I'm collecting mathematical definitions of numbers as well as numbers of mathematical interest.
This listing is the order I am presenting the numbers:
This listing is a tree of how the mathematical categories form a pseudo-tree:
Major categories are represented as a capital letter in bold or in a fancy font called Blackboard bold [W].
P ⊆ N ⊆ Z ⊆ Q ⊆ R ⊆ C ⊆ H ⊆ O ⊆ S.
ℙ ⊆ ℕ ⊆ ℤ ⊆ ℚ ⊆ ℝ ⊆ ℂ ⊆ ℍ.
Natural numbers (ℕ or N) is either the set of positive integers {1,2,3,...} or the set of non-negative integers {0,1,2,3,...}.
A perfect number is a natural number that is also equal to the sum of all of its divisors. EG: 6 is equal to the sum of its divisors: 1 + 2 + 3. The next six perfect numbers are 28, 496, 8128, 130816, 2096128, and 33550336.
A prime number (ℙ or P) is a natural number greater than 1 that can be divided evenly only by 1 and itself. Thus the first few prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...
A number that can be written as a product of prime numbers is composite. Thus there are three types of natural numbers: primes, composites, and 1.
The unproven Riemann Hypothesis basically states that that there is no pattern to prime numbers. Proving or disproving the theory would provide great insight into prime numbers. It is a "Holy Grail" of mathematic just as Fermat's Last Theorem was.
Fun fact: The prime #37 can be divided wholly into 111, 222, 333, 444, 555, 666, 777, 888, and 999
For a text file of the first 78,498 prime numbers, click here.
For m to be prime, p itself must be prime, but m must also pass other tests that verify a prime number. EG: Although 11 is a prime number, it does not produce a Mersenne prime number:m = 2^p - 1
As of March 2000 only 38 Mersenne prime numbers are known:2^11 -1 = 2047 = 23 x 89
The largest prime number known is a Mersenne prime number. Mersenne prime numbers have their own web site (Mersenne.org) which is dedicated to a netwide search for Mersenne prime numbers and related tasks.p = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593.
Integer numbers (ℤ or Z) include all natural numbers, but also negative equivalents, i.e. {...,-3,-2,-1,0,1,2,3,...}.
Just for fun: Supposedly you can add the reverse to any integer (except for 1) to derive a palindromic integer. EG:
69 + 96 = 165 165 + 561 = 726 726 + 627 = 1353 1353 + 3531 = 4884
A rational numbers (ℚ or Q) can be expressed as a fraction p/q where q is non-zero and p and q are both integers, i.e. {p/q: p,q ∈ ℤ, q≠0}.
An irrational number is a real number that is not a rational number. That is, they cannot be expressed as fractions or decimal fractions that do not repeat infinitely.
Real numbers (ℝ or R) are are points on the number line from, either rational and irrational numbers.
i
. Complex numbers are of the form:z = a + bi
where a
and b
are real numbers and i
, called the imaginary unit, is the square root of negative 1.
i^2 = -1
a_{0}x^{n} + a_{1}x^{n-1} + ... + a_{n-1}x + a_{n} = 0
Where the a's are integers and r satisfies no similar equation of degree <n, then r is an algebraic number of degree n. If a is in the polynomial equation are algebraic numbers, then any root of that equation is also an algebraic number.
Transcendental numbers are real numbers that are not algebraic numbers. All transcendental numbers are irrational numbers. A transcendental number is not the root of any polynomial equation with integer coefficients. EG: e and pi are transcendental but are also real and irrational. Here are a few transcendental numbers:
These famous numbers are irrational but not transcendental numbers:
See also my section on Measurements for a discussion on SI/metric prefixes for decimal powers (EG: 10e6 = Mega).
Numbers related to 1 in 10^n are commonly used in areas such as QoS (Quality of Service), 6 sigma, defining miracles, and such. See also Measurements.
