## TYPES OF NUMBER## abundant numberA number that is smaller than the sum of its aliquot parts (proper divisors). Twelve is the smallest abundant number – the sum of its aliquot parts is 1 + 2 + 3 + 4 + 6 = 16 – followed by 18, 20, 24, and 30.A weird number is an abundant number that is not semiperfect;
in other words, n is weird if the sum of its divisors is greater
than n, but n is not equal to the sum of any subset of its
divisors. The first few weird numbers are 70, 836, 4030, 5830, and 7192.
It isn't known if there are any odd weird numbers. A deficient number is one that is greater than the sum
of its aliquot parts. The first few deficient numbers are 1, 2, 3, 4, 5,
7, 8, and 9. Any divisor of a deficient (or perfect) number is deficient.
A number that is not abundant or deficient is known as a perfect
number. ## alephArithmetica infinitorum but didn't appear in print until the Ars conjectandi by Jakob Bernoulli,
published posthumously in 1713 by his nephew Nikolaus Bernoulli. ## algebraic numberA real number that is a root of a polynomial equation with integer coefficients. For example, any rational numbera/b,
where a and b are non-zero integers, is an algebraic number
of degree one, because it is a root of the linear equation bx - a = 0. The square root of two is
an algebraic number of degree two because it is a root of the quadratic equation x^{2} - 2 = 0. If a real number is not algebraic,
then it is a transcendental number.
Almost all real numbers are transcendental because, whereas the set of algebraic
numbers is countably infinite, the set of transcendental numbers is uncountably
infinite (see infinity). ## almost perfect numberA description sometimes applied to the powers of 2 because the aliquot parts (proper divisors) of 2 sum to 2^{n} - 1. So a power
of 2 is a deficient number (one that is less than the sum of its proper
divisors), but only just. It isn't known whether there is an odd number ^{n}n whose divisors (excluding itself) sum to n - 1. ## amicable numbersA pair of numbers, also known asfriendly numbers,
each of whose aliquot parts add to give
the other number. (An aliquot part is any divisor that doesn't include the
number itself). The smallest amicable numbers are 220 (aliquot parts 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, with a sum of 284) and 284 (aliquot parts 1, 2, 4, 71, and 142, with a sum of 220). This pair was known to the ancient Greeks, and the Arabs found several more. In 1636 Pierre de Fermat rediscovered the amicable pair 17296 and 18416; two years later René Descartes rediscovered a third pair, 9363584 and 9437056. In the 18th century Leonhard Euler drew up a list of more than 60. Then, in 1866, B. Nicoló Paganini (not the violinist!), a 16-year-old Italian, startled the mathematical world by announcing that the numbers 1184 and 1210 were friendly. This second lowest pair of all had been completely overlooked. Today, the tally of known amicable numbers has grown to about two and half million. No amicable pair is known in which one of the two numbers is a square. An unusually high proportion of the numbers in amicable pairs ends in either 0 or 5. ## Arabic numeralNumerals written with Arabic digits alone: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or in combination: 10, 11, 12, ... 594, ... The system actually derives from Hindu mathematics.## automorphic numberAlso known as anautomorph, a number n whose square ends in n. For instance 5 is automorphic, because
5^{2} = 25, which ends in 5. A number n is called trimorphic if n^{3} ends in n. For example 49^{3},
= 117649, is trimorphic. Not all trimorphic numbers are automorphic. A number n is called tri-automorphic if 3n^{2} ends in n; for example 667 is tri-automorphic because 3 ×
667^{2}, = 1334667, ends in 667. ## Bell numberThe number of ways thatn distinguishable objects
(such as differently colored balls) can be grouped into sets (such as buckets)
if no set can be empty. For example, if there are three balls, colored red
(R), green (G), and blue (B), they can be grouped in five different ways:
(RGB), (RG)(B), (RB)(G), (BG)(R), and (R)(G)(B), so that the third Bell
number is 5. The sequence of Bell numbers, 1, 2, 5, 15, 52, 203, 877, 4140,
21147,..., can be built up in the form of a triangle, as follows. The first
row has just the number one. Each successive row begins with the last number
of the previous row and continues by adding the number just written down
to the number immediately above and to the right of it. 1 2 2 3 5 5 7 10 15 15 20 27 37 52 52 ... The Bell numbers appear down the left-hand side of the triangle. These normal Bell numbers contrast with ordered Bell numbers, which
count the number of ways of placing n distinguishable object (balls)
into one or more distinguishable sets (buckets) The ordered Bell numbers
are 1, 3, 13, 75, 541, 4683, 47293, 545835, ... Bell numbers, named after Eric Temple Bell, who was one of the first to analyze them in depth, are related to the Catalan numbers. See also combinatorics. ## Bernouilli numberA number of the type defined by Jakob Bernouilli in connection with evaluating sums of the form ∑ i^{k}. The
sequence B_{0}, B_{1}, B_{2},
... can be generated using the formulax/(e^{x} - 1) = ∑(B_{n} x^{n})/n!
though various different notations are used for them. The first few Bernoulli numbers are: B_{0} = 1, B_{1} = -1/2, B_{2} = 1/6 , B_{4} = -1/30 , B_{6} = 1/42 , ... They crop up in many diverse areas of mathematics including
the series expansions of tan x and Fermat's
last theorem. ## Betti numberAn important topological property of a surface, named after the Italian mathematician Enrico Betti (1823–1892). The Betti number is the maximum number of cuts that can be made without dividing the surface into two separate pieces. If the surface has edges, each cut must be a "crosscut," one that goes from a point on an edge to another point on an edge. If the surface is closed, like a sphere, so that it has no edges, each cut must be a "loop cut," a cut in the form a simple closed curve. The Betti number of a square is 0 because it is impossible to crosscut without leaving two pieces. However, if the square is folded into a tube, its topology changes – it now has two disconnected edges – and its Betti number changes to 1. A torus, or donut shape, has a Betti number of 2. See also chromatic number.## binarySee separate article on binary.## cardinal numberA number, often called simply acardinal, that is used
to count the objects or ideas in a set or collection: zero, one, two, ...
, eighty-three, and so on. The cardinality of a set is
just the number of elements the set contains. For finite sets this is always
a natural number. To compare the
sizes of two sets, X and Y, all
that's necessary is to pair off the elements of X with those of Y and see if there are any left over. This concept is obvious in the case
of finite sets but leads to some strange conclusions when dealing with infinite
sets (see infinity). For example, it is
possible to pair off all the natural numbers with all the even numbers,
with none left over; thus the set of natural numbers and the set of even
numbers have the same cardinality. In fact, an infinite set can be defined as any set that has a proper subset of the same cardinality. Every countable
set that is infinite has a cardinality of aleph-null;
the set of real numbers has cardinality
aleph-one. See also ordinal number. ## Carmichael numberAlso known as anabsolute pseudoprime, a number n that is a Fermat pseudoprime to any base, i.e., it divides (a^{n} - a) for any a. Another way of saying this is that a Carmichael
number is actually a composite number even though Fermat's little theorem suggests it is probably a prime number.
(Fermat's little theorem says that if P is a prime number then
for any number a, (a^{P} - a)
must be divisible by P. Carmichael numbers satisfy this condition to any
base despite being composite.) There are only seven Carmichael numbers under 10,000 – 561, 1105, 1729, 2465, 2821, 6601, and 8911 – and less than a quarter of a million of them under 10 ^{16}. Nevertheless, in 1994 it was proved that
there are infinitely many of them. All Carmichael numbers are the product
of at least three distinct primes, for example, 561 = 3 × 11 × 17. ## Catalan numberAny number,u, from the Catalan sequence defined by_{n}u = (2_{n}n)! / (n + 1)!n!
It begins: 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, ... The values of u represent the number of
ways a polygon with _{n}n + 2 sides can
be cut into n triangles using straight
lines joining vertices and are named after
the Belgian mathematician Eugène Catalan (1814-1894). They also arise
in other counting problems, for example in determining how many ways 2n beans can be divided into two containers if one container can never have
less than the second. ## chromatic number- In graph theory, the minimum
number of colors needed to color (the vertices of) a connected
graph so that no two adjacent vertices are colored the same. In
the case of simple graphs, this so-called
*coloring problem*can be solved by inspection. In general, however, finding the chromatic number of a large graph (and, similarly, an optimal coloring) is an NP-hard problem. - In topology, the maximum number of regions that can be drawn on a surface in such a way that each region has a border in common with every other region. If each region is given a different color, each color will border on every other color. The chromatic number of a square, tube, or sphere, for example, is 4; in other words, it is impossible to place more than four differently-colored regions on one of these figures so that any pair has a common boundary. "Chromatic number" also indicates the least number of colors needed to color any finite map on a given surface. Again, this is 4 in the case of the plane, tube, and sphere, as was proved quite recently in the solution to the four-color problem. The chromatic number, in both senses just described, is 7 for the torus, 6 for the Möbius band, and 2 for the Klein bottle. See also Betti number.
