This is a list of notable numbers and articles about them. The list does not contain all numbers in existence as most of the number sets are infinite. Numbers may be included in the list based on their mathematical, historical or cultural notability, but all numbers have qualities that could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property. This is known as the interesting number paradox.
The definition of what is classed as a number is rather diffuse and based on historical distinctions. For example, the pair of numbers (3,4) is commonly regarded as a number when it is in the form of a complex number (3+4i), but not when it is in the form of a vector (3,4). This list will also be categorized with the standard convention of types of numbers.
This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five (an abstract object equal to 2+3), and the numeral five (the noun referring to the number).
Natural numbers
Natural numbers are a subset of the integers and are of historical and pedagogical value as they can be used for counting and often have ethno-cultural significance (see below). Beyond this, natural numbers are widely used as a building block for other number systems including the integers, rational numbers and real numbers. Natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "sinal numbers" and words used for ordering are "ordinal numbers". Defined by the Peano axioms, the natural numbers form an infinitely large set. Often referred to as "the naturals", the natural numbers are usually symbolised by a boldface (or blackboard bold , Unicode ).
The inclusion of 0 in the set of natural numbers is ambiguous and subject to individual definitions. In set theory and computer science, 0 is typically considered a natural number. In number theory, it usually is not. The ambiguity can be solved with the terms "non-negative integers", which includes 0, and "positive integers", which does not.
Natural numbers may be used as cardinal numbers, which may go by various names. Natural numbers may also be used as ordinal numbers.
Table of notable natural numbers012345678910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697989910010110210310410510610710810911011111211311411511611711811912012112212312412512612712812913013113213313413513613713813914014114214314414514614714814915015115215315415515615715815916016116216316416516616716816917017117217317417517617717817918018118218318418518618718818919019119219319419519619719819920020120220320420520620720820921021121221321421521621721821922022122222322422522622722822923023123223323423523623723823924024124224324424524624724824925025125225325425525625725825926026126226326426526626726826927027127227327427527627727827928028128228328428528628728828929029129229329429529629729829930030130231834134334735335936036536638338440040542049549650050151151256156359360061061361666670070372077778680083684087788890010001001100210231024108910931105172217281729190020002520300035114000410450005040600061747000774478258000812881918192900098019855999910,00012,00016,80716,84320,00020,06730,00040,00050,00060,00064,07965,53565,53665,53770,00078,55780,00090,000105131,072142,857144,000294,001509,20310610743,112,6091081092,147,483,6474,294,967,29510101011101210131014larger numbers, including 10100 and 1010100
Mathematical significance Natural numbers may have properties specific to the individual number or may be part of a set (such as prime numbers) of numbers with a particular property.
Cultural or practical significance Along with their mathematical properties, many integers have cultural significance or are also notable for their use in computing and measurement. As mathematical properties (such as divisibility) can confer practical utility, there may be interplay and connections between the cultural or practical significance of an integer and its mathematical properties.
Classes of natural numbers Subsets of the natural numbers, such as the prime numbers, may be grouped into sets, for instance based on the divisibility of their members. Infinitely many such sets are possible. A list of notable classes of natural numbers may be found at classes of natural numbers.
Prime numbers
A prime number is a positive integer which has exactly two divisors: 1 and itself.
The first 100 prime numbers are:
Table of first 100 prime numbers 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97101103107109113 127131137139149151157163167173 179181191193197199211223227229 233239241251257263269271277281 283293307311313317331337347349 353359367373379383389397401409 419421431433439443449457461463 467479487491499503509521523541
Highly composite numbers
A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement.
The first 20 highly composite numbers are:
1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560
Perfect numbers
A perfect number is an integer that is the sum of its positive proper divisors (all divisors except itself).
The first 10 perfect numbers:
Integers
The integers are a set of numbers commonly encountered in arithmetic and number theory. There are many subsets of the integers, including the natural numbers, prime numbers, perfect numbers, etc. Many integers are notable for their mathematical properties. Integers are usually symbolized by a boldface (or blackboard bold , Unicode ); this became the symbol for the integers based on the German word for "numbers" (Zahlen).
Notable integers include −1, the additive inverse of unity, and 0, the additive identity.
As with the natural numbers, the integers may also have cultural or practical significance. For instance, −40 is the equal point in the Fahrenheit and Celsius scales.