1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|
n | 10^n | 1/(10^n) | % | 100%-% | US/Non-US Cardinal |
0 | 1 | 1 | 100% | 0% | 1 in one |
1 | 10 | 0.1 | 010% | 90% | 1 in ten |
2 | 100 | 0.01 | 001% | 99% | 1 in a hundred |
3 | 1 000 | 0.001 | 000.1% | 99.9% | 1 in a thousand |
4 | 10 000 | 0.000 1 | 000.01% | 99.99% | 1 in ten thousand |
5 | 100 000 | 0.000 01 | 000.001% | 99.999% | 1 in a hundred thousand |
6 | 1 000 000 | 0.000 001 | 000.000 1% | 99.999 9% | 1 in a million |
7 | 10 000 000 | 0.000 000 1 | 000.000 01% | 99.999 99% | 1 in ten million |
8 | 100 000 000 | 0.000 000 01 | 000.000 001% | 99.999 999% | 1 in a hundred million |
9 | 1 000 000 000 | 0.000 000 001 | 000.000 000 1% | 99.999 999 9% | 1 in a billion/milliard |
10 | 10 000 000 000 | 0.000 000 000 1 | 000.000 000 01% | 99.999 999 99% | 1 in ten billion/milliard |
11 | 100 000 000 000 | 0.000 000 000 01 | 000.000 000 001% | 99.999 999 999% | 1 in a hundred billion/milliard |
12 | 1 000 000 000 000 | 0.000 000 000 001 | 000.000 000 000 1% | 99.999 999 999 9% | 1 in a trillion/billion |
Linguistically cardinal numbers indicate quantity (EGs: 0, 1, 2, 3, 4, 500) as opposed to ordinal numbers which indicate order (EG: 0th, 1st, 2nd, 3rd, 4th, 500th).
The French and English used to agree that named cardinal numbers increase in magnitude by a thousand (10e3), however the French converted to the English and German system of increasing by a million (10e12). Because of this, I avoid cardinal numbers above a million. See Webster.com/mw/table/number.htm and io.com/~iareth/bignum.html
In a table its cardinal number would list as follows:
Cardinal Name | US Exponent | International Exponent |
---|---|---|
milliard | NA | 9 |
billion | 9 | 12 |
trillion | 12 | 18 |
quadrillion | 15 | 24 |
quintillion | 18 | 30 |
sextillion | 21 | 36 |
septillion | 24 | 42 |
octillion | 27 | 48 |
nonillion | 30 | 54 |
decillion | 33 | 60 |
undecillion | 36 | 66 |
duodecillion | 39 | 72 |
tredecillion | 42 | 78 |
quattuordecillion | 45 | 84 |
quindecillion | 48 | 90 |
sexdecillion | 51 | 96 |
septendecillion | 54 | 102 |
octodecillion or duodevigintillion |
57 | 108 |
novemdecillion or undevigintillion |
60 | 114 |
vigintillion | 63 | 120 |
... | ... | ... |
centillion | 303 | 600 |
... | ... | ... |
nongentinonagentanonillion or
undemillillion |
3000 | 5994 |
millilion | 3003 | 6000 |
... | ... | ... |
googolplex = 10^10^100 |
Cardinality in computers (especially regular expressions, database design, and UML) is notated as follows:
0
. Zero.1
. One.n
. Numerically specified. EG: 17
.w
. Whatever. Usually any non-negative integer.;
are used instead of ,
.[0,0]
= [0]
[0,1]
= ?
= Zero or one[0,n]
= Zero to n[0,w]
= [0,]
. *
= Zero or more in regular expressions. *
= Zero or more characters in MS Access. %
= Zero or more characters in MS SQL Server.[1,1]
= [1]
. .
= Any one character except for \n in regular expressions. ?
= Any one character in MS Access. _
= Any one character in MS SQL Server.[1,n]
= One to n[1,w]
= [1,]
= +
= One or more[n,n]
= [n]
[n,m]
= n to m[n,w]
= [n,]
= n or more...
are used instead of :
.0:1
. Zero-to-one.0:n
.0:*
.1:1
. One-to-one is very common.1:n
.1:*
. One-to-many is very common.n:m
.n:*
.*:*
. Many-to-many is very common.This of course leads right into set theory, so here some of the notation used in set theory:
Infinite cardinality is represented in math with a lemniscate (∞).
(x^2 + y^2)^2 = (x^2 + y^2)*a^2
.x221E = 8734
.∞
or ∞
.Renard Numbers (aka preferred numbers) were devised by French army engineer Col. Charles Renard (1847/1905). The idea was to find a standard way for dividing a range of values into standard intervals. Renard Numbers were adopted as ISO standard 3 in 1952.
The Renard Numbers are a kind of geometric progression or geometric sequence. A geometric progression takes this form:
ar^{0} = a, ar^{1}, ar^{2}, ar^{3}, ..., ar^{m}
where a is the scale factor and r ≠ 0 is the common ratio.
