less_retarded_wiki/number.md

Number

WIP kind of

{ There's most likely a lot of BS, math people pls send me corrections, thank u. ~drummyfish }

Numbers (from Latin numerus coming from a Greek word meaning "to distribute") are one of the most elementary mathematical objects, building stones serving most often as quantitative values (that is: telling count, size, length, order etc.) and labels, in higher math also used in much more abstract ways which have only distant relationship to traditional counting. Examples of numbers are minus one half, zero, pi or i. Numbers constitute the basis and core of mathematics and as such they sit almost at the lowest level of it, i.e. most other things such as algebra, functions and equations are built on top of numbers or require numbers to even be examined. In modern mathematics numbers themselves do not reside on the absolute bottom of the foundations though, they are themselves built on top of sets, 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 (as the topic gets closer to the fringes of mathematics and at times rather pertains to philosophy) -- 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 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). Interest in numbers isn't exclusive to mathematics -- numbers also play an important role in culture and religion for example; some even believe in "magical" power of numbers (see numerology).

Numbers must not be confused with digits or figures (numerals) -- a number is a purely abstract entity while digits serve as symbols for numbers so that they can be written down. The same number may be represented with many different symbols, using one of many numeral systems (Roman numerals, tally marks, Arabic numerals of different bases etc.), for example 4 stands for a number than can also be written as IV, four, IIII, 8/2, 16:4, 2^2, 4.00 or 0b100. There are also numbers which cannot be exactly expressed with our traditional numeral systems, for some of them we have special symbols -- most prominent example is of course pi whose digits in decimal expansion form an infinite series -- and there are even numbers lacking any symbolic representation, ones not well researched yet or not important enough, 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.

To embark on the history a bit, humans first started to use positive natural numbers (it seems as early as 30000 BC), 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 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, which in the beginning was a controversial discovery), expanding the set of known numbers further. A bit later (around 100 BC) negative numbers started to be used. Adoption of the number zero also took some time (1st use of true zero seem to be in 4th century BC), with it first just having a limited use as a mere placeholder digit. Since 16th century a highly abstract concept of complex numbers started to appear, which was later (19th century) expanded further to quaternions. 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" -- this can be exploited in both mathematics and programming (which are actually the same thing). Ways of encoding information as numbers may vary, for a mathematician it is natural to see any number as a multiset of its prime 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 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.

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 (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 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 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.

Order 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 facts about numbers:

• Some people associate numbers with colors, though what color each number has seems to be completely subjective. See synesthesia.
• There is a funny hypothetical number between 6 and 7 called thrembo.
• There exist illegal numbers, 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").
• ...

 quaternions                . imaginary line
            projected       : (imaginary numbers)
 projected   j line     2i ~+~ ~ ~ ~ ~+ 1 + 2i
  k line       :            :         ,
     ...        :_          :         ,             complex numbers
        \___      \_ j      :         ,
            \___    +_   i ~+~ ~ ~ ~ ~+ 1 + i
                +___  \_    :         ,
               k    \___\_  :         ,
                        \_\_:         1         2         3         4
  - - -~|~-~-~-~-~|~-~-~-~-~+~-~-|-~-~|~-~-~|~-~|~-~-~-|-~|~|~-~-~-~|~- - -
       -2        -1        0:   1/2   ,    phi         e    pi           real line
                = i^2       :  = 0.5  ,    ~=         ~=   ~= 3.14...  (real numbers)
                            :         ,   1.61...    2.71...
                        -i ~+~ ~ ~ ~ ~+
                            :           1 - i
                            .

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 -- of course, we have generated it with a program, so we also practice programming a bit ;) Here we just examine whole positive numbers (like number theorists would) up to 50 and take a look at some of their attributes -- we count each one's total number of divisors (excluding 1 and itself, 0 here means the number is prime except for 1, if the number is highest in the series so far the number is called "highly composite"), unique divisors (excluding itself), minimum divisor (excluding 1 except for 1), maximum divisor (excluding itself except for 1), 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.

