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number.md
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@ -14,6 +14,8 @@ Basically **anything can be encoded as a number** which makes numbers a universa
But what really is a number? What makes number a number? Where is the border between numbers and other abstract objects? Essentially number is an abstract mathematical object made to model something about [reality](irl.md) (most fundamentally the concept of counting, expressing amount) which only becomes meaninful and useful by its relationship with other similar objects -- other numbers -- that are parts of the same, usually (but not necessarily) infinitely large set. We create systems to give these numbers names because, due to there being infinitely many of them, we can't name every single one individually, and so we have e.g. the [decimal](decimal.md) system in which the name 12345 exactly identifies a specific number, but we must realize these names are ultimately not of mathematical importance -- we may call a number 1, I, 2/2, "one", "uno" or "jedna", it doesn't matter -- what's important are the relationships between numbers that create a STRUCTURE. I.e. a set of infinitely many objects is just that and nothing more; it is the relationships that allow us to operate with numbers and that create the difference between integers, real numbers or the set of colors. These relatinships are expressed by operations (functions, maps, ...) defined with the numbers: for example the comparison operation *is less than* (<) which takes two numbers, *x* and *y*, and always says either *yes* (*x* is smaller than *y*) or *no*, gives numbers order, it creates the number line and allows us to count and measure. Number sets usually have similar operations, typically for example addition and multiplication, and this is how we intuitively judge what numbers are: they are sets of objects that have defined operations similar to those of natural numbers (the original "cavemen numbers"). However some more "advanced" kind of numbers may have lost some of the simple operations -- for example [complex numbers](complex_number.md) are not so straightforward to compare -- and so they may get more and more distant from the original natural numbers. And this is why sometimes the border between what is and what isn't a number may be blurry -- for example it can't objectively be said if infinity is a number or not, simply because number sets that include infinity lose many of the nicely defined operations, the structure of the set changes a lot. So arguing about what is a number ultimately becomes subjective, it's similar to arguing about what is and isn't a planet.
An [interesting](interesting.md) remark: someone once plotted the number of occurrences of numbers in the online encyclopedia of integer series (OEIS) and discovered a curiosity (called Sloane's Gap). There is a clear gap separating numbers into two [clusters](cluster.md), one containing the "interesting numbers" and the other the rest ("boring numbers"). The interesting set contains primes, increments of powers of two and so on. This is partly [cultural](culture.md) (there is a bias towards base 10 for example), but it's very interesting the gap is so clear -- one would expect there would be a spectrum of how interesting numbers are, but it seems like it's just two clusters.
**[Order](order.md)** is an important concept related to numbers, we usually want to be able to compare numbers so apart from other operations such as addition and multiplication we also define the comparison operation. However note that not every order is total, i.e. some numbers may be incomparable (consider e.g. complex numbers).
Here are some [fun](fun.md) facts about numbers:
@ -224,14 +226,20 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| | 4.940656... * 10^-324| | smallest pos. number storable in IEEE-754 64 bit float |
| | 1.401298... * 10^-45 | | smallest pos. number storable in IEEE-754 32 bit float |
| | 1.616255... * 10^-35 | | Planck length in meters, smallest "length" in Universe |
| | 0.071111111111111... | 0x0.123456789abcdef101... | base 16 Champernowne constant |
