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 [1]mathematical 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 [2]abstract ways which have only distant relationship to traditional
counting. Examples of numbers are minus [3]one half, [4]zero, [5]pi or
[6]i. Numbers constitute the basis and core of mathematics and as such
they sit almost at the [7]lowest level of it, i.e. most other things such
as algebra, [8]functions and [9]equations 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 [10]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
-- 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 [11]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).
Let's not [12]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 [13]numeral systems (Roman numerals, tally marks,
Arabic numerals of different [14]bases 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 [15]pi 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 (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 [16]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 ([17]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 [18]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
[19]complex numbers started to appear, which was later (19th century)
expanded further to [20]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" -- we can exploit this in both mathematics and
programming (which are actually the same thing). Ways of encoding
[21]information in numbers may vary, for a mathematician it is natural to
see any number as a multiset of its [22]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
[23]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.
[24]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 [25]fun facts about numbers:
* Some people associate numbers with [26]colors, though what color each
number has seems to be completely subjective. See [27]synesthesia.
* There is a funny hypothetical number between 6 and 7 called
[28]thrembo.
* There exist [29]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. "[30]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 [31]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 [32]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 }
uniq. max div.
number divisors divisors min. max. divisor div. avg. div. div. range
uniq. div. div. sum sum spread (%) spread (%)
(%)
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 [33]data, let's for example notice we
can make a nice [34]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 [35]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 [36]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? If we write numbers like this, will some operations
maybe become easier to perform? You can just keep diving down rabbit holes
like this.
Numbers In Math
There are different types of numbers, in mathematics we classify them into
[37]sets (if we further also consider the operations we can perform with
numbers we also sort them into algebras and structures like [38]groups,
[39]fields or [40]rings). Though we can talk about finite sets of numbers
perfectly well (e.g. [41]modulo arithmetic, [42]Boolean algebra etc.), we
are firstly considering [43]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. [44]zero,
minus [45]infinity or aleph one.
* [46]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 [47]Universe.
* [48]noncomputable: Cannot be computed, i.e. any such number
has no [49]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).
* [50]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.
* [51]surreal numbers, *R: hyperreal numbers, superreal numbers,
...: Various extensions of real numbers, include also
infinitesimals and some transfinite numbers.
* [52]infinitesimals: Are closer to zero than any real number
without actually being zero, i.e. "infinitely small"
numbers, play big role in [53]calculus. E.g. 0.000...1 (with
infinitely many 0 digits before the 1).
* Qp: [54]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: [55]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: [56]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.
* [57]algebraic: Are roots of one variable
[58]polynomials with integer coefficients. E.g. 4/3,
the [59]golden ratio or square root of two.
* [60]transcendental: Aren't algebraic. E.g. [61]pi,
[62]sine of [63]e or two to the power of square root of
two.
* [64]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 ([65]i)
squared equals minus one. E.g. 0, 3i or -i.
* R: [66]real: Measure any continuous one dimensional
quantity (such as height or length), the line they form
is continuous. E.g. -0.3, [67]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. [68]pi, minus
[69]e or square root of 2.
* Q: [70]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: [71]whole (integers): Are [72]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: [73]natural (with zero): E.g. 0, 16
or 1000.
* [74]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.
* [75]modulo numbers: Finite sets of
numbers up to some N which are
allowed to "[76]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.
* [77]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.