{ Be warned the following is just me making some quick unoriginal antiresearch, I may mess something up, it's just to show the process of playing around with numbers. ~drummyfish }

number	divisors	divisors uniq.	min. div.	max. div.	divisor sum	uniq. div. sum	avg. div. spread (%)	max div. spread (%)	div. range (%)
1	0	1	1	1	0	1	0	0	-1
2	0	1	2	1	0	1	0	0	-1
3	0	1	3	1	0	1	0	0	-1
4	2	2	2	2	4	3	33	100	0
5	0	1	5	1	0	1	16	50	-1
6	2	3	2	3	5	6	43	100	16
7	0	1	7	1	0	1	24	66	-1
8	4	3	2	4	10	7	44	100	25
9	2	2	3	3	6	4	36	100	0
10	2	3	2	5	7	8	40	100	30
11	0	1	11	1	0	1	34	80	-1
12	5	5	2	6	17	16	53	100	33
13	0	1	13	1	0	1	35	83	-1
14	2	3	2	7	9	10	43	100	35
15	2	3	3	5	8	9	44	100	13
16	7	4	2	8	24	15	49	100	37
17	0	1	17	1	0	1	38	87	-1
18	5	5	2	9	23	21	47	100	38
19	0	1	19	1	0	1	42	88	-1
20	5	5	2	10	23	22	51	100	40
21	2	3	3	7	10	11	45	100	19
22	2	3	2	11	13	14	43	100	40
23	0	1	23	1	0	1	42	90	-1
24	8	7	2	12	39	36	55	100	41
25	2	2	5	5	10	6	45	100	0
26	2	3	2	13	15	16	45	100	42
27	4	3	3	9	18	13	44	100	22
28	5	5	2	14	29	28	49	100	42
29	0	1	29	1	0	1	45	92	-1
30	6	7	2	15	41	42	52	100	43
31	0	1	31	1	0	1	45	93	-1
32	9	5	2	16	42	31	48	100	43
33	2	3	3	11	14	15	45	100	24
34	2	3	2	17	19	20	47	100	44
35	2	3	5	7	12	13	48	100	5
36	10	8	2	18	65	55	54	100	44
37	0	1	37	1	0	1	45	94	-1
38	2	3	2	19	21	22	45	100	44
39	2	3	3	13	16	17	46	100	25
40	8	7	2	20	53	50	51	100	45
41	0	1	41	1	0	1	47	95	-1
42	6	7	2	21	53	54	51	100	45
43	0	1	43	1	0	1	46	95	-1
44	5	5	2	22	41	40	49	100	45
45	5	5	3	15	35	33	47	100	26
46	2	3	2	23	25	26	47	100	45
47	0	1	47	1	0	1	47	95	-1
48	12	9	2	24	85	76	53	100	45
49	2	2	7	7	14	8	48	100	0
50	5	5	2	25	47	43	49	100	46

Now we may start working with the data, let's for example notice we can make a nice tree of the numbers by assigning each number as its parent its greatest divisor:

                                     1
                                     |
 .----.-----------.------------.-----'--.-----.---.--.--.--.--.--.--.--.--.
 |    |           |            |        |     |   |  |  |  |  |  |  |  |  |
 2    3           5            7       11    13  17 19 23 29 31 37 41 43 47  <--- primes
 |    |           |            |        |     |   |  |  |
 |  .-'--.   .----+----. .---.-'-.--. .-'-. .-'-. |  |  |
 |  |    |   |    |    | |   |   |  | |   | |   | |  |  |
 4  6    9   10   15  25 14  21 35 49 22 33 26 39 34 38 46
 |  |    |   |    |    | |   |        |
 |  |  .-'-. |  .-'-.  | |   |        |
 |  |  |   | |  |   |  | |   |        |
 8  12 18 27 20 30 45 50 28  42       44
 |  |  |     |
 16 24 36    40
 |  |
 32 48

Here patterns start to show, for example the level one of the tree are all prime numbers. Also in this tree we can nicely find the greatest common divisor of two numbers as their closest common ancestor. Also if we go from low numbers to high numbers (1, 2, 3, ...) we see we go kind of in a zig-zag direction around the bottom-right diagonal -- what if we make a program that plots this path? Will we see something interesting? We could use this tree to encode numbers in an alternative way too, by indicating path to the number, for example 45 = {2,1,1}. Would this be good for anything? Would such a representation facilitate some operations? You can just keep diving down rabbit holes like this.