| one eight | 0.125 | 2^-3, 0b0.001, 0x0.2 | |
| | 0.163264812105216... | &0.1234567101112131415... | base 8 Champernowne constant |
| one fourth | 0.25 | 2^-2, 0b0.01, 0x0.4 | |
| one third | 0.333333... | 3^-1, 0b0.0101010..., ...1313132 (5-adic)| |
| one half | 0.5 | 2^-1, 0b0.1, 0x0.8 | |
| | 0.598958... | base3(0.121011122021221001...) | base 3 Champernowne constant |
| one over square root of two | 0.707106... | 1/sqrt(2), sin(pi/4), cos(pi/4), 2^(-1/2)| |
| | 0.862240124493837... | 0b0.110111001011101111... | base 2 Champernowne constant |
| [one](one.md) | 1 |2^0, 0!, 0.999..., sqrt(1), I, 0b1, cos(0)| NOT a prime, unit, multiplicative identity |
| | 0.123456789101112... | 0b0.000111111001101011... | base 10 Champernowne constant, normal number |
| [square root](sqrt.md) of two | 1.414213... | sqrt(2), 2^(1/2), 0b1.0110101 | irrational, diagonal of unit square, important in geom. |
| supergolden ratio | 1.465571... | solve(x^3 - x^2 - 1 = 0) | similar to golden ratio, bit more difficult to compute |
| Euler's constant | 0.577215... | 1 + 1/2 + 1/3 + 1/n ... - log(n) | some kinda deep and important constant `O_O` |
|phi ([golden ratio](golden_ratio.md))| 1.618033... | (1 + sqrt(5)) / 2, solve(x^2 - x - 1 = 0)| irrational, visually pleasant ratio, divine proportion |
| square root of three | 1.732050... | sqrt(3), 3^(1/2), 0b1.1011101 | irrational |
| square root of pi | 1.772453... | sqrt(pi) | |
@ -242,7 +250,7 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| [pi](pi.md) | 3.141592... | 2 * asin(1), 0b11.0010010 | circle circumference to its diameter, irrational |
| [four](four.md) | 4 | 2^2, 0b000100, IV, 0b100 |first composite number, min. needed to color planar graph|
| [five](five.md) | 5 | 3^2 - 2^2, V, 0b101, fib(5) | (twin, triplet) prime, number of platonic solids, Fib. |
| [six](six.md) (half dozen) | 6 | 3!, 1 * 2 * 3, 1 + 2 + 3, VI, 0b110 | highly composite number, perfect number |
| [six](six.md) (half dozen) | 6 | 3!, 1 * 2 * 3, 1 + 2 + 3, VI, 0b110 | highly composite number, 1st perfect number |
| [tau](tau.md) | 6.283185... | 2 * pi, 360 degrees | radians in full circle, defined mostly for convenience |
| [thrembo](thrembo.md) | ??? | | the hidden number |
| [seven](seven.md) | 7 | 2^3 - 1, VII, &7, 0b111 |(twin) prime, days in week, max. unsigned n. with 3 bits |
@ -265,6 +273,7 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| twenty four | 24 | 2 * 2 * 2 * 3, 4!, 0x18, XXIV |highly composite number, possible ways to order 4 objects|
| twenty five | 25 | 5^2, sqrt(625), 0x19, XXV | |
| twenty seven | 27 | 3^3, 0b11011, 0x1b, &33, 0x1b, XXVII | palindrome in base 2 and 8 |
| twenty eight | 28 | 0b11100, 0x1c, XXVIII | 2nd perfect number |
| twenty nine | 29 | 0b11101, &1002, 0x1d, XXIX | twin&sexy prime |
| thirty one | 31 | 2^5 - 1, 0b11111, &37, 0x1f, XXXI |max. unsigned number storable with 5 bits, Mersenne prime|
| [thirty two](thirty_two.md) | 32 | 2^5, 0b100000, &40, 0x20, XXXII | number of possible values storable with 5 bits |
@ -312,16 +321,19 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| three hundred sixty | 360 | 2 * 2 * 2 * 3 * 3 * 5, CCCLX | highly composite number, degrees in full circle |
| three hundred sixty five | 365 | 0x16d, CCCLXV | days in a year, binary palindrome |
| four hundred twenty | 420 | 0x1a4, CDXX | stoner shit (they smoke it at 4:20), divisible by 1 to 7|
| four hundred ninety six | 496 | 0x1f0, CDXCVI | 3rd perfect number |
| five hundred eleven | 511 | 2^9 - 1, DXI | largest number storable with 9 bits |
| five hundred twelve | 512 | 2^9, 2^(3^2), DXII | number of values storable with 9 bits |
| six hundred twenty five | 625 | 25^2, 5^4, DCXXV | |
| six hundred and sixty six | 666 | 0x29a, DCLXVI | number of the beast, palindromic |
| seven hundred twenty | 720 | 6!, 3!!, DCCXX | possible ways to order 6 objects, highly composite |
| seven hundred twenty nine | 729 | 3^6, (3^2)^3, DCCXXIX | |
| shitload | ??? | a lot | expressed a bigger quantity |
| one thousand (grand) | 1000 | 10^3, M, 0x3e8, 2^9.965... | |
| one thousand twenty three | 1023 | 2^10 - 1, &1777, 0x3ff, MXXIII | largest number storable with 10 bits |
