Home • About • Contact
Welcome • Page Content • Site Map

Search
GH.com   GH.com dir
Wikipedia   Web

I'm collecting mathematical definitions of numbers as well as numbers of mathematical interest.

This listing is the order I am presenting the numbers:

• By Mathematical Category
  • Natural Numbers ℕ
  • Perfect Numbers
  • Prime Numbers
  • Mersenne Prime Numbers
  • Integer Numbers ℤ
  • Rational Numbers ℚ
  • Irrational Numbers
  • Real Numbers ℝ
  • Complex Numbers ℂ
  • Algebraic Numbers
  • Transcendental Numbers
• Miscellany
  • 1 in 10^n
  • Cardinal Numbers
  • Renard Numbers
  • Roman Numerals

By Mathematical Category

This listing is a tree of how the mathematical categories form a pseudo-tree:

• Complex Numbers ℂ
  • Real Numbers ℝ
    • Rational Numbers ℚ
      • Integer Numbers ℤ
        • Natural Numbers ℕ
          • Perfect Numbers
          • Prime Numbers ℙ
            • Mersenne Prime Numbers
    • Irrational Numbers
      • Transcendental Numbers
    • Algebraic Numbers. All rational numbers are algebraic but not all irrational numbers are algebraic.

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

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,...}.

Perfect Numbers

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.

Prime Numbers

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.

Mersenne Prime Number

Mersenne prime numbers are prime numbers p, which can generate another prime number when used in this form:

m = 2^p - 1

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:

2^11 -1 = 2047 = 23 x 89

As of March 2000 only 38 Mersenne prime numbers are known:

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.

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.

Integer Numbers

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

Rational Numbers

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}.

Irrational Numbers

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

Real numbers (ℝ or R) are are points on the number line from, either rational and irrational numbers.

Complex Numbers

Complex numbers (ℂ or C) are a super set of real numbers by using the imaginary unit 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

Algebraic Numbers

Algebraic numbers are complex numbers that can be obtained as the root of a polynomial with integer coefficients. That is if r is a root of the polynomial equation

a₀xⁿ + a₁x^n-1 + ... + a_n-1x + 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

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:

• e = 2.71828... Euler's number or Napier's constant, the base of the natural logarithm. The number accurate to 10,000 digits. [W]
• π = pi = 3.14159...
  • Archimede's constant, the ratio of a circle's perimeter to its diameter.
  • The number accurate to 10,000 digits.
  • Don't for get to celebrate Pi Day [W] on 03-14 1:59:26.5 PM (or AM if you're hard core!). Incidentally, Albert Einstein was born 1879-03-14.
  • Pi [W]
  • Euler's Identity [W]. e^(i*pi) + 1 = 0.
• 2^sqrt(2) = 2^√2 =  2.6651441... Gelfond-Schneider constant [W]

These famous numbers are irrational but not transcendental numbers:

• = √2 = sqrt(2) = 1.41421... Pythagoras' constant, the square root of 2. The first known irrational number. The number accurate to 10,000 digits. [W].
• φ = phi = (sqrt(5)+1)/2 = 1.61803... The Golden Mean, a ratio of two numbers (u/v) such that u/(u+v) = v/u, i.e. "the whole is to the larger as the larger is to the smaller". The number accurate to 15,000 digitss. [W].

Miscellany

See also my section on Measurements for a discussion on SI/metric prefixes for decimal powers (EG: 10e6 = Mega).

1 in 10^n

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.

• When formatted the way I do, n also indicates the number of 0s in Columns 2-3, (for n>1) the number of 0s in Column 4, and the number of 9s in Column 5.
• For -n, simply switch the values in Columns 2 & 3, and use your brain for the other columns.

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

Cardinal Numbers

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).