SI prefixes One important use of integers is in orders of magnitude. A power of 10 is a number 10k, where k is an integer. For instance, with k = 0, 1, 2, 3, ..., the appropriate powers of ten are 1, 10, 100, 1000, ... Powers of ten can also be fractional: for instance, k = -3 gives 1/1000, or 0.001. This is used in scientific notation, real numbers are written in the form m × 10n. The number 394,000 is written in this form as 3.94 × 105.
Integers are used as prefixes in the SI system. A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or fraction of the unit. Each prefix has a unique symbol that is prepended to the unit symbol. The prefix kilo-, for example, may be added to gram to indicate multiplication by one thousand: one kilogram is equal to one thousand grams. The prefix milli-, likewise, may be added to metre to indicate division by one thousand; one millimetre is equal to one thousandth of a metre.
Value 1000m NameSymbol 10001Kilok 10002MegaM 10003GigaG 10004TeraT 10005PetaP 10006ExaE 10007ZettaZ 10008YottaY 10009RonnaR 100010QuettaQ
Rational numbers
A rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . Since may be equal to 1, every integer is trivially a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface (or blackboard bold , Unicode ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
Rational numbers such as 0.12 can be represented in infinitely many ways, e.g. zero-point-one-two (0.12), three twenty-fifths (), nine seventy-fifths (), etc. This can be mitigated by representing rational numbers in a canonical form as an irreducible fraction.
A list of rational numbers is shown below. The names of fractions can be found at numeral (linguistics).
+ Table of notable rational numbers Decimal expansion Fraction Notability 1.0One is the multiplicative identity. One is a rational number, as it is equal to 1/1. 1 −0.083 333... The value assigned to the series 1+2+3... by zeta function regularization and Ramanujan summation. 0.5 One half occurs commonly in mathematical equations and in real world proportions. One half appears in the formula for the area of a triangle: × base × perpendicular height and in the formulae for figurate numbers, such as triangular numbers and pentagonal numbers. 3.142 857... A widely used approximation for the number . It can be proven that this number exceeds . 0.166 666... One sixth. Often appears in mathematical equations, such as in the sum of squares of the integers and in the solution to the Basel problem.
Real numbers Real numbers are least upper bounds of sets of rational numbers that are bounded above, or greatest lower bounds of sets of rational numbers that are bounded below, or limits of convergent sequences of rational numbers. Real numbers that are not rational numbers are called irrational numbers. The real numbers are categorized as algebraic numbers (which are the root of a polynomial with rational coefficients) or transcendental numbers, which are not; all rational numbers are algebraic.
Algebraic numbers
Name Expression Decimal expansion NotabilityGolden ratio conjugate () Reciprocal of (and one less than) the golden ratio.Twelfth root of two Proportion between the frequencies of adjacent semitones in the 12-tone equal temperament scale.Cube root of two Length of the edge of a cube with volume two. See doubling the cube for the significance of this number.Conway's constant (cannot be written as expressions involving integers and the operations of addition, subtraction, multiplication, division, and the extraction of roots)Defined as the unique positive real root of a certain polynomial of degree 71. The limit ratio between subsequent numbers in the binary Look-and-say sequence ().Plastic ratioThe only real solution of .() The limit ratio between subsequent numbers in the Van der Laan sequence. () Square root of two = 2 sin 45° = 2 cos 45° Square root of two a.k.a. Pythagoras' constant. Ratio of diagonal to side length in a square. Proportion between the sides of paper sizes in the ISO 216 series (originally DIN 476 series).Supergolden ratioThe only real solution of .() The limit ratio between subsequent numbers in Narayana's cows sequence. ()Triangular root of 2 Golden ratio (φ) The larger of the two real roots of x = x + 1.Square root of three = 2 sin 60° = 2 cos 30° . A.k.a. the measure of the fish or Theodorus' constant. Length of the space diagonal of a cube with edge length 1. Altitude of an equilateral triangle with side length 2. Altitude of a regular hexagon with side length 1 and diagonal length 2.Tribonacci constantThe only real solution of .() The limit ratio between subsequent numbers in the Tribonacci sequence.() Appears in the volume and coordinates of the snub cube and some related polyhedra.Supersilver ratioThe only real solution of .() The limit ratio between subsequent numbers in the third-order Pell sequence. ()Square root of five Length of the diagonal of a 1 × 2 rectangle.Silver ratio (δ) The larger of the two real roots of x = 2x + 1. Altitude of a regular octagon with side length 1.Bronze ratio (S) The larger of the two real roots of x = 3x + 1.
Transcendental numbers
NameSymbol or