## circular numberAny number whose powers end on the same figure as it does itself. Circular numbers are numbers ending in 0, 1, 5, or 6.## common fractionA fraction that consists of the quotient of two integers.## composite numberA positiveinteger that can be factored into smaller positive
integers, neither of which is one. If a positive integer is not composite
(4, 6, 8, 9, 10, 12, ...) or one, then it is a prime
number (2, 3, 5, 7, 11, 13, 17, ...). As Karl Gauss put it in his Disquisitiones Arithmeticae (1801): "The problem
of distinguishing prime numbers from composite numbers and of resolving
the latter into their prime factors is known
to be one of the most important and useful in arithmetic." One reason it
is important today is that many secret codes and much of the security of
the Internet depends in part on the relative difficulty of factoring large
numbers. But more basic to a mathematician is that this problem has always
been central to number theory. Numbers that, for their size, have a lot of factors are sometimes referred to as highly composite numbers. Examples include 12, 24,
36, 48, 60, and 120. ## computable numberA real number for which there is an algorithm that, givenn, calculates
the nth digit. Alan Turing was the
first to define a computable number and the first to prove that almost all
numbers are uncomputable. An example of a number that, even though well-defined,
is uncomputable is Chaitin's constant. ## continued fractionA representation of a real number in the formwhich, mercifully for typesetters, can be written in compact notation as x = [a_{0}; a_{1}, a_{2}, a_{3}, ...]
where the integers a_{i} are called partial
quotients. Although rarely encountered in school and even college
math courses, continued fractions (CFs) provide one of the most powerful
and revealing forms of numerical expression. Numbers whose decimal expansions
look bland and unremarkable suddenly turn out, when unfolded as CFs, to
possess extraordinary symmetries and patterns. CFs also offer a way of constructing rational approximations to irrational
numbers and discovering the most irrational numbers. CFs first appeared in the works of the Indian mathematician Aryabhata in the sixth century, who used them to solve linear equations. They reemerged in Europe in the 15th and 16th centuries and Fibonacci attempted to define them in a general way. The actual term "continued fraction" first appeared in 1653 in an edition of Arithmetica Infinitorum by John Wallis. Their properties were also
studied by one of Wallis's English contemporaries, William Brouncker, who
along with Wallis, was one of the founders of the Royal Society. At about
the same time, in Holland, Christiaan Huygens made practical use of CFs in his designs of scientific instruments. Later,
in the 18th and early 19th centuries, Carl Gauss and Leonhard Euler delved into many of their
deeper properties.CFs can be finite or infinite in length. Finite CFs are unique so long as we don't allow a quotient of 1in the final entry in the bracket; for example, we should write 1/2 as [0; 2] rather than as [0; 1, 1]. We can always eliminate a 1 from the last entry by adding to the previous entry. If CFs are finite in length then they can be evaluated level by level (starting at the bottom) and will reduce always to a rational fraction; for example, the CF [1; 3, 2, 4] = 40/31. However, CFs can be infinite in length, in which case they produce representations of irrational numbers. Here are the leading terms from a few notable examples of infinite CF: e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]√2 = [1; 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...] √3 = [1; 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, ...] π = [3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, ...] These examples reveal a number of possibilities. All of the expansions have simple patterns except that for pi, which was first calculated by John Wallis in 1685 and has no obvious pattern at all. There also seems to be a preference for the quotients to be small numbers in these examples. The CF for e was first
calculated by Roger Cotes, the Plumian Professor of Experimental Philosophy
at Cambridge, in 1714.If an infinite CF is chopped off after a finite number of steps then the result is a rational approximation to the original irrational.
For example, in the case of π, chopping the CF at [3; 7] gives the familiar
rational approximation for π of 22/7 = 3.1428571... Retaining two more
terms leads to [3; 7, 15, 1] = 353/113 = 3.1415929..., which is an even
better approximation to the true value of π (3.14159265...) and one that
was known to the early Chinese. The more terms retained in the CF, the better
the rational approximation becomes. In fact, the CF provides the best possible
rational approximations to a general irrational number. Notice also that
if a large number occurs in the expansion of quotients, then truncating
the CF before that will produce an exceptionally good rational approximation.
Most CF quotients are small numbers (1 or 2), so the appearance in the CF
of π of a number as large as 292 so early in the expansion is unusual.
It also leads to an extremely good rational approximation to π = [3;
7, 15, 1, 292] = 103993/33102. ## cube numberA number formed by multiplying a digit by itself three times. Each cube number can be represented by a cube made up of unit cubes. ## Cullen numberA number of the form (n × 2^{n}) + 1, denoted C_{n}, and named after the Reverend James Cullen
(1867–1933), an Irish Jesuit priest and schoolmaster. Cullen noticed
that the first, C_{1} = 3, was a prime
number, but with the possible exception of the 53rd, the next 99 were
all composite. Soon afterward, Cunningham discovered that 5591 divides C_{53},
and noted that all the Cullen numbers are composite
numbers for n in the range 2 < n < 200, with the possible
exception of 141. Five decades later Robinson showed that C_{141} is a prime. Currently, the only known Cullen primes are those with n = 1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275,
and 481899. Although the vast majority of Cullen numbers are composite,
it has been conjectured that there are infinitely many Cullen primes. Whether n and C_{n} can simultaneously be prime
isn't known. Sometimes, the name "Cullen number" is extended to include
the Woodall numbers, W_{n} =
(n × 2^{n}) - 1. Finally, a few authors have
defined a number of the form (n × b^{n})
+ 1, with n + 2 > b, to be a generalized Cullen
number.< ## Cunningham chainA sequence of prime numbers in which each member is twice the previous one plus one. For example, {2, 5, 11, 23, 47} is the first Cunningham chain of length 5 and {89, 179, 359, 719, 1439, 2879} is the first of length 6. More generally, aCunningham
chain of length is a sequence of k of the first kindk prime numbers, each of which is twice the preceding one plus one. A Cunningham
chain of length is a sequence of k of the second kindk primes, each of which is twice the preceding one minus one. For
example, {2, 3, 5} is a Cunningham chain of length 3 of the second kind
and {1531, 3061, 6121, 12241, 24481} is a Cunningham chain of length 5 of
the second kind. Primes of both these forms are called complete chains if
they can't be extended by adding either the next larger or the next smaller
terms. See also Sophie
Germain prime. ## cute numberA numbern such that a square can
be cut into n squares of, at most, two different sizes. For example,
4 and 10 are cute numbers. ## cyclic numberA number withn digits, which, when multiplied by 1, 2, 3, ..., n produces the same digits in a different order. For example, 142857
is a cyclic number: 142857 × 2 = 285714; 142857 × 3 = 428571;
142857 × 4 = 571428; 142857 × 5 = 714285; 142857 × 6 = 857142,
and so on. It has been conjectured, but not yet proven, that an infinite
number of cyclic numbers exist. ## decimalThe commonly used number system, also known asdenary, in which each place has a value 10 times
the value of the place at its right. For example, 4327 in the decimal (base
10) system is shorthand for (4 × 10^{3}) + (3 × 10^{2})
+ (2 × 10^{1}) + (7 × 10^{0}), where 100 = 1.
Similarly, fractions may be expressed by
setting their denominators equal to
powers of 10. The word decimal comes from the Latin decimus for "tenth." The verb decimare, literally "to take a tenth of," was used to describe a
form of punishment applied to mutinous units in the Roman army. The men
were lined up and every tenth soldier was killed as a lesson to the rest.
