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{ 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, serving most often as quantitative values (i.e. 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 are 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 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 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.
@ -17,33 +17,6 @@ Here are some [fun](fun.md) facts about numbers:
- There exist [illegal numbers](illegal_number.md), owing to the above mentioned fact that any information can be encoded as a number along with the fact that some information is illegal (see e.g. "[intellectual property](intellectual_property.md)").
- ...
## 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:
- **all**: Anything conceivable as a number, even by stretch. E.g. [zero](zero.md), minus [infinity](infinity.md) or aleph one.
- **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.
- **[transcendental](transcendental_number.md)**: Aren't algebraic. E.g. [pi](pi.md), [sine](sin.md) of [e](e.md) or two to the power of square root of two.
- **[imaginary](imaginary_number.md)**: Have the same properties as real numbers but lie in another dimension, on a line perpendicular to the real number line, going through 0 -- they are connected to real numbers by the fact that imaginary unit ([i](i.md)) squared equals minus one. E.g. 0, 3i or -i.
- **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.
- **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.
- **even**: Are divisible by 2. E.g. -8, 0 or 1024.
- **odd**: Aren't even. E.g. 1, -13 or 1023.
- **N0: [natural](natural_number.md) (with zero)**: E.g. 0, 16 or 1000.
- **[Fibonacci](fibonacci.md)**: Are part of a sequence that starts with 0 and 1 and continues with numbers each of which is the sum of previous two. E.g. 0, 3 or 89.
- **N: natural (without zero)**: "Caveman numbers", the kind of numbers people started to use first. E.g. 1, 10 or 945.
- **[prime](prime.md)**: Are only divisible by 1 and themselves, excluding 1. E.g. 2, 7 or 809.
- **composite**: Aren't primes, excluding 1. For example 4, 22 or 150.
One of the most interesting and mysterious number sets are the [prime numbers](prime.md), in fact many number theorists dedicate their whole careers solely to them. Primes are the kind of thing that's defined very simply but give rise to a whole universe of mysteries and whys, there are patterns that seem impossible to describe, conjectures that look impossible to prove and so on.
```
quaternions . imaginary line
projected : (imaginary numbers)
@ -66,6 +39,89 @@ One of the most interesting and mysterious number sets are the [prime numbers](p
*Number lines and some notable numbers -- the horizontal line is real line, the vertical is imaginary line that adds another dimension and reveals complex numbers. Further on we can see quaternion lines projected, hinting on the existence of yet higher dimensional numbers (which however cannot properly be displayed using mere two dimensions here).*
The following is a table demonstrating just one way of how you can play around with numbers -- here we just examine whole positive numbers (like number theorists would) up to 50, count each one's total number of divisors (excluding 1 and itself, 0 here means the number is [prime](prime.md) except for 1, if the number is highest in the series so far the number is called "highly composite"), unique divisors (excluding itself), sum of total and unique divisors (if the number equal sum of unique divisors, it is said to be a "perfect number"), average "dividing spread" (distance of each tested potential divisor's remainder after division from half of this tested potential divisor, kind of "amount of not dividing the number") in percents, maximum dividing spread and normalized range between smallest and biggest divisor expressed in percents (-1 if there are none). You can make quite interesting graphs from similar data and discover cool and interesting patterns.
| number | divisors |divisors uniq.|divisor sum|uniq. div. sum|avg. div. spread (%)|max div. spread (%)|div. range (%)|
| -------- | -------- | ------------ | --------- | ------------ | ------------------ | ----------------- | ------------ |
| 1 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 2 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 3 | 0 | 1 | 0 | 1 | 0 | 0 | -1 |
| 4 | 2 | 2 | 4 | 3 | 33 | 100 | 0 |
| 5 | 0 | 1 | 0 | 1 | 16 | 50 | -1 |
| 6 | 2 | 3 | 5 | 6 | 43 | 100 | 16 |
| 7 | 0 | 1 | 0 | 1 | 24 | 66 | -1 |
| 8 | 4 | 3 | 10 | 7 | 44 | 100 | 25 |
| 9 | 2 | 2 | 6 | 4 | 36 | 100 | 0 |
| 10 | 2 | 3 | 7 | 8 | 40 | 100 | 30 |
| 11 | 0 | 1 | 0 | 1 | 34 | 80 | -1 |
| 12 | 5 | 5 | 17 | 16 | 53 | 100 | 33 |
| 13 | 0 | 1 | 0 | 1 | 35 | 83 | -1 |
| 14 | 2 | 3 | 9 | 10 | 43 | 100 | 35 |
| 15 | 2 | 3 | 8 | 9 | 44 | 100 | 13 |
| 16 | 7 | 4 | 24 | 15 | 49 | 100 | 37 |
| 17 | 0 | 1 | 0 | 1 | 38 | 87 | -1 |
| 18 | 5 | 5 | 23 | 21 | 47 | 100 | 38 |
| 19 | 0 | 1 | 0 | 1 | 42 | 88 | -1 |
| 20 | 5 | 5 | 23 | 22 | 51 | 100 | 40 |
| 21 | 2 | 3 | 10 | 11 | 45 | 100 | 19 |
| 22 | 2 | 3 | 13 | 14 | 43 | 100 | 40 |
| 23 | 0 | 1 | 0 | 1 | 42 | 90 | -1 |
| 24 | 8 | 7 | 39 | 36 | 55 | 100 | 41 |
| 25 | 2 | 2 | 10 | 6 | 45 | 100 | 0 |
| 26 | 2 | 3 | 15 | 16 | 45 | 100 | 42 |
