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number.md
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number.md
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# Number
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WIP
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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 }
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WIP kind of
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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).
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@ -103,11 +101,11 @@ Now we may start working with the [data](data.md), let's for example notice we c
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```
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1
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.---.------------.------------.-----'--.-----.---.--.--.--.--.--.--.--.--.
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2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 <--- primes
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| .'---. .----+----. .---.-'-.--. .-'-. .-'-. | | |
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.----.-----------.------------.-----'--.-----.---.--.--.--.--.--.--.--.--.
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2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 <--- primes
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| .-'--. .----+----. .---.-'-.--. .-'-. .-'-. | | |
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4 6 9 10 15 25 14 21 35 49 22 33 26 39 34 38 46
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@ -127,7 +125,9 @@ Here patterns start to show, for example the level one of the tree are all prime
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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):
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- **all**: Anything conceivable as a number, even by stretch. E.g. [zero](zero.md), minus [infinity](infinity.md) or aleph one.
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- **[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.
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- **[transfinite (infinite) 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, beth two or aleph one.
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- **[surreal numbers](surreal_number.md)**, **\*R: hyperreal numbers**, **superreal numbers**, ...: Various extensions of real numbers, include also infinitesimals and some transfinite numbers.
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- **[infinitesimals](infinitesimal.md)**: Are closer to zero than any real number without actually being zero, i.e. "infinitely small" numbers, play big role in [calculus](calculus.md). E.g. 0.000...1 (with infinitely many 0 digits before the 1).
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- **Qp: [p-adic numbers](p_adic_number.md)**: Alternative way of generalizing rational numbers; p-adics are quite mindblowing as they may have infinitely many digits to the left side (for which they are sometimes called *leftist numbers*), there are numbers that are their own squares without either being 1 or 0, they also contain negative numbers and fractions without having to add extra symbols. There are different kinds of p-adic number sets for different *p*s, e.g. 10-adic, 3-adic and so on (prime number *p*s are chosen for good properties). E.g. (10-adic) ...333.33, ...87187, ...11112 etc.
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- **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.
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- **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.
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@ -145,6 +145,7 @@ There are different types of numbers, in mathematics we classify them into [sets
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- **odd**: Aren't even. E.g. 1, -13 or 1023.
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- **N0: [natural](natural_number.md) (with zero)**: E.g. 0, 16 or 1000.
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- **[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.
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- **[modulo](mod.md) numbers**: Finite sets of numbers up to some *N* which are allowed to "[overflow](overflow.md)", basic operations like subtraction and multiplication are still well defined. Numbers in computer mostly behave this way. E.g. numbers modulo 5 are 0, 1, 2, 3 and 4.
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- **N: natural (without zero)**: "Caveman numbers", the kind of numbers people started to use first. E.g. 1, 10 or 945.
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- **[prime](prime.md)**: Are only divisible by 1 and themselves, excluding 1. E.g. 2, 7 or 809.
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- **composite**: Aren't primes, excluding 1. For example 4, 22 or 150.
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@ -263,9 +264,9 @@ Here is a table of some notable numbers, mostly important in math and programmin
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| [Graham's number](grahams_number.md)| | g64 | extremely, unimaginably large number, > googolplex |
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| TREE(3) | unknown | | yet even larger number, > Graham's number |
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| [infinity](infinity.md) | | | not always considered a number, largest possible value |
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| [aleph](aleph.md) zero | | cardinality(N) | infinite cardinal number, "size" of the set of nat. num.|
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| [aleph](aleph.md) zero | | beth zero, cardinality(N) | infinite cardinal number, "size" of the set of nat. num.|
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| [i](i.md) (imaginary unit) | | j * k | part of complex numbers and quaternions |
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| [j](j.md) | | k * i | one of quaternion units |
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| [k](k.md) | | i * j | one of quaternion units |
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TODO: add some p-adic number to the table
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TODO: add some p-adic and infinitesimal
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