Numbers In Math

There are countless different types of numbers, in mathematics we classify them into sets (and if we additionally consider operations with numbers too, we also sort them into algebras and structures such as groups, fields or rings). Although we can talk about finite sets of numbers perfectly well (e.g. modulo arithmetic, Boolean algebra etc.), we are often examining and using infinite 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, minus infinity or aleph one.
  • unknowable: Cannot be known for some reason, e.g. being non-computable or requiring more energy for their computation than will ever be present in our Universe.
    • noncomputable: Cannot be computed, i.e. any such number has no Turing machine which when passed N on input would output Nth digit of the number in finite time. E.g. Chaitin's constant (probability that a randomly generated program will halt).
  • transfinite (infinite) numbers: 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.
  • surreal numbers, *R: hyperreal numbers, superreal numbers, ...: Various extensions of real numbers, include also infinitesimals and some transfinite numbers.
    • infinitesimals: Are closer to zero than any real number without actually being zero, i.e. "infinitely small" numbers, play big role in calculus. E.g. 0.000...1 (with infinitely many 0 digits before the 1).
  • Qp: p-adic numbers: 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 ps, e.g. 10-adic, 3-adic and so on (prime number ps are chosen for good properties). E.g. (10-adic) ...333.33, ...87187, ...11112 etc.
  • H: quaternions: 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: A sum of real and imaginary number, forming a number in two dimensional plane. E.g. 3 + 2i, 0.5 - 13i or 100i.
      • complex integers: Complex numbers with both real and imaginary component being integer. E.g. 13 - 2i, 44i or 0.
      • algebraic: Are roots of one variable polynomials with integer coefficients. E.g. 4/3, the golden ratio or square root of two.
      • transcendental: Aren't algebraic. E.g. pi, sine of e or two to the power of square root of two.
      • imaginary: Are similar to 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) squared equals minus one. E.g. 0, 3i or -i.
      • R: real: Measure any continuous one dimensional quantity (such as height or length), the line they form is continuous. E.g. -0.3, pi or cube root of 10000.
        • negative: Smaller than zero. E.g. -1, -123 or -1000.
        • R0+: non-negative: Aren't negative. E.g. 0, 1 or 1000.
        • R+: positive: Greater than zero. E.g. 1, 456 or 1000.
        • irrational: Aren't rational. E.g. pi, minus e or square root of 2.
        • Q: rational: "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): Are discrete, 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 (with zero): E.g. 0, 16 or 1000.
              • Fibonacci: 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.
              • modulo numbers: Finite sets of numbers up to some N which are allowed to "overflow", 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.
              • N: natural (without zero): "Caveman numbers", the kind of numbers people started to use first. E.g. 1, 10 or 945.
                • prime: 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.
                  • highly composite: Composite numbers that have more divisors than any lower number. E.g. 4, 36 or 1260.
                  • perfect: Equal to the sum of its divisors. E.g. 6, 28 or 8128.

One of the most interesting and mysterious number sets are the prime numbers, 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.

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.

Worthy of mentioning is also linear algebra which treats vectors and matrices like elementary algebra treats numbers -- though vectors and matrices aren't usually seen as numbers, they may be seen as an extension of the concept.

Numbers are awesome, just ask any number theorist (or watch a numberphile video for that matter). Normal people perceive numbers just as boring, soulless quantities but the opposite is true for that who studies them with love -- the world of numbers is without a doubt staggeringly beautiful, their study runs to depths without end, possibly as far as humans can ever hope to get a glimpse of the mechanisms behind the curtains of our Universe, and oftentimes once you pay a closer attention to a seemingly innocently looking detail, you reveal a breathtaking pattern and 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. 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 of all time, embedded right in this Universe, in logic 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 }

There is a famous encyclopedia of integer sequences at https://oeis.org/, made by number theorists -- it's quite minimalist, now also free licensed (used to be proprietary, they seem to enjoy license hopping). At the moment it contains more than 370000 sequences; by browsing it you can get a glimpse of how deep the study of numbers goes. These people are also somewhat funny, they give numbers entertaining names like happy numbers (adding its squared digits eventually gives 1), polite numbers, friendly numbers, cake numbers, lucky numbers or weird numbers.

Some numbers cannot be computed, i.e. there exist noncomputable numbers. This follows from the existence of noncomputable functions (such as that representing the halting problem). For example let's say we have a real number x, written in binary as 0. d0 d1 d2 d3 ..., where dn is nth digit (1 or 0) after the radix point. We can define the number so that dn is 1 if and only if a Turing machine represented by number n halts. Number x is noncomputable because to compute the digits to any arbitrary precision would require being able to solve the unsolvable halting problem.

