A triangular number or triangle number counts objects arranged in an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...
. Triangular numbers are a type of
figurate number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean
* polygon ...
, other examples being
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s and
cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the
natural number
In mathematics, the natural numbers are those numbers 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").
Numbers used for counting are called ''Cardinal n ...
s from 1 to . The
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of triangular numbers, starting with the
0th triangular number, is
(This sequence is included in the On-Line Encyclopedia of Integer Sequences .)
Formula
The triangular numbers are given by the following explicit formulas:
where
is a
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two".
The first equation can be illustrated using a
visual proof
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered mo ...
. For every triangular number
, imagine a "half-square" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions
, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or:
. The example
follows:
This formula can be proven formally using
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
. It is clearly true for
:
Now
assume that, for some natural number
,
. Adding
to this yields
so if the formula is true for
, it is true for
. Since it is clearly true for
, it is therefore true for
,
, and ultimately all natural numbers
by induction.
The German mathematician and scientist,
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, is said to have found this relationship in his early youth, by multiplying pairs of numbers in the sum by the values of each pair .
However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the
Pythagoreans
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...
in the 5th century BC. The two formulas were described by the Irish monk
Dicuil
Dicuilus (or the more vernacular version of the name Dícuil) was an Irish monk and geographer, born during the second half of the 8th century.
Background
The exact dates of Dicuil's birth and death are unknown. Of his life nothing is known exce ...
in about 816 in his
Computus
As a moveable feast, the date of Easter is determined in each year through a calculation known as (). Easter is celebrated on the first Sunday after the Paschal full moon, which is the first full moon on or after 21 March (a fixed approxi ...
. An English translation of Dicuil's account is available.
The triangular number solves the handshake problem of counting the number of handshakes if each person in a room with people shakes hands once with each person. In other words, the solution to the handshake problem of people is . The function is the additive analog of the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
function, which is the ''products'' of integers from 1 to .
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
In the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
, the ratio between the two numbers, dots and line segments is
Relations to other figurate numbers
Triangular numbers have a wide variety of relations to other figurate numbers.
Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically,
This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square:
The double of a triangular number, as in the visual proof from the above section , is called a
pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...
.
There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula:
with
''All''
square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are infinitely many square triangular numbers; the first few are:
:0, 1, 36, , , , , , ,
Expl ...
s are found from the recursion
with
and
Also, the
square of the th triangular number is the same as the sum of the cubes of the integers 1 to . This can also be expressed as
The sum of the first triangular numbers is the th
tetrahedral number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is,
...
:
More generally, the difference between the th
-gonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
Definition and examples
...
and the th -gonal number is the th triangular number. For example, the sixth
heptagonal number A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula
:H_n=\frac.
The first few heptagonal numbers are:
: 0, 1, 7, 18, 34, 55, 81, 112 ...
(81) minus the sixth
hexagonal number
A hexagonal number is a figurate number. The ''n''th hexagonal number ''h'n'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular hexagons with sides up to n dots, when the hexagons are overlaid so ...
(66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any
centered polygonal number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side i ...
; the th centered -gonal number is obtained by the formula
where is a triangular number.
The positive difference of two triangular numbers is a
trapezoidal number
In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite... The impolite numbers are exactly the powers of two, an ...
.
The pattern found for triangular numbers
and for tetrahedral numbers
which uses
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s, can be generalized. This leads to the formula:
Other properties
Triangular numbers correspond to the first-degree case of
Faulhaber's formula
In mathematics, Faulhaber's formula, named after the early 17th century mathematician Johann Faulhaber, expresses the sum of the ''p''-th powers of the first ''n'' positive integers
:\sum_^n k^p = 1^p + 2^p + 3^p + \cdots + n^p
as a (''p''&nb ...
.
Alternating triangular numbers (1, 6, 15, 28, ...) are also hexagonal numbers.
Every even
perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
is triangular (as well as hexagonal), given by the formula
where is a
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
. No odd perfect numbers are known; hence, all known perfect numbers are triangular.
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.
The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
In
base 10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
, the
digital root
The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit su ...
of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
There is a more specific property to the triangular numbers that aren't divisible by 3; that is, they either have a remainder 1 or 10 when divided by 27. Those that are equal to 10
mod
Mod, MOD or mods may refer to:
Places
* Modesto City–County Airport, Stanislaus County, California, US
Arts, entertainment, and media Music
* Mods (band), a Norwegian rock band
* M.O.D. (Method of Destruction), a band from New York City, US ...