From this custom comes our word "decimate," which we use more loosely –
in fact, incorrectly – to indicate near-total destruction. The Latin decimare was also used in a less ferocious sense to mean "to tax
to the amount of one tenth." However, the usual word describing a one-tenth
tax in English is tithe, which comes from the Old English teogotha,
a form of tenth. ## decimal fractionA number consisting of an integer part, which may be zero, and a decimal part less than unity that follows the decimal marker (which may be a point or a comma). A finite or terminating decimal fraction has a sequence of decimals with a definite break-off point after which all the places are zeros. Other fractions produce endless sequences of decimals that are periodic non-terminating.## denominatorIn a rational number, the number that appears below the fraction bar, i.e., the divisor of a common fraction. The denominator indicates how many parts the whole is divided into. For example, in the fraction 3/5, the denominator is 5; 3 is the numerator.## duodecimal
^{3}) + (4 ×
12^{2}) + (5 × 12^{1}) + (0 × 12^{0})
or, in duodecimals, 2450. Fractions are expressed similarly. Two extra symbols
are needed for this system to represent the numbers 10 and 11; these are
generally accepted as X (dek) and Σ (el) respectively. The advantage of this system can be realized by consideration of the integral factors of 10 and 12: 10 has two (2, 5) while 12 has four (2, 3, 4, 6). The fractions one-third and two-thirds are 0.4 and 0.8, respectively, in duodecimal – much simpler than the recurring expressions 0.3333... and 0.6666... in the decimal system. The most common examples of everyday use of this system are the setting of 12 inches to the foot and 12 months to the year. ## economical numberA number that has no more digits than there are digits in its prime factorization (including powers). If a number has fewer digits than are in its prime factorization it is known as afrugal number. The smallest frugal is
125, which has three digits, but can be written as 5^{3}, which
has only two. The next few frugals are 128 (2^{7}), 243 (3^{5}),
256 (2^{8}), 343 (7^{3}), 512 (2^{9}), 625 (5^{4}),
and 729 (3^{6}). An equidigital number is an economical
number that has the same number of digits as make up its prime factorization.
The smallest equidigitals are 1, 2, 3, 5, 7, and 10 (= 2 × 5). All prime numbers are equidigital. An extravagant
number is one that has fewer digits than are in its prime factorization.
The smallest extravagant number is 4 (= 2^{2}), followed by 6, 8,
and 9. There are infinitely many of each of these kinds of numbers. Are
there also arbitrarily long sequences of consecutive ones? Strings of consecutive
economical numbers of length seven start at each of 157, 108749, 109997,
121981 and 143421. On the other hand, the longest string of consecutive
frugal numbers up to 1000000 is just two (for example, 4374 and 4375). Even
so, it has been proved that, if a certain conjecture about prime numbers,
known as Dickson's Conjecture is true, then there are arbitrarily
long strings of frugals. ## equivalent numbersNumbers such that the sums of their aliquot parts (proper divisors) are the same. For example, 159, 559, and 703 are equivalent numbers because their aliquot parts all sum to 57.## factorianA natural number that equals of the sums of the factorials of its digits in a given base. The only known decimal factorions are 1 = 1!, 2 = 2!, 145 = 1! + 4! + 5!, and 40585 = 4! + 0! + 5! + 8! + 5!.## Fermat numberA number defined by the formulaF_{n} = 2^{2^n} + 1 and named after Pierre Fermat who conjectured,
wrongly, that all such numbers would be prime. The first five Fermat numbers, F_{0} = 3, F_{1} = 5, F_{2} = 17, F_{3} = 257, and F_{4} = 65,537,
are prime. However, in 1732, Leonhard Euler discovered that 641 divides F_{5}.
It takes only two trial divisions to find this factor because Euler showed
that every factor of a Fermat number F with _{n}n greater than 2 has the form k × 2^{n+2} + 1.
In the case of F_{5} this is 128k + 1, so we would
try 257 and 641 (129, 385, and 513 are not prime). It is likely that there
are only finitely many Fermat primes. Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g.,
by straightedge and compass) if and only
if n is a power of two times a product of distinct Fermat primes. ## Fibonacci sequenceSee separate article on Fibonacci sequence.## figurate numberA number sequence found by creating consecutive geometrical figures from arrangements of equally spaced points. Here is an example:
The points can be arranged in one, two, three, or more dimensions. There are many different kinds of figurate numbers, such as polygonal numbers and tetrahedral numbers. ## fractionA rational number that represents a part, or several equal parts, of a whole; examples include one-half, two-thirds, and three-fifths. The word comes from the Latinfrangere, meaning
"to break." A simple, common, or vulgar fraction is of
the form a/b, where a (the numerator) may be
any integer and b (the denominator)
may be any integer greater than 0. If a < b, the fraction
is said to be proper ("bottom heavy"); otherwise it is improper ("top heavy"). A decimal
fraction has a denominator of 10, 100, 1000, etc. In an algebraic fraction, the denominator, or the numerator and denominator, are algebraic expressions, for example x/(x^{2} + 2). In a composite fraction, both the numerator and denominator are themselves fractions. Two fractions a/b and c/d can be added, subtracted, multiplied, and divided according to the rules:
## happy numberIf you iterate the process of summing the squares of the decimal digits of a number and if this process terminates in 1, then the original number is called a happy number. For example 7 (7^{2}) 49 (4^{2} + 9^{2}) 97 (9^{2} + 7^{2}) 130 (1^{2} + 3^{2})10 1. See also amicable numbers. ## Harshad numberA number that is divisible by the sum of its own digits. For example, 1729 is a Harshad number because 1 + 7 + 2 + 9 = 19 and 1729 = 19 × 91. Harshad numbers are also known asNiven numbers.A Harshad amicable pair is an amicable
pair (m, n) such that both m and n are Harshad
numbers. For example, 2620 and 2924 are a Harshad amicable pair because
2620 is divisible by 2 + 6 + 2 + 0 = 10 and 2924 is divisible by 2 + 9 +
2 + 4 = 17 (2924/17 = 172). There are 192 Harshad amicable pairs in first
5,000 amicable pairs. ## hexadecimalA number system in which each place has a value 16 times the value of the place at its right. Hexadecimal numbers are written using the symbols 0–9 and A–F (or a–f). The hexadecimal number 35B, for example, is equivalent to (3 × 256) + (5 × 16) + (11 × 1) = 859 in decimal.Hexadecimal provides a convenient way to express binary numbers in modern computers in which a byte is almost always defined as containing eight binary digits. When showing the contents of computer storage – for example, when getting a core dump of storage in order to debug a new computer program or when expressing a string of text characters or a string of binary values – one hexadecimal digit can represent the arrangement of four binary digits. Two hexadecimal digits can represent eight binary digits, or a byte. ## highest common fcatorOf two or more integers, the largest integer which they share as a factor. For example, the highest common factor (hcf) of 15 and 17 is 1, since 17 is a prime number. In algebra, the hcf of two or more algebraic expressions may be found by examination of the factors of each: hence the hcf of 9ax^{2} (= 3.3.a.a.x.x) and 3ax^{2} (= 3.a.a.x)
is 3a.x. ## hyperfactorialA number such as 108, which is equal to 3^{3} × 2^{2} ×
1^{1}. In general, the n-th hyperfactorial H(n)
is given byH(n) = n^{n} (n-1)^{n-1} ... 3^{3} 2^{2} 1^{1}
The first eight hyperfactorials are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, and 55696437941726556979200000. See also large numbers and superfactorials. ## hyperreal numberAny of a colossal set of numbers, also known asnonstandard reals,
that includes not only all the real numbers but also certain classes of infinitely large (see infinity)
and infinitesimal numbers as well.
Hyperreals emerged in the 1960s from the work of Abraham Robinson who showed how infinitely large and infinitesimal numbers can be rigorously
defined and developed in what is called nonstandard
analysis. Because hyperreals represent an extension of the real numbers, R, they are usually denoted by *R. Hyperreals include all the reals (in the technical sense that they form an ordered field containing the reals as a subfield) and also contain infinitely many other numbers that are either infinitely large (numbers whose absolute value is greater than any positive real number) or infinitely small (numbers whose absolute value is less than any positive real number). No infinitely large number exists in the real number system and the only real infinitesimal is zero. But in the hyperreal system, it turns out that that each real number is surrounded by a cloud of hyperreals that are infinitely close to it; the cloud around zero consists of the infinitesimals themselves. Conversely, every (finite) hyperreal number x is infinitely close to exactly
one real number, which is called its standard part, st(x).
In other words, there exists one and only one real number st(x) such
that x – st(x) is infinitesimal. ## integerAny positive or negative whole number or zero: ...-3, -2, -1, 0, 1, 2, 3, ... "Integer" is Latin for "whole" or "intact." The set of all integers is denoted byZ, which stands for Zahlen (German for "number").
The integers are an extension of the natural
numbers to include negative numbers and so make possible the solution of all equations of the form a + x = b, where a and b are natural numbers.