• 0 = zero = nil = love (from the French l'oeuf for egg) = nada = cero (Spanish) = nada (Spanish) = null (but null is not the same as 0 or empty string ("") in databases and math) = zip = zilch = ling2 (Chinese)
• 1/10 = tithe = deci-
• 1/2 = semi- = hemi- = demi-
• 1 = one = uni- = I (Roman) = uno (Spanish) = ichi (Japanese) = yi1 (Chinese) = isa (Tagalog)
• 2 = two = pair = couple = brace = bi- = di- = II (Roman) = dos (Spanish) = ni (Japanese) = er4 (Chinese) = dalawa (Tagalog)
• 3 = three = tri- = ter- = III or IV (Roman) = tres (Spanish) = san (Japanese) = san1 (Chinese) = tatlo (Tagalog)
• 4 = four = tetra- = tetr- = tessera- = quadri- = quadr- = IV (Roman) = quatro (Spanish) = shi (Japanese) = si4 (Chinese) = apat (Tagalog)
• 5 = five = pent- = penta- = quinqu- = quinque- = quint- = V (Roman) = cinco (Spanish) = go (Japanese) = wu3 (Chinese) = lima (Tagalog)
• 6 = six = half a dozen = sex- = sexi- = hex- = hexa- = VI (Roman) = seis (Spanish) = rokku (Japanese) = liu1 (Chinese) = anim (Tagalog)
• 7 = seven = hept- = hepta- = sept- = septi- = septem- = VII (Roman) = siete (Spanish) = sichi (Japanese) = qi1 (Chinese) = pito (Tagalog)
• 8 = eight = oct- = octa- = octo- = VIII (Roman) = ocho (Spanish) = hachi (Japanese) = ba1 (Chinese) = walo (Tagalog)
• 9 = nine = non- = nona- = ennea- = IX (Roman) = nueve (Spanish) = ku (Japanese) = jiu3 (Chinese) = siyam (Tagalog)
• 10 = ten = dec- = deca- = X (Roman) = dies (Spanish) = ju (Japanese) = shi2 (Chinese) = sampu (Tagalog)
• 11 = eleven = hendeca- = undec- = undeca- = XI (Roman) = once (Spanish) = labing-isa (Tagalog)
• 12 = twelve = dozen = dodeca- = XII (Roman) = doce (Spanish) = labindalawa (Tagalog)
• 13 = thirteen = baker's dozen = XIII (Roman) = trece (Spanish) = labintatlo (Tagalog)
• 14 = fourteen = XIV (Roman) = catorce (Spanish)
• 15 = fifteen = quindeca- = XV (Roman) = quince (Spanish)
• 16 = sixteen = XVI (Roman) = dieciseis (Spanish)
• 17 = seventeen = XVII (Roman) = diecisiete (Spanish)
• 18 = eighteen (there is only 1 "t") = XVIII (Roman) = dieciocho (Spanish)
• 19 = nineteen = XIX (Roman) = diecenueve (Spanish)
• 20 = twenty = score = icos- = icosa- = icosi- = XX (Roman) = veinte (Spanish)
• 50 = fifty = L (Roman)
• 100 = one hundred = cent- = centi- = centum = C (Roman)
• 144 = gross = a dozen dozen = CXLIV (Roman)
• 500 = five hundred = D (Roman)
• 1,000 = thousand = mille = M (Roman) = kilo- = K (SI)
• 1,000,000 = million = M (Roman) = mega- = M (SI)

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 the US, a billion is a thousand million = 10e3 x 10e6 = 10e9, but elsewhere a billion is a million million = 10e6 x 10e6 =10e12.
• In the US, a trillion is a thousand billion = 10e3 x 10e12 = 10e15, but elsewhere a trillion is a million billion = 10e6 x 10e12 =10e18.
• etc.