There are four series of Renard Numbers, Rn, where a = 1 and r = 10^{1/n}. For each series, values are calculated from 1 to 10, and then rounded. Hence:
R5: r = 10^{1/5}. Hence: 10^{0/5} = 1, 10^{1/5} = 1.584... ~ 1.6, 10^{2/5} = 2.511... ~ 2.5, 10^{3/5} = 3.981... ~ 4.0, 10^{4/5} = 6.309... ~ 6.3, 10^{5/5} = 10
EG: If you have screws from 12 mm to 270 mm long, then the R5 series would yield 7 screws of sizes: 16 mm, 25 mm, 40 mm, 63 mm, 100 mm, 160 mm, and 250 mm. If you used the R10 series you would get the R5 screws plus 6 more intermediate screws of sizes: 20 mm, 32 mm, 50 mm, 80 mm, 125 mm, and 200 mm.
Here are the four series of Renard Numbers and their more rounded variants as of ISO 3:
m | R5 | R5' | m | R10 | R10' | R10'' | m | R20 | R20' | R20'' | m | R40 | R40' |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1.00 | 1 | 0 | 1.00 | 1 | 1 | 0 | 1.00 | 1 | 1 | 0 | 1.00 | 1 |
1 | 1.06 | 1.05 | |||||||||||
1 | 1.12 | 1.1 | 1.1 | 2 | 1.12 | 1.1 | |||||||
3 | 1.18 | 1.2 | |||||||||||
1 | 1.25 | 1.25 | 1.2 | 2 | 1.25 | 1.25 | 1.2 | 4 | 1.25 | 1.25 | |||
5 | 1.32 | 1.3 | |||||||||||
3 | 1.40 | 1.4 | 1.4 | 6 | 1.40 | 1.4 | |||||||
7 | 1.50 | 1.5 | |||||||||||
1 | 1.60 | 1.5 | 2 | 1.60 | 1.6 | 1.5 | 4 | 1.60 | 1.6 | 1.6 | 8 | 1.60 | 1.6 |
9 | 1.70 | 1.7 | |||||||||||
5 | 1.80 | 1.8 | 1.8 | 10 | 1.80 | 1.8 | |||||||
11 | 1.90 | 1.9 | |||||||||||
3 | 2.00 | 2 | 2 | 6 | 2.00 | 2 | 2 | 12 | 2.00 | 2 | |||
13 | 2.12 | 2.1 | |||||||||||
7 | 2.24 | 2.2 | 2.2 | 14 | 2.24 | 2.2 | |||||||
15 | 2.36 | 2.4 | |||||||||||
2 | 2.50 | 2.5 | 4 | 2.50 | 2.5 | 2.5 | 8 | 2.50 | 2.5 | 2.5 | 16 | 2.50 | 2.5 |
17 | 2.65 | 2.6 | |||||||||||
9 | 2.80 | 2.8 | 2.8 | 18 | 2.80 | 2.8 | |||||||
19 | 3.00 | 3 | |||||||||||
5 | 3.15 | 3.2 | 3 | 10 | 3.15 | 3.2 | 3 | 20 | 3.15 | 3.2 | |||
21 | 3.35 | 3.4 | |||||||||||
11 | 3.55 | 3.6 | 3.5 | 22 | 3.55 | 3.6 | |||||||
23 | 3.75 | 3.8 | |||||||||||
3 | 4.00 | 4 | 6 | 4.00 | 4 | 4 | 12 | 4.00 | 4 | 4 | 24 | 4.00 | 4 |
25 | 4.25 | 4.2 | |||||||||||
13 | 4.50 | 4.5 | 4.5 | 26 | 4.50 | 4.5 | |||||||
27 | 4.75 | 4.8 | |||||||||||
7 | 5.00 | 5 | 5 | 14 | 5.00 | 5 | 5 | 28 | 5.00 | 5 | |||
29 | 5.30 | 5.3 | |||||||||||
15 | 5.60 | 5.6 | 5.5 | 30 | 5.60 | 5.6 | |||||||
31 | 6.00 | 6 | |||||||||||
4 | 6.30 | 6 | 8 | 6.30 | 6.3 | 6 | 16 | 6.30 | 6.3 | 6 | 32 | 6.30 | 6.3 |
33 | 6.70 | 6.7 | |||||||||||
17 | 7.10 | 7.1 | 7 | 34 | 7.10 | 7.1 | |||||||
35 | 7.50 | 7.5 | |||||||||||
9 | 8.00 | 8 | 8 | 18 | 8.00 | 8 | 8 | 36 | 8.00 | 8 | |||
37 | 8.50 | 8.5 | |||||||||||
19 | 9.00 | 9 | 9 | 38 | 9.00 | 9 | |||||||
39 | 9.50 | 9.5 | |||||||||||
5 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 10 | 10 | 10 | 40 | 10 | 10 |
In real life, all sorts of other preferred numbers are used. See also Preferred number [W].
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