Integers can be added and subtracted, multiplied, and compared. Like the
natural numbers, the integers form a countably infinite set. However, the
integers don't form a field since, for instance,
there is no integer x such that 2x = 1; the smallest field
containing the integers is that of the rational
numbers. An important property of the integers is division with
remainder: given two integers a and b with b not equal to 0, it is always possible to find integers q and r such that a = bq + r, and such that 0 r < |b|. q is called the quotient and r is called the remainder resulting from division
of a by b. The numbers q and r are uniquely
determined by a and b. From this follows the fundamental
theorem of arithmetic, which states that integers can be written
as products of prime numbers in an essentially
unique way. ## interesting numbersClearly some numbers are of greater interest (at least to mathematicians) than are others. The number pi, for instance, is far more interesting than 1.283 – or virtually any other number for that matter. Confining our attention to integers, can there be such a thing as an uninteresting number? It is easy to show that the answer must be "no." Suppose there were a set U of uninteresting integers. Then it must contain a least member,u. But the property of being the smallest
uninteresting integer makes u interesting! As soon as u is removed from U, there is a new smallest uninteresting integer, which
must then also be excluded. And so the argument could be continued until
U was empty. Given that all integers are interesting can they be ranked
from least to most interesting? Again, no. To be ranked as "least interesting"
is an extremely interesting property, and thus leads to another logical
contradiction! When Srinivasa Ramanujan, the great Indian mathematician, was ill with tuberculosis in a London hospital, his colleague G. H. Hardy went to visit him. Hardy opened the conversation with: "I came here in taxi-cab number 1729. That number seems dull to me which I hope isn't a bad omen." Ramanujan replied, without hesitation: "Nonsense, the number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." (1729 = 1 ^{3} + 12^{3} and 9^{3} + 10^{3}.) ## irrational numberA real number that can't be written as one whole number divided by another; in other words, a real number that isn't a rational number. The decimal expansion of an irrational numbers doesn't come to an end or repeat itself (in equal length blocks), though it may have a pattern such as 0.101001000100001... The vast majority of real numbers are irrational, so that if you were to pick a single point on the real number line at random the chances are overwhelmingly high that it would be irrational. Put another way, whereas the set of all rationals is countable, the irrationals form an uncountable set and therefore represent a larger kind of infinity. Indeed, as the Harvard logician Willard Van Orman Quine pointed out: "The irrationals exist in such variety ... that no notation whatever is capable of providing a separate name for each of them."There are two types of irrational number: algebraic numbers, such as the square root of 2, which are the roots of algebraic equations, and the transcendental numbers, such as π and e, which aren't. In some cases it
isn't known if a number is irrational or not; undecided cases include 2,
π^{e}, and the Euler-Mascheroni
constant, γ (gamma). ^{e}An irrational number raised to a rational power can be rational; for instance, √2 to the power 2 is 2. Also, an irrational number to an irrational power can be rational. What kind of number is √2 ^{√2}?
The answer is irrational. This follows from the so-called Gelfond-Schneider
theorem, which says that if A and B are roots of polynomials, and A is not 0
or 1 and B is irrational, then AB must be irrational (in
fact, transcendental). ## Kaprekar numberTake a positive whole numbern that has d number of digits.
Take the square n and separate the result into two pieces: a right-hand
piece that has d digits and a left-hand piece that has either d or d-1 digits. Add these two pieces together. If the result is n, then n is a Kaprekar number. Examples are 9 (9^{2} = 81, 8 + 1 = 9), 45 (45^{2} = 2025, 20 + 25 = 45), and 297 (297^{2} = 88209, 88 + 209 = 297). The first 20 Kaprekar numbers according to this definition are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, and 99999. Kaprekar numbers can also be defined by higher powers. For example, 45 ^{3} = 91125, and 9 + 11 + 25 = 45. The first ten numbers with this property
are: 1, 8, 10, 45, 297, 2322, 2728, 4445, 4544, and 4949. For fourth powers,
the sequence begins 1, 7, 45, 55, 67, (100), 433, 4950, 5050, 38212, 65068.
Notice that 45 is a Kaprekar number for second, third, and fourth powers
(45^{4} = 4100625, and 4 + 10 + 06 + 25 = 45) – the only number
in all three Kaprekar sequences, up to at least 400,000. See also unique number. ## Liouville numberA transcendental number that can be approximated very closely by a rational number. Normally, proving that any given number is transcendental is difficult. However, the French mathematician Joseph Liouville (1809–1882) showed the existence of a large (in fact, infinitely large) class of transcendentals whose nature is easy to ascertain. A typical Liouville number is 0.101 001 000 000 100 000 000 000 000 000 000 000 01... in which the successive groups of zeros are of length 1!, 2!, 3!, 4!, and so on.## lowest common multipleThe lowest number of which two or more positive whole numbers are factors. For example, the lowest common multiple of 2, 4, and 6 is 12.## lucky numberA number in a sequence, first identified and named around 1955 by Stanislaw Ulam, that evades a particular type of number "sieve" (similar to the famous Sieve of Eratosthenes), which works as follows. Start with a list of integers, including 1, and cross out every second number: 2, 4, 6, 8, ... The second surviving integer is 3. Cross out every third number not yet eliminated. This removes 5, 11, 17, 23, ... The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Repeat this process indefinitely and the numbers that survive are the "lucky" ones:1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195 ...Amazingly, though generated by a sieve based entirely on a number's position in an ordered list, the luckies share many properties with prime numbers. For example, there are 25 primes less than 100 and 23 luckies less than 100. In fact, it turns out that primes and luckies crop up about equally often within given ranges of integers. Also, the gaps between successive primes and the gaps between successive luckies widen at roughly the same rate as the numbers increase, and the number of twin primes – primes that differ by 2 – is close to the number of twin luckies. The luckies even have their own equivalent of the famous (still unsolved) Goldbach Conjecture, which states that every even number greater than 2 is the sum of two primes. In the case of luckies, it is conjectured that every even number is the sum of two luckies; no exception has yet been found. Another unresolved problem is whether there are an infinite number of lucky primes. See also Ulam Spiral. ## Mersenne numberA number of the form 2^{n} - 1 (one less than a power of
2), where n is a positive integer. Mersenne numbers are named after
Marin Mersenne who wrote about them in his Cogita
Physica-Mathematica (1644) and wrongly conjectured that they were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, and 257, and composite for n > 257. See also Mersenne prime. ## narcissistic numberAlso known as anArmstrong number or a plus perfect
number, an n-digit number equal to the sum of its digits
raised to the nth power. For instance, 371 is narcissistic because
3^{3} + 7^{3} + 1^{3} = 371, and 9474 is narcissistic
because 9^{4} + 4^{4} + 7^{4} + 4^{4} =
9474. The smallest narcissistic number of more than one digit is 153 = 1^{3} + 5^{3} + 3^{3}. The largest narcissistic number (in base
10) is 115132219018763992565095597973971522401, which is the sum of the
39th powers of its digits. The reason there are no larger numbers is related
to the fact that, as the number of digits increases, more and more 9's are
required to get a sum that has n digits. For example, 10^{70}-1
is a number consisting of 70 nines in a row, and the sum of the 70th powers
of its digits is 70 × 9^{70} is approximately equal to 4.386051
× 10^{68}, which is only 69 digits long. So there is no way any
70-digit number can be equal to the sum of the 70th powers of its digits.
The reason we see the last number occur at 39 digits is because, as the
limit is approached, the number of big digits like 8's and 9's has to increase
to make sure the sum will be big enough, but this means that there are a
lot fewer combinations of digits to choose from. ## natural numberA number used for counting: 1, 2, 3, .... The debate about whether zero should also be included as a natural number has been going on for hundreds of years, and there's no general agreement even today. To avoid confusion, 0, 1, 2, 3, ..., are often referred to asnon-negative integers or whole numbers, while 1, 2, 3, ..., are called positive
integers. Adding or multiplying natural numbers always produces other natural numbers. However, subtracting them can produce zero or negative integers, while dividing them produces rational numbers. An important property of the natural numbers is that they are well-ordered, in other words, every set of natural numbers has a smallest element. The deeper properties of the natural numbers, such as the distribution of prime numbers, are studied in number theory. Natural numbers can be used for two purposes: to describe the position of an element in an ordered sequence, which is generalized by the concept of ordinal number, and to specify the size of a finite set, which is generalized by the concept of cardinal number. In the finite world, these two concepts coincide; however, they differ when it comes to infinite sets (see infinity). ## negative numbersThe introduction of the negative numbers is due to the need for subtraction to be performable without restriction. In the domain of positive numbers the subtractiona - b = c can only be carried out if a > b (a is greater than b). If, on the other hand, a < b (a is less than b) we define c = -(b - a), e.g., 5 - 7 = (-2). Here the "- sign" on the left hand
side of the equation represents an operation; on the right hand side it
forms part of the number itself. In the case of positive numbers the associated
sign (+) may be omitted, but not in the case of negative numbers. Representation
of the numbers on a straight line clarifies the notion of a negative number. Long denied legitimacy in mathematics, negative numbers are nowhere to be found in the writings of the Babylonians, Greeks, or other ancient cultures. On the contrary, because Greek mathematics was grounded in geometry, and the concept of a negative distance is meaningless, negative numbers seemed to make no sense. They surface for the first time in in bookkeeping records seventh-century India and in a chapter of a work by the Hindu astronomer Brahmagupta. Their earliest documented use in Europe is in 1545 in the Ars magna of Girolamo Cardano.