| 27 | 4 | 3 | 18 | 13 | 44 | 100 | 22 |
| 28 | 5 | 5 | 29 | 28 | 49 | 100 | 42 |
| 29 | 0 | 1 | 0 | 1 | 45 | 92 | -1 |
| 30 | 6 | 7 | 41 | 42 | 52 | 100 | 43 |
| 31 | 0 | 1 | 0 | 1 | 45 | 93 | -1 |
| 32 | 9 | 5 | 42 | 31 | 48 | 100 | 43 |
| 33 | 2 | 3 | 14 | 15 | 45 | 100 | 24 |
| 34 | 2 | 3 | 19 | 20 | 47 | 100 | 44 |
| 35 | 2 | 3 | 12 | 13 | 48 | 100 | 5 |
| 36 | 10 | 8 | 65 | 55 | 54 | 100 | 44 |
| 37 | 0 | 1 | 0 | 1 | 45 | 94 | -1 |
| 38 | 2 | 3 | 21 | 22 | 45 | 100 | 44 |
| 39 | 2 | 3 | 16 | 17 | 46 | 100 | 25 |
| 40 | 8 | 7 | 53 | 50 | 51 | 100 | 45 |
| 41 | 0 | 1 | 0 | 1 | 47 | 95 | -1 |
| 42 | 6 | 7 | 53 | 54 | 51 | 100 | 45 |
| 43 | 0 | 1 | 0 | 1 | 46 | 95 | -1 |
| 44 | 5 | 5 | 41 | 40 | 49 | 100 | 45 |
| 45 | 5 | 5 | 35 | 33 | 47 | 100 | 26 |
| 46 | 2 | 3 | 25 | 26 | 47 | 100 | 45 |
| 47 | 0 | 1 | 0 | 1 | 47 | 95 | -1 |
| 48 | 12 | 9 | 85 | 76 | 53 | 100 | 45 |
| 49 | 2 | 2 | 14 | 8 | 48 | 100 | 0 |
| 50 | 5 | 5 | 47 | 43 | 49 | 100 | 46 |
## 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):
- **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*).
- **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.
- **[transcendental](transcendental_number.md)**: Aren't algebraic. E.g. [pi](pi.md), [sine](sin.md) of [e](e.md) or two to the power of square root of two.
- **[imaginary](imaginary_number.md)**: Have the same properties as real numbers but lie in another dimension, on a line perpendicular to the real number line, going through 0 -- they are connected to real numbers by the fact that imaginary unit ([i](i.md)) squared equals minus one. E.g. 0, 3i or -i.
- **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.
- **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.
- **even**: Are divisible by 2. E.g. -8, 0 or 1024.
- **odd**: Aren't even. E.g. 1, -13 or 1023.
- **N0: [natural](natural_number.md) (with zero)**: E.g. 0, 16 or 1000.
- **[Fibonacci](fibonacci.md)**: Are part of a sequence that starts with 0 and 1 and continues with numbers each of which is the sum of previous two. E.g. 0, 3 or 89.
- **N: natural (without zero)**: "Caveman numbers", the kind of numbers people started to use first. E.g. 1, 10 or 945.
- **[prime](prime.md)**: Are only divisible by 1 and themselves, excluding 1. E.g. 2, 7 or 809.
- **composite**: Aren't primes, excluding 1. For example 4, 22 or 150.
One of the most interesting and mysterious number sets are the [prime numbers](prime.md), in fact many number theorists dedicate their whole careers solely to them. Primes are the kind of thing that's defined very simply but give rise to a whole universe of mysteries and whys, there are patterns that seem impossible to describe, conjectures that look impossible to prove and so on. Another similar type of numbers are the [perfect numbers](perfect_number.md).
Of course there are countless other number sets, especially those induced by various number sequences and functions of which there are whole encyclopedias. Another possible division is e.g. to *cardinal* and *ordinal* numbers: ordinal numbers tell the order while cardinals say the size (cardinality) of a set -- when dealing with finite sets the distinction doesn't really have to be made, within natural numbers the order of a number is equal to the size of a set of all numbers up to that number, but with infinite sets this starts to matter -- for example we couldn't tell the size of the set of natural numbers by ordinals as there is no last natural number, but we can assign the set a cardinal number (aleph zero) -- this gives rise to new kind of numbers.
**Numbers are awesome**, just ask any number theorist (or watch a numberphile video for that matter). Normal people see numbers just as boring soulless quantities but the opposite is true for that who studies them -- study of numbers goes extremely deep, possibly as deep as humans can go and once you get a closer look at something, you discover the art of nature. Each number has its own unique set of properties which give it a kind of "personality", different sets of numbers create species and "teams" of numbers. Numbers are intertwined in intricate ways, there are literally infinitely many patterns that are all related in weird ways -- normies think that mathematicians know basically everything about numbers, but in higher math it's the exact opposite, most things about number sequences are mysterious and mathematicians don't even have any clue about why they're so, many things are probably even [unknowable](knowability.md). Numbers are also self referencing which leads to new and new patterns appearing without end -- for example prime numbers are interesting numbers, but you may start counting them and a number that counts numbers is itself a number, you are getting new numbers just by looking at other numbers. The world of numbers is like a whole universe you can explore just in your head, anywhere you go, it's almost like the best, most free video [game](game.md) of all time, embedded right in this [Universe](universe.md), in [logic](logic.md) itself. Numbers are like animals, some are small, some big, some are hardly visible, trying to hide, some can't be overlooked -- they inhabit various areas and interact with each other, just exploring this can make you quite happy. { Pokemon-like game with numbers when? ~drummyfish }