* [78]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 [79]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 [80]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 [81]linear algebra which treats [82]vectors
and [83]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 [84]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 [85]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 [86]game
of all time, embedded right in this [87]Universe, in [88]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 [89]encyclopedia of integer sequences at
https://oeis.org/, made by number theorists -- it's quite [90]minimalist,
now also [91]free licensed (used to be [92]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 [93]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 [94]noncomputable
numbers. This follows from the existence of noncomputable functions (such
as that representing the [95]halting problem). For example let's say we
have a real number x, written in [96]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 [97]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 [98]natural numbers are [99]interesting: there is a [100]fun
[101]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 kind of
"weird" hypothetical numbers like hyperreals and transcendentals,
[102]programmers still mostly work with "normal", practical numbers 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 [103]modulo arithmetic with some
high power of two modulo, e.g. 2^32 or 2^64, which is a [104]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 [105]approximations where they can -- for example
programmers normally approximate [106]real numbers with [107]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
[108]underflows and [109]overflows (i.e. a value wrapping around from
lowest to highest value and vice versa), limited and sometimes non-uniform
precision resulting in [110]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 [111]group,
however signed integers in [112]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 [113]floating point
types we see values like plus/minus [114]infinity, [115]negative zero or
[116]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 [117]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, [118]two's complement, [119]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 [120]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 that can store any kind of
number, automatically choosing best representation for it etc.
* [121]unsigned: Don't allow negative values -- this is sufficient
in many cases, simpler to implement and can offer higher range in
the positive direction.
* [122]signed: Allow also negative values which brings up the issue
of what representation to use -- nowadays the most common is
[123]two's complement.
* fixed size: Most common, each number takes some fixed size in
memory, expressed in [124]bits or [125]bytes -- this of course
determines the maximum number of values and so for example the
minimum and maximum storable number.
* 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.
* ...
* [126]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, may be
considerably slower.
* [127]integer: Integer values, most common, usually using direct
or [128]two's complement representation.
* fractional: Have higher precision than integers, allow storing
fractions, are often used to [129]approximate real numbers.
* [130]fixed point: Are represented by a number with radix
point in fixed place, have uniform precision.
* [131]floating point: Have movable radix point which is more
[132]complicated but allows for representing both very high
and very small values due to non-uniform precision.
* [133]complex: Analogous to mathematical complex numbers.
* [134]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.
* ...
Notable Numbers
Here is a table of some notable numbers, mostly important in math and
programming but also some famous ones from [135]physics and popular
culture (note: the order is rougly 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).
number value equal to notes
not always
minus considered a
[136]infinity number, smallest
possible value
minus/negative -1 i^2, j^2, k^2
one
"[137]negative non-mathematical,
zero" "-0" zero sometimes used in
programming
[138]zero 0 negative zero, "nothing"
e^(i * pi) + 1
infinitesimal,
epsilon 1 / omega "infinitely small"
non-zero
smallest number
4.940656... * 10^-324 storable in
IEEE-754 64 binary
float
smallest number
1.401298... * 10^-45 storable in
IEEE-754 32 binary
float
Planck length in
1.616255... * 10^-35 meters, smallest
"length" in
Universe
one eight 0.125 2^-3
one fourth 0.25 2^-2
one third 0.333333... ...1313132
(5-adic)
one half 0.5 2^-1
[139]one 1 2^0, 0!, NOT a prime
0.999...
irrational,
[140]square root 1.414213... 2^(1/2) diagonal of unit
of two square, important
in geom.
similar to golden
supergolden ratio 1.465571... solve(x^3 - x^2 ratio, bit more
- 1 = 0) difficult to
compute
(1 + sqrt(5)) / irrational,
phi ([141]golden 1.618033... 2, solve(x^2 - visually pleasant
ratio) x - 1 = 0) ratio, divine
proportion
[142]two 2 2^1, 0b000010 (only even) prime
1 + sqrt(2), similar to golden
[143]silver ratio 2.414213... solve(x^2 - 2 * ratio
x - 1 = 0)
[144]e (Euler's 2.718281... base of natural
number) [145]logarithm
prime, max.