By the early 17th century, Renaissance mathematicians were explicitly using
negative numbers but also meeting with heavy opposition. René Descartes called negative roots "false roots", and Blaise Pascal was convinced that numbers "less than zero" couldn't exist. Gottfried Leibniz admitted that they could lead to some absurd conclusions, but defended then
as useful aids in calculation. By the 18th century, negative numbers had
become an indispensable part of algebra. ## Niven numberAny whole number that is divisible by the sum of its digits. For example, 126 is a Niven number because, the sum of its digits 1 + 2 + 6, is 9, and 9 goes into 126 exactly 14 times. Niven numbers are name after the number theorist Ivan Niven who, in 1977, gave a talk at a conference in which he mentioned integers which are twice the sum of their digits. Then in a 1982 article, the mathematician Robert Kennedy christened numbers which are divisible by their digital sum in honor of Niven. They are also known asHarshad
numbers. ## normal numberA number in which digit sequences of the same length occur with the same frequency. A constant is considered normal to base 10 if any single digit in its decimal expansion appears one-tenth of the time, any two-digit combination one-hundredth of the time, any three-digit combination one-thousandth of the time, and so on. In the case of pi, the digit 7 is expected to appear 1 million times among the first 10 million digits of its decimal expansion. It actually occurs 1,000,207 times – very close to the expected value. Each of the other digits also turns up with approximately the same frequency, showing no significant departure from predictions.A number is said to be absolutely normal if its digits
are normal not only to base 10 but also to every integer base greater than
or equal to 2. In base 2, for example, the digits 1 and 0 would appear equally
often. Émile Borel introduced the concept of normal numbers in 1909 as a way to characterize the resemblance between the digits of a mathematical constant such as pi and a sequence of random numbers. He quickly established that there are lots of normal numbers, though finding a specific example of one proved to be a major challenge. The first to be found was Champernowne's Number, which is normal to base 10. Analogous normal numbers can be created for other bases. To date, no specific "naturally occurring" real number has been proved to be absolutely normal, even though it is known that almost all real numbers are absolutely normal! However, in 2001, Greg Martin of the University of Toronto found some examples of the opposite extreme – real numbers that are normal to no base whatsoever. ^{1} To start with, he noted that every rational
number is absolutely abnormal. For example, the fraction 1/7 can be
written in decimal form as 0.1428571428571... The digits 142857 just repeat
themselves. Indeed, an expansion of a rational number to any base b or b eventually repeats. Martin then focused on constructing
a specific ^{k}irrational absolutely abnormal number. He nominated the
following candidate, expressed in decimal form, for the honor:a = 0.6562499999956991999999...9999998528404201690728...- Martin, G. "Absolutely abnormal numbers."
*American Mathematical Monthly*, 108: 746-754, 2001.
## numeralA symbol, or combination of symbols, that describes a number. Arabic numerals are the 10 digits from 0 to 9. Roman numerals, as normally used today, consist of seven letters or marks (I=1, V=5, X=10, L=50, C=100, D=500, and M=1,000). The formation of numbers from numerals depends on the number system used.
## numerator## oblong numberAny number that is the product of two consecutive integers. Oblong numbers are also known aspronic numbers. The first
few of them are: 0, 2, 6, 12, 20, and 30. ## octonionAlso known as aCayley number. Octonions are a non-associative generalization of the quaternions and
the complex numbers involving numbers
with one real coefficient and seven
imaginary coefficients. ## ordered pairA collection of two objects such that one can be distinguished as thefirst
element and the other as the second element. An ordered pair
with first element a and second element b is usually written
as (a, b). Two such ordered pairs (a_{1}, b_{1})
and (a_{2}, b_{2}) are equal if and only if a_{1} = a_{2} and b_{1} = b_{2}. Ordered triples and ordered n-tuples (ordered
lists of n terms) are defined in the same way. An ordered triple
(a, b, c) can be defined as (a , (b, c)), i.e. as two nested pairs. ## ordinal numberA number used to give the position in an ordered sequence: first, second, third, fourth ... Ordinal numbers are distinct from cardinal numbers (one, two, three, four, ...), which describe the size of a collection. The mathematician Georg Cantor showed in 1897 how to extend the concept of ordinals beyond the natural numbers to the infinite and how to do arithmetic with the resultingtransfinite ordinals (see infinity). ## palindromic numberA number such as 1234321 that reads the same forwards and backwards; more generally, a symmetrical number written in some basea as a_{1} a_{2} a_{3} ...|... a_{3} a_{2} a_{1}.
In the familiar base 10 system, there are nine two-digit palindromic numbers:
11, 22, 33, 44, 55, 66, 77, 88, 99; there are 90 palindromics with three
digits: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919,
929, 939, 949, 959, 969, 979, 989, 999; and there are 90 palindromics with
four digits: 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991,
..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999, giving
a total of 199 palindromic numbers below 10^{4}. Below 10^{5} there are 1099 palindromics and for other exponents of 10^{n} there are 1999, 10999, 19999, 109999, 199999, 1099999, ... It is conjectured,
but has not been proved, that there are an infinite number of palindromic prime numbers. With the exception of 11, palindromic primes must have an odd number of digits. A normally quick way to produce a palindromic number is to pick a positive integer of two or more digits, reverse the digits, and add to the original, then repeat this process with the new number, and so on. For example, 3462 gives the sequence 3462, 6105, 11121, 23232. Does the series formed by adding a number to its reverse always end in a palindrome? It used to be thought so. However, this conjecture has been proved false for bases 2, 4, 8, and other powers of 2, and seems to be false for base 10 as well. Among the first 100,000 numbers, 5,996 numbers are known that have not produced palindromic numbers by the add-and-reverse method in calculations carried out to date. The first few of these are 196, 887, 1675, 7436, 13783, 52514, ... A proof that these numbers never produce palindromes, however, has yet to be found. The largest known palindromic prime, containing 30,913 digits was found by David Broadhurst in 2003. ## pandigital numberAn integer that contains each of the digits from 0 to 9 and whose leading digit is nonzero. The first few pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, and 1023456978. A ten-digit pandigital number is always divisible by 9. If zeros are excluded, the first few "zeroless" pandigital numbers are 123456789, 123456798, 123456879, 123456897, 123456978, and 123456987, and the first few zeroless pandigital primes are 1123465789, 1123465879, 1123468597, 1123469587, and 1123478659. The sum of the first 32423 (a palindromic number) consecutive primes is 5897230146, which is pandigital. No other palindromic number shares this property. Examples of palindromic numbers that are the product of pandigital numbers are 2 970 408 257 528 040 792 (= 1 023 687 954 × 2 901 673 548) and 5 550 518 471 748 150 555 (= 1 023 746 895 × 5 421 768 309), both found in 2001. Apandigital
product is a product in which the digits of the multiplicand, multiplier,
and product, taken together, form a pandigital number. ## partition numberA number that gives the number of ways of placingn indistinguishable
balls into n indistinguishable urns. For example:
1: (*)The sequence runs: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, ... If the urns are distinguishable, the number of ways is 2 n. If the
balls are distinguishable, the number of ways is given by the nth Bell number. ## Pell numbersNumbers that are similar to the Fibonacci numbers and are generated by the formulaA_{n} = 2A_{n-1} + A_{n-2}.