All natural numbers are interesting: there is a fun proof by contradiction of this. Suppose there exists a set of uninteresting numbers which is a subset of natural numbers; then the smallest of these numbers is interesting by being the smallest uninteresting number -- we've arrived at contradiction, therefore a set of uninteresting numbers cannot exist.

TODO: what is the best number? maybe top 10? would 10 be in top 10? what's the first number that's in top itself?

Numbers In Programming/Computers

While mathematicians work mostly with infinite number sets and all kinds of "weird" hypothetical numbers like hyperreals and transcendentals, programmers still typically deal with "normal" numbers pertaining to practical applications, and have to limit themselves to finite number sets because, of course, computers have limited memory and can only store limited number of numeric values -- computers typically work with modulo arithmetic with some high power of two, e.g. 2^32 or 2^64, which is a good enough approximation of an infinite number set. Mathematicians are as precise with numbers as possible as they're interested in structures and patterns that numbers form, programmers just want to use numbers to solve problems, so they mostly use approximations where they can -- for example programmers normally approximate real numbers with floating point numbers that are really just a subset of rational numbers. This isn't really a problem though, computers can comfortably work with numbers large and precise enough for solving any practical problem -- a slight annoyance is that one has to be careful about such things as underflows and overflows (i.e. a value wrapping around from lowest to highest value and vice versa), limited and sometimes non-uniform precision resulting in error accumulation, unlinearization of linear systems and so on. Programmers also don't care about strictly respecting some properties that certain number sets must mathematically have, for example integers along with addition are mathematically a group, however signed integers in two's complement aren't a group because the lowest value doesn't have an inverse element (e.g. on 8 bits the lowest value is -128 and highest 127, the lowest value is missing its partner). Programmers also allow "special" values to be parts of their number sets, especially e.g. with the common IEEE floating point types we see values like plus/minus infinity, negative zero or NaN ("not a number") which also break some mathematical properties and creates situations like having a number that says it's not a number, but again this really doesn't play much of a role in practical problems. Numbers in computers are represented in binary and programmers themselves often prefer to write numbers in binary, hexadecimal or octal representation -- they also often meet powers of two rather than powers of ten or primes or other similar limits (for example the data type limits are typically limited by some power of two). There also comes up the question of specific number encoding, for example direct representation, sign-magnitude, two's complement, endianness and so on. Famously programmers start counting from 0 (they go as far as using the term "zeroth") while mathematicians rather tend to start at 1. Just as mathematicians have different sets of numbers, programmers have an analogy in numeric data types -- a data type defines a set of values and operations that can be performed with them. The following are some of the common data types and representations of numbers in computers:

• numeric: Anything considered a number. In very high level languages there may be just one generic "number" type capable of storing any kind of number, automatically choosing the best representation behind the scenes, dynamically allocating memory as it changes size etc.
  • unsigned: Don't allow negative values -- this is sufficient in many situations, more straightforward to implement and can reach higher values in the positive direction.
  • signed: Permit both positive and negative values which brings up the question of what representation to choose -- for integers the most common one nowadays is two's complement.
  • fixed size: Most common, each number takes some fixed size in memory, expressed in bits or bytes -- this naturally determines the maximum number of possible values and thus also the minimum and maximum representable number. Going beyond or below the range typically results in an overflow.
    • 8bit: Can store 256 value (e.g. integers from 0 to 255 or -128 to 127).
    • 16bit: Can store 65536 values.
    • 32bit: Can store 4294967296 values.
    • ...
  • arbitrary size: Can store arbitrarily high/low and/or precise value, take variable amount of memory depending on how much is needed, used only in very specialized cases when absurdly high numbers may appear, may be considerably slower due to the overhead and lack of direct hardware support for extremely large numbers.
  • integer: Integer values, most common, usually using direct or two's complement representation.
  • fractional: Have higher precision than integers, allow storing fractions, are often used to approximate real numbers.
    • fixed point: Are represented by a number with radix point in fixed place, have uniform precision.
    • floating point: Have movable radix point which is more complicated but allows for representing both very high and very small values due to non-uniform precision.
  • complex: Analogous to mathematical complex numbers.
  • quaternion: Analogous to mathematical quaternions.
  • symbolic: Used in some specialized mathematical software to perform symbolic computation, i.e. computation done in a human-like way, by manipulating symbols without using concrete values that would have to resort to approximation.
  • ...

