.md) (if we further also consider the operations we can perform with numbers we also sort them into algebras and structures like [groups](group.md), [fields](field.md) or [rings](ring.md)). Though we can talk about finite sets of numbers perfectly well (e.g. [modulo](mod.md) arithmetic, [Boolean algebra](boolean_algebra.md) etc.), we are firstly considering [infinite](infinity.md) sets (curiously some of these infinite sets can still be considered "bigger" than other infinite sets, e.g. by certain logic there is more real numbers than rational numbers, i.e. "fractions"). Some of these sets are subsets of others, some overlap and so forth. Here are some notable number sets (note that a list can potentially not capture all relationships between the sets):
 
 - **all**: Anything conceivable as a number, even by stretch. E.g. [zero](zero.md), minus [infinity](infinity.md) or aleph one.
-  - **[p-adic numbers](p_adic_number.md)**: Another way of generalizing rational numbers, they may (a bit mindblowingly) have infinitely many digits to the left (for which they are sometimes called *leftist numbers*).
+  - **[transfinite numbers](transfinite_number.md)**: Numbers that are in a sense "infinite", used to compare objects that are infinite in size (e.g. number sets themselves). E.g. omega, omega plus ten or aleph one. 
+  - **Qp: [p-adic numbers](p_adic_number.md)**: Another way of generalizing rational numbers, they may (a bit mindblowingly) have infinitely many digits to the left (for which they are sometimes called *leftist numbers*). E.g. (10-adic) ...333.33, ...87187, ...11112 etc. 
   - **H: [quaternions](quaternion.md)**: A sum of real number, imaginary number and two other kinds of numbers, forming a number in four dimensional space. E.g. 1 + i + j - k, 50 - 0.6k or 2i + 7j.
     - **C: [complex](complex_number.md)**: A sum of real and imaginary number, forming a number in two dimensional plane. E.g. 3 + 2i, 0.5 - 13i or 100i.
       - **[algebraic](algebraic_number.md)**: Are roots of one variable [polynomials](polynomial.md) with integer coefficients. E.g. 4/3, the [golden ratio](golden_ratio.md) or square root of two.
@@ -108,7 +111,7 @@ There are different types of numbers, in mathematics we classify them into [sets
       - **R: [real](real_number.md)**: Measure any continuous one dimensional quantity (such as height or length), the line they form is continuous. E.g. -0.3, [pi](pi.md) or cube root of 10000.
         - **negative**: Smaller than zero. E.g. -1, -123 or -1000.
         - **non-negative**: Aren't negative. E.g. 0, 1 or 1000.
-        - **positive**: Greater than zero. E.g. 1, 456 or 1000.
+        - **R+: positive**: Greater than zero. E.g. 1, 456 or 1000.
         - **irrational**: Aren't rational. E.g. [pi](pi.md), minus [e](e.md) or square root of 2.
         - **Q: [rational](rational_number.md)**: "Fractions", countable set, can be written as a fraction of two integers; between any two there is always another one, so they are very densely "packed", though the line they form is not truly continuous. E.g. -2/3, 0.12345 or 2135.
           - **Z: [whole (integers)](integer.md)**: Are [discrete](discrete.md), starting at zero, extending in positive and negative direction, all neighbors are spaced by the same distance of one unit. E.g. -5123, 32 or 0.
@@ -238,3 +241,5 @@ Here is a table of some notable numbers, mostly important in math and programmin
 | [i](i.md) (imaginary unit)          |                     | j * k                                    | part of complex numbers and quaternions                 |
 | [j](j.md)                           |                     | k * i                                    | one of quaternion units                                 |
 | [k](k.md)                           |                     | i * j                                    | one of quaternion units                                 |
+
+TODO: add some p-adic number to the table