The sequence runs: 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, ... The ratio of successive terms approaches 1 plus the square root of 2. ## perfect cubeAn integer of the formm^{3} where m is an integer. ## perfect numberA whole number that is equal to the sum of all its factors except itself. For example, 6 is a perfect number because its factors, 1, 2, and 3 add to give 6. The next smallest is 28 (the sum of 1 + 2 + 4 + 7 + 14). The ancient Christian scholar Augustine explained that God could have created the world in an instant but chose to do it in a perfect number of days, 6. Early Jewish commentators felt that the perfection of the universe was shown by the Moon's period of 28 days. The next in line are 496, 8128, and 33,550,336. As René Descartes pointed out: "Perfect numbers like perfect men are very rare." All end in 6 or 8, though what seems to be an alternating pattern of 6's and 8's for the first few perfect numbers doesn't continue. All are of the form 2^{n-1}(2-1),
where 2^{n}-1 is a Mersenne prime, so that the search for
perfect numbers is the search for Mersenne primes. The largest one known,
as of March 2003, is 4.27764198 × 108107891. It isn't known if there are
infinitely many perfect numbers or if there are any odd perfect numbers. ^{n}A pseudoperfect number or semi-perfect number is a number equal to the sum of some of its divisors, e.g. 12 = 2 + 4 +
6, 20 = 1 + 4 + 5 + 10. An irreducible semi-perfect number is a semi-perfect number, none of whose factors is semi-perfect, e.g. 104.
A quasi-perfect number would be a number n whose divisors
(excluding itself) sum to n + 1, but it isn't known if such a number
exists. A multiply perfect number is a number n whose divisors sum to a multiple of n. An example is 120, whose divisors
(including itself) sum to 360 = 3 × 120. If the divisors sum to 3n, n is called multiply perfect of order 3, or tri-perfect.
Ordinary perfect numbers are multiply perfect of order 2. Multiply perfect
numbers are known of order up to 8. ## perfect powerAn integer of the formm^{n} where m and n are integers and n > 1. ## perfect squareA number that is the product of two equal whole numbers, e.g. 1 = 1 × 1, 4 = 2 × 2, 9 = 3 × 3, 16 = 4 × 4, 25 = 5 × 5.## permutable primeAlso known as anabsolute prime, a prime
number with at least two distinct digits which remains prime on every
rearrangement (permutation) of the digits. For example, 337 is a permutable
prime because each of 337, 373, and 733 are prime. Most likely, in base
ten, the only permutable primes are 13, 17, 37, 79, 113, 199, 337, and their
permutations. Obviously permutable primes cannot have any of the digits
2, 4, 6, 8 or 5. They also cannot have all four of the digits 1, 3, 7, and
9 simultaneously. ## polygonal numberThe number of equally spaced dots needed to draw a polygon. Polygonal numbers, which are a type of figurate number, include square numbers, triangular numbers, pentagonal numbers, and hexagonal numbers.Polygonal numbers form an arithmetic sequence of the second order. The general term of these sequences is where d is one of the numbers 1, 2, 3, ...With d = 1 we obtain the triangular
numbers: 1, 3, 6, 10, 15, ... With d = 2 we obtain the square
numbers: 1, 4, 9, 16, 25, ... With d = 3 we obtain the pentagonal numbers: 1,
5, 12, 22, 35, ... ## powerful numberAlso known as asquarefull number, a positive whole number n such that for every prime number p dividing n, p^{2} also divides n.
Every powerful number can be written as a^{2}b^{3},
where a and b are positive integers. The first 20 powerful
numbers 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125,
128, 144, and 169. There are pairs of consecutive powerful numbers, such
as (8,9), (288,289), and (675,676). However, no three consecutive powerful
numbers are known and, in 1978, Paul Erdös conjectured that none exist. ## practical numberA numbern such that every positive integer less than n is either a divisor or a sum of distinct divisors of n. The first
few practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, ... All perfect
numbers are practical. ## prime numberSee separate article for prime number.## pronic numberAlso known as arectangular or oblong number,
a number that is the product of two consecutive integers: 2 (1 × 2),
6 (2 × 3), 12 (3 × 4), 20 (4 × 5), ... The pronic numbers
are twice the triangular numbers,
and represent the lengths that produce the musical intervals: octave (1:2),
fifth (2:3), fourth (3:4), major third (4:5) .... Pronic seems to be a misspelling
of promic, from the Greek promekes, for rectangular or
oblong; however, the "n" form goes back at least far as Leonhard
Euler who used it in series one, volume fifteen of his Opera. ## pseudoprimeA number that passes the test of Fermat's little theorem (FLT) for prime numbers but actually isn't a prime. FLT says that ifp is prime and a is coprime to p, then a^{p-1} - 1 is divisible by p. If a number x is not prime, a is coprime to x, and x divides a^{x-1} - 1, then x is called a pseudoprime to base a. A number x that is a pseudoprime for all values of a that
are coprime to x is called a Carmichael
number. The smallest pseudoprime for in base 2 is 341. It isn't prime
because 341 = 11 × 31; however, it satisfies FLT: 2^{340} -
1 is divisible by 341. ## pyramidal numberThe number of dots that may be arranged in a pyramid with a regular polygon as base.## Pythagorean tripletAlso called aPythagorean triple, a set of three whole
numbers that satisfies Pythagoras's
theorem, i.e. the squares of two of the numbers add up to the square
of the third number. Examples include (3, 4, 5), (5, 12, 13), and (7, 24,
25). These are called primitive triplets because they have
no common divisors. If the members of a primitive triplet are multiplied
by the same integer the result is a new (but not primitive) triplet. In any primitive Pythagorean triplet, one, and only one, of the three numbers must be even (but can't equal 2); the other two numbers are relatively prime. There are infinitely many such triplets, and they are easy to generate using a classic formula, known since ancient times. If the numbers in the triplet are a, b, and c, then: a = n^{2} - m^{2}, b = 2mn, c = m^{2} + n^{2}, where m and n are two integers and m is less than n. Because
the square root of two is irrational,
there can't be any Pythagorean triplets (a, a, c).
However, there are an infinite number of triplets (a, a + 1, c), the first three of which (apart from the trivial (0, 1,
1)) are (3, 4, 5), (20, 21, 29), and (119, 120, 169). There are also an
infinite number of Pythagorean quartets (a, b, c, d) such that a^{2} + b^{2} + c^{2} = d^{2}. This is simply the three-dimensional
form of Pythagoras's theorem and can be interpreted as the fact that the
point in three-dimensions with Cartesian coordinates (a, b, c) lies an integer distance d from the origin. A formula that generates Pythagorean quartets is: a = m^{2}, b = 2mn, c = 2n^{2}; d = (m^{2} + 2n^{2}) = a + c.
Also note that b^{2} = 2ac. When m =1
and n = 1, we get the quarter (1, 2, 2, 3) – the simplest
example. Although there are an infinite number of Pythagorean triplets, Fermat's last theorem, which
is now know to be true, ensures that there are no triplets for higher powers. See also Euler's conjecture and multigrade. ## quaternion
a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary
numbers defined by the equations i^{ 2} = j^{ 2} = k ^{2} = -1 and -ji = k. Quaternions
are similar to complex numbers, but
whereas complex numbers can be represented by points of a two-dimensional plane, quaternions can be viewed as points
in four-dimensional space (see fourth
dimension). For a while, quaternions were very influential: they were taught in many mathematics departments in the United States in the late 1800s, and were a mandatory topic of study at Dublin, where Hamilton ran the observatory. William Clifford developed the theory of them further. But then they were driven out by the vector notation of William Gibbs and Oliver Heaviside. Had quaternions come along later, when theoretical physicists were trying to understand patterns among subatomic particles, they may have found a place in modern science; after all, the unit quaternions form the group SU(2), which is perfect for studying spin-½ particles. But the way things turned out, quaternions had fallen from favor by the 20th century and Wolfgang Pauli used 2 × 2 complex matrices instead to describe the generators of SU(2). ## rational numberA number that can be written as an ordinary fraction – a ratio,a/b,
of two integers, a and b, where b isn't zero –
or as a decimal expansion that either stops (like 4.58) or is periodic (like
1.315315...). Other examples include 1, 1.2, 385.66, and 1/3. Rational numbers are countable, which means that, although there are infinitely many of them, they can always be put in a definite order, from smallest to largest, and can thus be counted. They also form what's called a densely
ordered set; in other words, between any two rationals there always
sits another one – in fact infinitely many other ones. The rational numbers are a subset of the real numbers; real numbers that aren't rational are called, rationally enough, irrational numbers. Although rationals are dense on the real number line, in the sense that any open set contains a rational, they're pretty sparse in comparison with the irrationals. One way to think of this is that the infinity of rationals (which, strangely enough, is exactly the same size as the infinity of whole numbers) is smaller than the infinity of irrational numbers. Another way to grasp the scarcity versus density issue, is to realize that the rationals can be covered by a set whose "length" is arbitrarily small. In other words, given a string of any positive length, no matter how short, it will still be long enough to cover all the rationals. In mathematical parlance, the rationals are a measure zero set. The irrationals, by contrast,
are a measure one set. This difference in measure means that the rationals
and irrationals are quite different even though a rational can always be
found between any two irrationals, and an irrational exists between any
two rationals. For more, see the separate article on rational numbers. ## real numberAny number that can be represented as a, possibly infinitely long and non-repeating, decimal. Real numbers stand in one-to-one correspondence with the points on a continuous line, known as thereal number line, that
stretches from zero to infinity in both directions. The set of real numbers
contains the set of all rational numbers and the set of all irrational numbers.