However some programming languages, such as Lisp, sometimes treat numbers in very abstract, more mathematical ways (for the price of some performance loss and added complexity) such as exactly handling rational numbers with arbitrary precision, distinguishing between exact and inexact numbers etc.

Notable Numbers

Here is a table of some numbers and "number like objects" worthy of mention, mostly relevant in math and programming but also some famous ones from physics and popular culture (note: the order is roughly from lower numbers to higher ones, however not all of these numbers can be compared easily or at all, so the ordering isn't strictly correct; notes: & means base 8, b3 means base 3).

number	value	equal to, AKA	notes
not a number (NaN, undefined, ...)	none	1/0, 0^0, tan(pi/2)	error value
minus infinity			not always considered a number, smallest possible value
	-1.797693... * 10^308		smallest number storable in IEEE-754 64 bit float
	-3.402823... * 10^38		smallest number storable in IEEE-754 32 bit float
	-9223372036854776000	-1 * 2^64 / 2	minimum two's complement signed 64 bit number
	-2147483648	-1 * 2^32 / 2	minimum two's complement signed 32 bit number
minus thirty two thousand seven ...	-32768	-1 * 2^16 / 2	minimum two's complement signed 16 bit number
minus one hundred twenty eight	-128	-1 * 2^7	minimum value of signed byte (two's complement)
minus/negative one	-1	i^2, j^2, k^2
minus one twelfth	-0.08333...	-1/12	by some methods the result of 1 + 2 + 3 + ...
"negative zero"	"-0"	0	non-mathematical, sometimes used in programming
zero (none, nil)	0	"-0", e^(i * pi) + 1, lim x->inf 1/x	"nothing", additive identity
epsilon		1 / omega	infinitesimal, "infinitely small" non-zero
	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
one eight	0.125	2^-3, 0b0.001, 0x0.2
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
one over square root of two	0.707106...	1/sqrt(2), sin(pi/4), cos(pi/4), 2^(-1/2)
one	1	2^0, 0!, 0.999..., sqrt(1), I, 0b1, cos(0)	NOT a prime, unit, multiplicative identity
square root 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
phi (golden ratio)	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)
two (couple, pair)	2	2^1, 2!, 2!!, 0b000010, II, 0b10	(only even) prime, base of binary system
silver ratio	2.414213...	1 + sqrt(2), solve(x^2 - 2 * x - 1 = 0)	similar to golden ratio
e (Euler's number)	2.718281...	0b10.1011011	base of natural logarithm
three	3	2^2 - 1, III, Ob11, 2^1.584...	prime, max. number on 2 bits, regular plane tilings
pi	3.141592...	2 * asin(1), 0b11.0010010	circle circumference to its diameter, irrational
four	4	2^2, 0b000100, IV, 0b100	first composite number, min. needed to color planar graph
five	5	3^2 - 2^2, V, 0b101, fib(5)	(twin, triplet) prime, number of platonic solids, Fib.
six (half dozen)	6	3!, 1 * 2 * 3, 1 + 2 + 3, VI, 0b110	highly composite number, perfect number
tau	6.283185...	2 * pi, 360 degrees	radians in full circle, defined mostly for convenience
thrembo	???		the hidden number
seven	7	2^3 - 1, VII, &7, 0b111	(twin) prime, days in week, max. unsigned n. with 3 bits
eight	8	2^3, 0b001000, VIII, &10, 0b1000, fib(6)	base of octal system, 7th Fibonacci number
nine	9	3^3, 1^3 + 2^3, sqrt(81), IX, 0b1001
pi squared	9.869604...	pi^2
ten	10	10^1, 1 + 2 + 3 + 4, X, 0b1010, 2^3.321...	your IQ? :D base of our decimal system
eleven	11	0xb, b3(102), &13, 0b1011, XI	palindromic twin prime
twelve (dozen)	12	2 * 2 * 3, 0xc, 0b1100, XII	highly composite number
thirteen (long or devil's dozen)	13	fib(7), 0xd, 0b1101, XIII	prime considered unlucky (in west and China), Fib. num.
fourteen	14	&112, 0b1110, 0xe, XIV
fifteen	15	2^4 - 1, 0b1111, 0xf, 1 + 2 + 3 + 4 + 5	maximum unsigned number storable with 4 bits
sixteen	16	2^4, 4^2, 2^2^2, 0b010000, &20, 0x10, XVI	base of hexadecimal system
seventeen	17	0b10001, &21, 0x11, XVII	twin&sexy prime, binary palindrome