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.
• Non-negative integer intervals. The second specified number is always greater than or equal to the first specified number. Sometimes other characters such as ; are used instead of ,.
  1. [0,0] = [0]
  2. [0,1] = ? = Zero or one
  3. [0,n] = Zero to n
  4. [0,w] = [0,]. * = Zero or more in regular expressions. * = Zero or more characters in MS Access. % = Zero or more characters in MS SQL Server.
  5. [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.
  6. [1,n] = One to n
  7. [1,w] = [1,] = + = One or more
  8. [n,n] = [n]
  9. [n,m] = n to m
  10. [n,w] = [n,] = n or more
• Relations. Sometimes other characters such as ... 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:

• {x} = A set. EG: N = {1,2,3,...}
• {x : f(x)} = {x | f(x)} Set builder. EG: N = {|a| : a ∈ Z}. EG: The set of even numbers = {k : for each N, k = 2n}
• (a,b) = {x | a < x < b }. A parentheses indicates an exclusive end of an interval set.
• [a,b] = {x | a <= x <= b }. A square bracket indicates an inclusive end of an interval set.
• ∅ = {}.  Empty set.
• d ∈ D. This dog is a member or element of the set of Dogs.
• d ∉ D. This dog is not a member or element of the set of Dogs.
• A ⊆ F. Apples are a subset of Fruits.
• F ⊇ A. Fruits are a superset of Apples.
• A ⊂ F. Apples are a subset of Fruits, but the set of Apples is not the same as the set of Fruit.
• F ⊃ A. Fruits are a superset of Apples, but the set of Fruits is not the same as the set of Apples.
• M ∪ W. The large set formed by the union of Men and Women.
• M ∩ W. The small set formed by the intersection of Men who are also Women.
• M \ W. The medium set formed by the complement of Men who are not Women.
• ∀ x : f(x). For all x, f(x) is true. EG: ∀ n ∈ N : n*n ≥ n.
• ∃ x : f(x). There exists an x where f(x) is true. EG: ∃ n ∈ N : n is even.
• ∃! x : f(x). There exists exactly one x where f(x) is true. EG: ∃ n ∈ N : n + 7 = 2n.

Infinite cardinality is represented in math with a lemniscate (∞).

• The lemniscate is often described as the figure eight (8) on its side.
• The lemniscate has the Cartesian equation of (x^2 + y^2)^2 = (x^2 + y^2)*a^2.
• The lemniscate has the Unicode codepoint of x221E = 8734.
• The lemniscate as HTML NCR is ∞ or &#8734.
• Lemniscate [W]

Renard Numbers

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:

ar⁰ = a,  ar¹, ar², ar³, ..., 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].

• 2: 1, 5.
• 3: 1, 2, 5. This is like the Euro: 1, 2, 5, 10, 20, 50 cent coins; 1, 2 Euro coins; 5, 10, 20, 50, 100, 200, 500 bills.
• 4: 1, 2.5, 5, 7.5.
• 4x: 1, 2, 2.5, 5. This is like the U.S. Dollar: 1, 5, 10, 25, 50 cent coins; 1 dollar coins; 1, 2, 5, 10, 20, 50, 100 bills.
• 5: 1, 2, 4, 6, 8.
• 6: 1/3, 2/3, 3/3.
• 9: 1, 2, 3, 4, 5, 6, 7, 8, 9.
• 12: 1/6, 1/3, 3/6, 2/3, 5/6, 6/6.
• 18: 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5.
• Liter drinks: 0.1, 0.25, 0.375, 0.5, 0.75, 1, 1.5, 2, 3, and 5 liter.

Roman Numerals

• I = 1; I = 1,000; |I| = 100,000
• V = 5; V = 5,000;  |V|= 500,000
• X = 10; X = 10,000; |X| = 1,000,000
• L = 50; L = 50,000; |L| = 5,000,000
• C = 100; C = 100,000; |C| = 10,000,000. C is the initial of centum.
• D = 500; D = 500,000; |D| = 50,000,000. Derived by graphically halving the phi glyph.
• M = 1,000; M = 1,000,000; |M| = 100,000,000. M is the initial of mille. This was originally represented by the Greek letter phi (Φ).
• Largest numbers come first.
• Roman numerals use subtractive notation. If larger number is preceded by a smaller number, then the combined value is the difference. EGs: XIV = 14 and MIIM = 1998.
• A later Roman practiced added an overbar to indicate a multiple of 1,000.
• Another later Roman added an overbar and vertical lines indicate a multiple of 100,000.

Page Modified: (Hand noted: 2007-10-23 23:30:17Z) (Auto noted: 2010-12-24 22:58:48Z)