The name "real number" is a retronym, coined by René Descartes in response to the concept of imaginary
numbers. Number systems that are even more general that the real numbers
include the complex numbers and, of much more recent discovery, hyperreal
numbers and surreal numbers. ## rectangular numberA number that can be represented in the shape of a rectangle, for example, 6 (2 × 3) and 15 (3 × 5). More generally, a rectangular number can be written in the formm × n, where m and n are positive integers greater than 1. If m = n, then the number is also a square number. ## relatively primeTwo integers are relatively prime if they share no common positive factors (divisors) except 1. For example, 12 and 13 are relatively prime, but 12 and 14 are not. See also coprime.## repdigitA number composed of repetition of a single digit in a given base, generally taken as base 10 unless otherwise specified. For example, the Beast Number 666 is a (base-10) repdigit.## rep-unitA number whose digits are all units; the rep-unit (repeated unit) withn digits is denoted R_{n}. For example, R_{1}=1, R_{2}=11, R_{3}=111, and R_{n}=(10 - 1)/9. ^{n}R_{n} divides R_{m} whenever n divides m. No repunit can be a square number,
but it is not known if one can be a cube. Repunit primes are repunits that are prime numbers.
The only known repunit primes are R_{2} (11), R_{23}, R_{19}, R_{317}, and R_{1031},
though R_{49081} and R_{86453} are suspected
primes. ## Roman numeralsA number system in which each symbol represents a fixed value regardless of its position; this differs from the place-value system of Arabic numerals. The earliest form of the Roman system was, however, decimal. In this primitive version a series of I's represented any number from 1 to 9, and a new symbol was introduced for each higher power of 10: X for 10, C for 100, and M for 1,000. The symbols V, L, and D, which stand for 5, 50, and 500, are thought to have been introduced by the Etruscans. For example, VIII is 8, XL is 40, MCD is 1400, and MCDXLVIII is 1448.A common remark is that multiplication and division using Roman numerals is so awkward as to be totally impractical. However, an article by James G. Kennedy in The American Mathematical Monthly in 1980 gives algorithms
for these operations that are actually more straightforward in the Roman
system than in the Arabic. In multiplication the first step is to rewrite
the numbers in a simple place-value notation. Seven columns are set up,
headed by the symbols M, D, C, L, X, V, and I, and tallies are marked in
each column corresponding to the number of times that symbol appears in
the multicand. For example, if the multiplicand is XIII (13), one tally
is marked in the X column and three tallies are marked in the I column.
The multiplier is written in the same way. The multiplication itself is
done by forming partial products according to two simple rules. In most
cases the partial product given by any one tally in the multiplier is simply
the set of tallies that represents the multiplicand, shifted to the left
an appropriate number of columns. If the multiplier digit is I, the multiplicand
is not shifted at all; the multiplicand is shifted one place to the left
for V, two places for X, three places for L, and so on. The second rule
is applied only when one Etruscan character is multiplied by another. In
such cases the tallies representing the multiplicand digit are written twice
in the appropriately shifted column and an additional tally is written one
column to the right. Once a partial product has been formed for every tally
in the multiplier, the tallies in each column are accumulated and replaced
by the Roman symbol at the head of the column, giving the final answer.
Only a slight change in the method is needed for Roman numerals in "subtractive
notation," where 10 is written as IX, and so on. If all this sounds not
quite so simple, the method for multiplying Arabic numbers is just as involved
if they are written in explicit form. Furthermore, Arabic operations require
a multiplication table giving the 100 products of all the possible pairs
of Arabic digits. No comparable table is needed with Roman numerals, where
all arithmetical operations can be defined in terms of shifting rules, addition,
and subtraction. ## sexagesimalOf, relating to, or based upon the number sixty. Sexagesimal refers especially to the number system with base 60. The Babylonians began using such a scheme around the beginning of the second millennium B.C. in what was the first example of a place-value system. Our degree of 60 minutes, minute of 60 seconds (in both time and angle measure), and hour of 60 minutes hark back to this ancient method of numeration. Quite why the Babylonians counted using sexagesimal isn't known, but 60 certainly has more factors than any other number of comparable size.## Sierpinski numberA positive, odd integerk such that k times 2^{n} + 1 is never a prime number for any
value of n. In 1960 Waclaw Sierpinski showed that there were infinitely many such numbers (though he didn't give
a specific example.) This a strange result. Why should it be that while
the vast majority of expressions of the form m times 2^{n} + 1 eventually produce a prime, some don't? For now, mathematicians are
focused on a more manageable problem posed by Sierpinski: What is the smallest
Sierpinski number? In 1962, John Selfridge discovered what is still the
smallest known Sierpinski number, k = 78557. The next largest is
271129. Is there a smaller Sierpinski number? No one yet knows. However,
to establish that 78557 is really the smallest, it would be sufficient to
find a prime of the form k(2^{n} + 1) for every
value of k less than 78557. In early 2001, there were only 17 candidate
values of k left to check: 4847, 5359, 10223, 19249, 21181, 22699,
24737, 27653, 28433, 33661, 44131, 46157, 54767, 55459, 65567, 67607, and
69109. In March 2002, Louis Helm of the University of Michigan and David
Norris of the University of Illinois started a project called "Seventeen
or Bust," the goal of which is to harness the computing power of a worldwide
network of hundreds of personal computers to check for primes among the
remaining candidates. The team's effort have so far eliminated five candidates
– 46157, 65567, 44131, 69109, and 54767. Despite this encouraging
start, it may take as long as a decade, with many additional participants,
to check the dozen remaining candidates. ## Smith numberA composite number, the sum of whose digits equals the sum of the digits of its prime factors. The name stems from a phone call in 1984 by the mathematician Albert Wilansky to his brother-in-law, called Smith, during which Wilansky noticed that the phone number, 4937775, obeyed the condition just mentioned. Specifically:4937775 = 3 × 5 × 5 × 65837Trivially, all prime numbers have this property, so they are excluded. The Smith numbers less than 1000 are: 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483, 517, 526, 535, 562, 576, 588, 627, 634, 636, 645, 648, 654, 663, 666, 690, 706, 728, 729, 762, 778, 825, 852, 861, 895, 913, 915, 922, 958, and 985.In 1987, Wayne McDaniel proved that there are infinitely many Smiths. ## Sophie Germain primeAny prime numberp such that
2p + 1 is also prime; the smallest examples are 2, 3, 5, 11, 23,
29, 41, 53, 83, 89, 113, and 131. Around 1825 Sophie Germain proved that the first case of Fermat's
last theorem (FLT) is true for such primes. Soon after, Adrien-Marie
Legendre began to generalize this by showing the first case of FLT also
holds for odd primes p such that kp + 1 is prime, k = 4, 8, 10, 14, and 16. In 1991 Fee and Granville extended this to k < 100, where k not a multiple of three. Many similar results were
also shown, but now that FLT has been proven correct, they are of less interest. ## square freeAn integer that is not divisible by a perfect square,n^{2}, for n > 1. ## square numberA number that is the product of two equal integers, e.g. 49 is a square number because 49 = 7 × 7. Square numbers are a type of polygonal number.## standard formScientific method of writing numbers, especially very large or very small numbers. The first non-zero digit of the number is placed before the decimal point and all other digits are placed in order after the decimal point. This decimal is multiplied by an appropriate power of 10. For example, 1,721,448 is written as 1.721 448 × 10^{6} in standard form and 0.003378159
is written as 3.378 159 × 10^{-3}. ## strobogrammatic primeA prime number that remains unchanged when rotated through 180°. An example is 619, which looks the same when read upside-down. To be strobogrammatic, a prime cannot contain digits other than 0, 1, and 8, which have a horizontal line of symmetry (ignoring font variations), and 6 and 9, which are vertical reflections of each other. Aninvertible prime is one that yields a different prime
when the digits are inverted. Of course, these definitions are not taken
seriously by mathematicians! ## sublime numberA number such that both the sum of its divisors and the number of its divisors are perfect numbers. The smallest sublime number is 12. There are 6 divisors of 12 – 1, 2, 3, 4, 6, and 12 – the sum of which is 28. Both 6 and 28 are perfect. The second sublime number begins 60865..., ends ...91264, and has a total of 76 digits! It is not known if there are larger even sublime numbers, nor if there are any odd sublime numbers.## superfactorialThe product of the firstn factorials (definition by Neil Sloane and Simon Plouffe in 1995). For example: The sequence of superfactorials starts (from n = 0) as
1, 1, 2, 12, 288, 34560, 24883200, ...This idea was extended in 2000 by Henry Bottomley to the superduperfactorial as the product of the first n superfactorials, starting (from n = 0) as
1, 1, 2, 24, 6912, 238878720, 5944066965504000, ... ## surdA now little-used term meaning the square root of a whole number that has an irrational value. Some examples are √2 (see square root of 2), √3, and √10.A surd can often be simplified using the fact that √( ab)
= √a × √b. For example, √12 = √(4
× 3) = √4 × √3 = 2√3. Surds may be used to express some results in exact form, such as sin 60° = √3/2. ## surreal numberA member of a mind-bogglingly vast class of numbers that includes all of the real numbers, all of Georg Cantor's infinite ordinal numbers (different kinds of infinity), a set of infinitesimals (infinitely small numbers) produced from these ordinals, and strange numbers that previously lived outside the known realm of mathematics. Each real number, it turns out, is surrounded by a "cloud" of surreals that lie closer to it than do any other real number. One of these surreal clouds occupies the curious space between zero and the smallest real number greater than zero and is made up of the infinitesimals.Surreal numbers were invented or discovered (depending on your philosophy) by John Conway to help with his analysis of certain kinds of games. The idea came to him after watching the British Go champion playing in the mathematics department at Cambridge. Conway noticed that endgames in Go tend to break up into a sum of games, and that some positions behaved like numbers. He then found that, in the case of infinite games, some positions behaved like a new kind of number – the surreals. The name "surreal" was introduced by Donald Knuth in his 1974 book Surreal Numbers: How Two Ex-Students Turned on
to Pure Mathematics and Found Total Happiness.^{1} This novelette
is notable as being the only instance where a major mathematical idea has
been first presented in a work of fiction. Conway went on to describe the
surreal numbers and their use in analyzing games in his 1976 book On
Numbers and Games.^{2} The surreals are similar to the hyperreal
numbers, but they are constructed in a very different way and the class
of surreals is larger and contains the hyperreals as a subset. - Knuth, Donald.
*Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness*. Reading, Mass.: Addison-Wesley, 1974. - Conway, John Horton.
*On Numbers and Games*. New York: Academic Press, 1976.
## tetrahedral numberA number that can be made by considering a tetrahedral pattern of beads in three dimensions. For example, if a triangle of beads is made with three beads to a side, and on top of this is placed a triangle with two beads to a side, and on top of that a triangle with one bead to a side, the result is a tetrahedron of beads. In this case the total number of beads is (3rd triangular number) + (2nd triangular number) + (1st triangular number) = 6 + 3 + 1 = 10. In general thenth
tetrahedral number is equal to the sum of the first n triangular
numbers. This is the same as the 4th number from the left in the (n + 3)th row of Pascal's triangle.
We can use the binomial formula for numbers in Pascal's triangle to show
that the nth tetrahedral number is ^{n+2}C_{3},
or (n + 2)(n + 1)n/6. The only numbers that are
both tetrahedral and square are 4 (= 2^{2} = T_{2})
and 19600 (= 140^{2} = T_{48}). ## transcendental numberA number that can't be expressed as the root of a polynomial equation with integer coefficients. Transcendental numbers are one of the two types of irrational number, the other being algebraic numbers. Their existence was proved in 1844 by the French mathematician Joseph Liouville (1809–1882). Georg Cantor showed (1874) that there are more transcendental than algebraic numbers; the set of all algebraic numbers is enumerable (like the integers); the set of transcendental numbers is not.Although transcendentals make up the vast majority of real numbers, it is often surprisingly hard, and may even be impossible, to tell whether a certain number is transcendental or algebraic. For example it is known that both π and e are transcendental and also that at least one of π + e and π
× e must be transcendental, but it is not known which. It is also
known that e^{π} is transcendental. This follows from
the Gelfond-Schneider theorem, which says that if a and b are algebraic, a is not 0 or 1, and b is not
rational, then a is transcendental. Using Euler's formula, ^{b}e^{iπ}= -1, and taking both sides to the power
-i gives (-1)^{-i} = (e^{iπ})^{-i} = e^{π}. Since the theorem tells us that the left hand
side is transcendental, it follows that the right hand side is too. (It
also follows that e × π and e + π are not both algebraic,
because if they were then the equation x^{2} + x(e + π) + eπ = 0 would have roots e and π, making both numbers
algebraic.) But although it is known that e^{π} is transcendental,
the status of e, π^{e}^{e}, and π^{π} remains uncertain. ## transfinite numberAny of the infinite ordinal numbers first described by Georg Cantor.## triangular numberAny number that can be represented by a triangular array of dots. 1, 3, 6, 10, ... are triangular numbers. The nth triangular number isn(n + 1)/2. Every integer is the sum of at most three triangular numbers. Every
even triangular number is a perfect number.
If T is a triangular number, 8T + 1 is a perfect square
and 9T + 1 is another triangular number. The square of the nth triangular number is equal to the sum of the first n cubes. Certain triangular numbers are also
squares, but no triangular number can be a third, fourth or fifth power,
nor can one end in 2, 4, 7, or 9. ## truncatable primeA prime numbern that remains
a prime when digits are deleted from it one at a time. For example 410256793
is a truncatable prime because each number created by the removal of the
digit underlined produces a new prime: 410256793, 41256793, 4125673, 415673,
45673, 4567, 467, 67, 7. It is conjectured that there are infinitely many
of these primes. If the digits from a prime can be deleted only from the
right to leave a prime, then n is called a right truncatable
prime. If they can be deleted only from the left to leave a prime,
then n is a left truncatable prime. The list of
primes from which any digit can be deleted at each step to leave a prime
is very short indeed, because it demands that each digit be a prime and
also that no digit occurs twice. Only these numbers satisfy this requirement:
2, 3, 5, 7, 23, 37, 53 and 73. ## undulating numberAn integer whose digits, in a given base, alternate – that is, one written in the formababab..., where a and b are digits. For example, 434343 and 101010101
are undulating numbers. ## unique numberThe constantU that results if a number _{n}A consisting of _{n}n consecutive digits, in ascending order, is subtracted
from the number A' obtained by reversing the
digits of _{n}A. For example, a three-digit number
345, if subtracted from its reverse 543, yields a difference of 198. Any
other three-digit number subtracted from its reverse gives the same difference.
Thus _{n}U = 198. Similarly for a number with
four consecutive digits, the unique number _{3}U = 3087. The first ten unique numbers are: _{4}U = 0, _{1}U = 9, _{2}U = 198, _{3}U = 3087, _{4}U = 41976, _{5}U = 530865, _{6}U = 6419754, _{7}U = 75308643, _{8}U = 864197532, and _{9}U = 9753086421. Unique numbers
are related to Kaprekar numbers, _{10}K, by the formula
_{n}For example, when n = 4, K = 6174, _{4}K' = 4716, _{4}U = 3087, _{4}U' = 7803, and
_{4}3087 + 7803 = 10890 = 6174 + 4716. - Gupta, S. S. "Unique Numbers",
*Science Today*, January 1988, India.
## unit fractionA fraction whose numerator (number on top) is 1.## untouchable numberA number that is not the sum of the aliquot parts of any other number. The first few untouchable numbers are 2, 5, 52, and 88.## vampire numberA natural numberx that
can be factorized as y × z in such a way that the
number of occurrences of a particular digit in the representation of x in a given base (say 10) appears the
same number of times in the representations in that same base of y and z together. For example, 2187 is a vampire number since 2187
= 21 × 87 ; similarly 136948 is a vampire because 136948 = 146 ×
938. Vampire numbers are a whimsical idea that was introduced by Clifford
Pickover in 1995. ^{1}- Pickover, Clifford.
*Keys to Infinity*. New York: W. H. Freeman, 1995.
## vigesimalOf, relating to, or based on the number 20; the term comes from the Latinvigesimus for "twentieth." Mayan arithmetic, which took account
of all the toes as well as the fingers, used a vigesimal system. In place
of the multiples of 10 used in the decimal system, 1, 10, 100, 1000, 10000,
..., the Mayans dealt in multiples of 20: 1, 20, 400, 8000, 160000, ... ## winding numberThe number of times a closed curve in the plane passes around a given point in the counterclockwise direction.## Related categories | |||||||||||||||||||||||||||||||||||||||

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