eighteen	18	0b10010, &22, 0x12, XVIII
nineteen	19	0b10011, &23, 0x13, XIX	twin&sexy prime
twenty	20	0b10100, &24, 0x14, XX
twenty one	21	0b10101, 0x15, BB(3), fib(8), 0x15, XXI	maximum number of 1s produced by 3 state Turing Machine
twenty three	23	0b10111, &27, 0x17,	sexy prime
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 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	32	2^5, 0b100000, &40, 0x20, XXXII	number of possible values storable with 5 bits
thirty three	33	1! + 2! + 3! + 4!, XXXIII
thirty four	34	fib(9), 0x22, XXXIV	Fibonacci number
thirty six	36	2 * 2 * 3 * 3, XXXVI	highly composite number
thirty seven	37	0b100101, 0x25, XXXVII	most commonly picked 1 to 100 "random", permutable prime
forty one	41	0b101001, 0x29, XLI	twin&sexy prime
forty two	42	XLII	cringe number, answer to some stuff, unlucky in Japan
forty three	43	0b101011, 0x2b, XLIII	twin&sexy prime
forty seven	47	0b101111, 0x2f, XLVII	sexy prime
forty eight	48	2^5 + 2^4, 2 * 2 * 2 * 2 * 3, XLVIII, 0x30	highly composite number
forty nine	49	7^2
fifty	50	L
fifty three	53	0b110101, 0x35, LIII	sexy prime
fifty five	55	fib(10), 1 + 2 + ... + 10, LV	sum of numbers up to 10, 11th Fibonacci number
fifty nine	59	0b111011, 0x3b, LIX	twin&sexy prime
sixty	60	0x3c, LX	highly composite number, used in time measuring
sixty one	61	0x3d, LXI	twin&sexy prime
sixty three	63	2^6 - 1, 0b111111, &77, 0x3f, LXIII	maximum unsigned number storable with 6 bits
sixty four	64	2^6, 0b1000000, &100, 0x40, LXIV	number of squares on a chess board
sixty seven	67	0x43, LXVII	sexy prime
sixty nine	69	0x45, LXIX	sexual position
seventy one	71	0x47, LXXI	twin prime
seventy three	73	0b1001001, 0x49, LXXIII	twin&sexy prime, binary palindrome
seventy nine	79	0x4f, LXXIX	sexy prime
eighty one	81	3^4, 9*9, XXCI
eighty three	83	LXXXIII	sexy prime
eighty nine	89	fib(11), 0x59, LXXXIX	Fibonacci number, sexy prime
ninety six	96	2^5 + 2^6, 5! - 4!, 0x60	alternative sexual position
ninety seven	97	XCVII	sexy prime
one hundred	100	10^2, 0x64, C, 2^6.643...
one hundred seven	107	BB(4), CVII	maximum number of 1s produced by 4 state Turing machine
one hundred twenty	120	5!, C(10,3), CXX	possible ways to order 5 objects, highly composite
one hundred twenty one	121	11^2, CXXI	palindromic
one hundred twenty five	125	5^3, CXXV
one hundred twenty seven	127	2^7 - 1, 0b01111111, &177, 0x7f, CXXVII	maximum value of signed byte, Mersenne prime
one hundred twenty eight	128	2^7, 0x80, &200, CXXVIII, 10^2.107...	number of values storable with 7 bits
one hundred forty four (gross)	144	12^2, fib(12), CXLIV	13th Fibonacci number, 12 dozen
one hundred sixty eight	168	24 * 7, CLXVIII	hours in week
two hundred forty three	243	3^5, 0xf3, CCXLIII
two hundred fifty five	255	2^8 - 1, 0b11111111, &377, 0xff, CCLV	maximum value of unsigned byte, hex palindrome
two hundred fifty six	256	2^8, 4^4, 16^2, 0x100, ((2^2)^2)^2, CCLVI	number of values that can be stored in one byte
three hundred forty three	343	7^3, CCCXLIII	palindrome
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
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
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
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
three thousand one hundred ...	3125	5^5, MMMCXXV
three thousand nine hundred ...	3999	MMMCMXCIX	largest number that can be written with Roman numerals
four thousand ninety five	4095	2^12 - 1, &7777, 0xfff	maximum unsigned integer storable with 12 bits
four thousand ninety six	4096	2^12, 2^(3^4), &10000, 0x1000	number of values storable with 12 bits
five thousand forty	5040	7!, 1 * 2 * ... * 7	possible ways to order 7 objects
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
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
nineteen thousand six hundred ...	19683	3^9, 3^(3^3), 0x4ce3
thirty two thousand seven hundred ...	32767	2^16 / 2 - 1, 0x7fff	maximum two's complement signed 16 bit number