| one thousand twenty four | 1024 |2^10, 4^5, &2000, 0x400, MXXIV, 10^3.01...| number of values storable with 10 bits |
| one thousand six hundred eighty | 1680 | 0x690, MDCLXXX | highly composite, often used as horizontal resolution |
| one thousand seven hundred ... | 1729 | 0x6c1, MDCCXXIX | Ramanujan number, taxican number, part of math lore |
| two thousand forty eight | 2048 | 2^11, 0x800, MMXLVIII | number of values storable with 11 bits |
|two thousand one hundred eighty seven| 2187 | 3^7, 0x88b, MMCLXXXVII | |
| two thousand four hundred one | 2401 | 7^4, MMCDI | |
@ -333,6 +345,7 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| five thousand fifty | 5050 | 1 + 2 + ... + 100 | sum of numbers up to 100 |
| six thousand five hundred sixty one | 6561 | 3^8, 3^(2^3) | |
|six thousand seven hundred sixty five| 6765 | fib(20), 0x1a6d | Fibonacci number |
|eight thousand one hundred ... | 8128 | 0x1fc0 | 4th perfect number |
| ten thousand (myriad) | 10000 | 10^4, 100^2, 2^13.287... | |
| fifteen thousand six hundred ... | 15625 | 5^6, 0x3d09 | |
| sixteen thousand eight hundred ... | 16807 | 7^5, 0x41a7 | |
@ -357,16 +370,19 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| | 43046721 | 3^16 | |
| | 47176870 | BB(5) | maximum number of 1s produced by 5 state Turing machine |
| | 31556926 | | seconds in a year |
| | 33550336 | | 5th perfect number |
| | 39916800 | 11!, 1 * 2 * ... * 11 | possible ways to order 11 objects |
| | 479001600 | 12!, 1 * 2 * ... * 12 | possible ways to order 12 objects |
| one [billion](billion.md) | 1000000000 | 10^9, milliard, 0x3b9aca00, 2^29.897... | |
| | 9876543210 | 0x4cb016ea | all decimal digits from highest to lowest |
| | 2147483647 | 2^32 / 2 - 1 | maximum two's complement signed 32 bit number |
| | 3735928559 | 0xdeadbeef | one of famous hexadeciaml constants, spells out DEADBEEF|
| | 4294967295 | 2^32 - 1, 0xffffffff | maximum unsigned number storable with 32 bits |
| | 4294967296 | 2^32, ((((2^2)^2)^2)^2)^2, 0x100000000 | number of values storable with 32 bits |
| | 6227020800 | 13!, 1 * 2 * ... * 13 | possible ways to order 13 objects |
| | 8589869056 | 0x1ffff0000 | 6th perfect number |
| | 9876543210 | 0x24cb016ea | all decimal digits from highest to lowest |
| | 87178291200 | 14!, 1 * 2 * ... * 14 | possible ways to order 14 objects |
| | 137438691328 | 0x1ffffc0000 | 7th perfect number |
| | 500000500000 | 1 + 2 + ... + 1000000 | sum of numbers up to 1000000 |
| one trillion | 1000000000000 | 10^12, billion (LS) | |
| | 1307674368000 | 15! | possible ways to order 15 objects |
@ -379,6 +395,7 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| | 9007199254740992 | | precision of IEEE double falls below 1 after this num. |
| | 121645100408832000 | 19! | possible ways to order 19 objects |
| quintillion | 1000000000000000000 | 10^18 | |
| | 2305843008139952128 | 0x1fffffffc0000000 | 8th perfect number |
| | 2432902008176640000 | 20! | possible ways to order 20 objects |
| | 9223372036854776000 | 2^64 / 2 - 1 | maximum two's complement signed 64 bit number |
| | 18364758544493064000 | 0xfedcba9876543210 | all hexadecimal digits from highest to lowest |
@ -387,9 +404,12 @@ Here is a table of some numbers and "number like objects" worthy of mention, mos
| | 43252003274489856000 | | number of possible Rubik's cube configurations |
| |2015099950053364471960| | number of possible chess games after 15 half moves |
| |6670903752021072936960| | possible valid filled [sudoku](sudoku.md) grids |
| | 1.000000... * 10^30 | 1000000000000066600000000000001 | Belphegor's prime, evil (666, 13 zeroes), palindromic |
| | 1.267650... * 10^30 | 2^100 | number of values storable with 100 bits |
| | 2.658455... * 10^36 | | 9th perfect number |
| | 3.402823... * 10^38 | (2 - 2^(-23)) * 2^127 | largest number storable in IEEE-754 32 bit float |
| | 3.402823... * 10^38 | 2^128 | number of values storable with 128 bits |
| | 1.915619... * 10^53 | | 10th perfect number |
| | 1.157920... * 10^77 | 2^256 | number of values storable with 256 bits |
| | 10^80 | | approx. number of atoms in observable universe |
| [googol](googol.md) | 10^100 | | often used big number |