[146]three 3 2^2 - 1 unsigned number
with 2 bits
circle
[147]pi 3.141592... circumference to
its diameter,
irrational
first composite
[148]four 4 2^2, 0b000100 number, min.
needed to color
planar graph
(twin) prime,
[149]five 5 number of platonic
solids
3!, 1 * 2 * 3, highly composite
[150]six 6 1 + 2 + 3 number, perfect
number
radians in full
[151]tau 6.283185... 2 * pi circle, defined
mostly for
convenience
[152]thrembo ??? the hidden number
(twin) prime, days
[153]seven 7 2^3 - 1 in week, max.
unsigned n. with 3
bits
[154]eight 8 2^3, 0b001000
[155]nine 9
[156]ten 10 10^1, 1 + 2 + 3 your IQ? :D
+ 4
twelve, dozen 12 2 * 2 * 3 highly composite
number
2^4 - 1, maximum unsigned
fifteen 15 0b1111, 0x0f, 1 number storable
+ 2 + 3 + 4 + 5 with 4 bits
[157]sixteen 16 2^4, 2^2^2,
0b010000
twenty four 24 2 * 2 * 2 * 3, highly composite
4! number
max. unsigned
thirty one 31 2^5 - 1 number storable
with 5 bits,
Mersenne prime
[158]thirty two 32 2^5, 0b100000
thirty six 36 2 * 2 * 3 * 3 highly composite
number
most commonly
thirty seven 37 chosen 1 to 100
"random" number
cringe number,
[159]fourty two 42 answer to some
stuff
fourty eight 48 2^5 + 2^4, 2 * highly composite
2 * 2 * 2 * 3 number
maximum unsigned
sixty three 63 2^6 - 1 number storable
with 6 bits
[160]sixty four 64 2^6
[161]sixty nine 69 sexual position
ninety six 96 2^5 + 2^6 alternative sexual
position
one hundred 100 10^2
one hundred 121 11^2
twenty one
one hundred maximum value of
twenty seven 127 2^7 - 1 signed byte,
Mersenne prime
one hundred 128 2^7
twenty eight
one hundred 144 12^2
fourty four
one hundred sixty 168 24 * 7 hours in week
eight
two hundred fifty 2^8 - 1, maximum value of
five 255 0b11111111, unsigned [162]byte
0xff
two hundred fifty 2^8, 16^2, number of values
six 256 ((2^2)^2)^2 that can be stored
in one byte
three hundred 2 * 2 * 2 * 3 * highly composite
sixty 360 3 * 5 number, degrees in
full circle
stoner shit (they
four hundred 420 smoke it at 4:20),
twenty divisible by 1 to
7
five hundred 512 2^9
twelve
six hundred and 666 number of the
sixty six beast
one thousand 1000 10^3
one thousand 1024 2^10
twenty four
two thousand 2048 2^11
fourty eight
four thousand 4096 2^12
ninety six
ten thousand 10000 10^4, 100^2
maximum unsigned
... (enough lol) 65535 2^16 - 1 number storable
with 16 bits
2^16, 256^2, number of values
65536 2^(2^(2^2)) storable with 16
bits
80085 looks like BOOBS
hundred thousand 100000 10^5
one [163]million 1000000 10^6
one [164]billion 1000000000 10^9
one of famous
3735928559 0xdeadbeef hexadeciaml
constants, spells
out DEADBEEF
2^32 - 1, maximum unsigned
4294967295 0xffffffff number storable
with 32 bits
number of values
4294967296 2^32 storable with 32
bits
one trillion 1000000000000 10^12
maximum unsigned
18446744073709551615 2^64 - 1 number storable
with 64 bits
number of values
18446744073709551616 2^64 storable with 64
bits
largest number
3.402823... * 10^38 storable in
IEEE-754 32 binary
float
approx. number of
10^80 atoms in
observable
universe
[165]googol 10^100 often used big
number
religious number,
[166]asankhyeya 10^140 often used in
[167]Buddhism
approx. number of
4.65... * 10^185 Planck volumes in
observable
universe
largest number
1.797693... * 10^308 storable in
IEEE-754 64 binary
float
another large
[168]googolplex 10^(10^100) 10^googol number, number of
genders in 21st
century
extremely,
[169]Graham's g64 unimaginably large
number number, >
googolplex
yet even larger
TREE(3) unknown number, > Graham's
number
not always
[170]infinity considered a
number, largest
possible value
infinite cardinal
[171]aleph zero beth zero, number, "size" of
cardinality(N) the set of nat.
num.