forty thousand three hundred twenty	40320	8!, 1 * 2 * ... * 8, 0x9d80	possible ways to order 8 objects
... (enough lol)	59049	3^10, 0xe6a9
	65504		largest number storable in IEEE-754 16 bit float
	65535	2^16 - 1, &177777, 0xffff	maximum unsigned number storable with 16 bits
	65536	2^16, 256^2, &200000, 0x10000, 2^(2^(2^2))	number of values storable with 16 bits
	72078		number of possible chess positions after 4 half moves
	80085		looks like BOOBS
	86400	60 * 60 * 24	seconds in a day
hundred thousand	100000	10^5, 2^16.609...
	362880	9!, 1 * 2 * ... * 9	possible ways to order 9 objects
	500500	1 + 2 + ... + 1000	sum of numbers up to 1000
one million	1000000	10^6, 0xf4240, 2^19.931...
	3197281		number of possible chess games after 4 half moves
	3628800	10!, 1 * 2 * ... * 10	possible ways to order 10 objects
	16777216	2^24, 16^6, 0xffffff	number of distinct 24 bit values, no. of RGB24 colors
	16777217	2^24 + 1, 0x1000000	min. pos. int. unstorable in 32b float (prec. falls < 1)
	43046721	3^16
	47176870	BB(5)	maximum number of 1s produced by 5 state Turing machine
	31556926		seconds in a year
	39916800	11!, 1 * 2 * ... * 11	possible ways to order 11 objects
	479001600	12!, 1 * 2 * ... * 12	possible ways to order 12 objects
one billion	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
	87178291200	14!, 1 * 2 * ... * 14	possible ways to order 14 objects
	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
	20922789888000	16!	possible ways to order 16 objects
thirty trillion	30000000000000		approximate number of cells in human body
	355687428096000	17!	possible ways to order 17 objects
bazillion	???		used to just express a very large value
quadrillion	1000000000000000	10^15
	6402373705728000	18!	possible ways to order 18 objects
	9007199254740992		precision of IEEE double falls below 1 after this num.
	121645100408832000	19!	possible ways to order 19 objects
quintillion	1000000000000000000	10^18
	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
	18446744073709551615	2^64 - 1, 0xffffffffffffffff	maximum unsigned number storable with 64 bits
	18446744073709551616	2^64	number of values storable with 64 bits
	43252003274489856000		number of possible Rubik's cube configurations
	2015099950053364471960		number of possible chess games after 15 half moves
	6670903752021072936960		possible valid filled sudoku grids
	1.267650... * 10^30	2^100	number of values storable with 100 bits
	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
	1.157920... * 10^77	2^256	number of values storable with 256 bits
	10^80		approx. number of atoms in observable universe
googol	10^100		often used big number
Shannon number	10^120		estimated number of possible games in chess
asankhyeya	10^140		religious number, often used in Buddhism
	1.340780... * 10^154	2^512	number of values storable with 512 bits
	9.332621... * 10^157	100!	possible ways to order 100 objects
	4.65... * 10^185		approx. number of Planck volumes in observable universe
	1.797693... * 10^308		largest number storable in IEEE-754 64 bit float
	1.797693... * 10^308	2^1024	number of values storable with 1024 bits
	3.231700... * 10^616	2^2048	number of values storable with 2048 bits
	4.023872... * 10^2567	1000!	possibe ways to order 1000 objects
googolplex	10^(10^100)	10^googol	another large number, number of genders in 21st century
Graham's number		g64	extremely, unimaginably large number, > googolplex
TREE(3)	unknown		yet even larger number, > Graham's number
infinity		lim x->0 1/x, 1 + 1 + 1 + ...	not always considered a number, largest possible value
aleph zero		beth zero, cardinality(N)	infinite cardinal number, "size" of the set of nat. num.
i (imaginary unit)		j * k	part of complex numbers and quaternions
j		k * i	one of quaternion units
k		i * j	one of quaternion units

See Also

• real number
• pseudonumber
• not a number
• illegal number
• offensive number
• binary numbers