[172]i (imaginary part of complex
unit) j * k numbers and
quaternions
[173]j k * i one of quaternion
units
[174]k i * j one of quaternion
units
See Also
* [175]offensive number
Links:
1. math.md
2. abstraction.md
3. one.md
4. zero.md
5. pi.md
6. i.md
7. low_level.md
8. function.md
9. equation.md
10. set.md
11. number_theory.md
12. often_confused.md
13. numeral_system.md
14. base.md
15. pi.md
16. abstraction.md
17. irrational_number.md
18. zero.md
19. complex_number.md
20. quaternion.md
21. information.md
22. prime.md
23. binary.md
24. order.md
25. fun.md
26. color.md
27. synesthesia.md
28. thrembo.md
29. illegal_number.md
30. intellectual_property.md
31. programming.md
32. prime.md
33. data.md
34. tree.md
35. gcd.md
36. interesting.md
37. set.md
38. group.md
39. field.md
40. ring.md
41. mod.md
42. boolean_algebra.md
43. infinity.md
44. zero.md
45. infinity.md
46. knowability.md
47. universe.md
48. computability.md
49. turing_machine.md
50. transfinite_number.md
51. surreal_number.md
52. infinitesimal.md
53. calculus.md
54. p_adic_number.md
55. quaternion.md
56. complex_number.md
57. algebraic_number.md
58. polynomial.md
59. golden_ratio.md
60. transcendental_number.md
61. pi.md
62. sin.md
63. e.md
64. imaginary_number.md
65. i.md
66. real_number.md
67. pi.md
68. pi.md
69. e.md
70. rational_number.md
71. integer.md
72. discrete.md
73. natural_number.md
74. fibonacci.md
75. mod.md
76. overflow.md
77. prime.md
78. perfect_number.md
79. prime.md
80. perfect_number.md
81. linear_algebra.md
82. vector.md
83. matrix.md
84. awesome.md
85. knowability.md
86. game.md
87. universe.md
88. logic.md
89. encyclopedia.md
90. minimalism.md
91. free_culture.md
92. proprietary.md
93. fun.md
94. computability.md
95. halting_problem.md
96. binary.md
97. turing_machine.md
98. natural_number.md
99. interesting.md
100. fun.md
101. proof.md
102. programming.md
103. mod.md
104. good_enough.md
105. approximation.md
106. real_number.md
107. float.md
108. underflow.md
109. overflow.md
110. error.md
111. group.md
112. twos_complement.md
113. float.md
114. infinity.md
115. negative_zero.md
116. nan.md
117. binary.md
118. twos_complement.md
119. byte_sex.md
120. data_type.md
121. unsigned.md
122. signed.md
123. twos_complement.md
124. bit.md
125. byte.md
126. arbitrary_size_int.md
127. int.md
128. twos_complement.md
129. approximation.md
130. fixed_point.md
131. float.md
132. bloat.md
133. complex_number.md
134. quaternion.md
135. physics.md
136. infinity.md
137. negative_zero.md
138. zero.md
139. one.md
140. sqrt.md
141. golden_ratio.md
142. two.md
143. silver_ratio.md
144. e.md
145. log.md
146. three.md
147. pi.md
148. four.md
149. five.md
150. six.md
151. tau.md
152. thrembo.md
153. seven.md
154. eight.md
155. nine.md
156. ten.md
157. sixteen.md
158. thirty_two.md
159. 42.md
160. sixty_four.md
161. 69.md
162. byte.md
163. million.md
164. billion.md
165. googol.md
166. asankhyeya.md
167. buddhism.md
168. googolplex.md
169. grahams_number.md
170. infinity.md
171. aleph.md
172. i.md
173. j.md
174. k.md
175. offensive_number.md