27 are also equal to 10 mod 81.
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
If is a triangular number, then is also a triangular number, given is an odd square and . Note that
will always be a triangular number, because , which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for given is an odd square is the inverse of this operation.
The first several pairs of this form (not counting ) are: , , , , , , ... etc. Given is equal to , these formulas yield , , , , and so on.
The sum of the
reciprocals of all the nonzero triangular numbers is
This can be shown by using the basic sum of a
telescoping series
In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n).
As a consequence the partial sums only consists of two terms of (a_n) after ca ...
:
Two other formulas regarding triangular numbers are
and
both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.
In 1796, Gauss discovered that every positive integer is representable as a sum of three triangular numbers (possibly including = 0), writing in his diary his famous words, "
ΕΥΡΗΚΑ! ". This theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10 + 0), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the
Fermat polygonal number theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most -gonal numbers. That is, every positive integer can be written as the sum of three or fewer triangular numbers, and as the sum ...
.
The largest triangular number of the form is
4095 (see
Ramanujan–Nagell equation In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be ...
).
Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in
geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For ex ...
. It was conjectured by Polish mathematician
Kazimierz Szymiczek
Kazimierz (; la, Casimiria; yi, קוזמיר, Kuzimyr) is a historical district of Kraków and Kraków Old Town, Poland. From its inception in the 14th century to the early 19th century, Kazimierz was an independent city, a royal city of the C ...
to be impossible and was later proven by Fang and Chen in 2007.
Formulas involving expressing an integer as the sum of triangular numbers are connected to
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s, in particular the
Ramanujan theta function
In mathematics, particularly -analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant fo ...
.
Applications
A
fully connected network
Network topology is the arrangement of the elements ( links, nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and contr ...
of computing devices requires the presence of cables or other connections; this is equivalent to the handshake problem mentioned above.
In a tournament format that uses a round-robin
group stage
A tournament is a competition involving at least three competitors, all participating in a sport or game. More specifically, the term may be used in either of two overlapping senses:
# One or more competitions held at a single venue and concent ...
, the number of matches that need to be played between teams is equal to the triangular number . For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.
One way of calculating the
depreciation
In accountancy, depreciation is a term that refers to two aspects of the same concept: first, the actual decrease of fair value of an asset, such as the decrease in value of factory equipment each year as it is used and wear, and second, the a ...
of an asset is the
sum-of-years' digits method, which involves finding , where is the length in years of the asset's useful life. Each year, the item loses , where is the item's beginning value (in units of currency), is its final salvage value, is the total number of years the item is usable, and the current year in the depreciation schedule. Under this method, an item with a usable life of = 4 years would lose of its "losable" value in the first year, in the second, in the third, and in the fourth, accumulating a total depreciation of (the whole) of the losable value.
Triangular roots and tests for triangular numbers
By analogy with the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
of , one can define the (positive) triangular root of as the number such that :
which follows immediately from the
quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, g ...
. So an integer is triangular
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
is a square. Equivalently, if the positive triangular root of is an integer, then is the th triangular number.
Alternative name
An alternative name proposed by
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
, by analogy to
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
s, is "termial", with the notation ? for the th triangular number.
[Donald E. Knuth (1997). ''The Art of Computer Programming: Volume 1: Fundamental Algorithms''. 3rd Ed. Addison Wesley Longman, U.S.A. p. 48.] However, although some other sources use this name and notation,
they are not in wide use.
See also
*
1 + 2 + 3 + 4 + ⋯
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1 ...
*
Doubly triangular number, a triangular number whose position in the sequence of triangular numbers is also a triangular number
*
Tetractys
The tetractys ( el, τετρακτύς), or tetrad, or the tetractys of the decad is a triangular number, triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical repr ...
, an arrangement of ten points in a triangle, important in Pythagoreanism
References
External links
*
Triangular numbersat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
There exist triangular numbers that are also squareat
cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Math ...
*
Hypertetrahedral Polytopic Rootsby Rob Hubbard, including the generalisation to ''triangular cube roots'', some higher dimensions, and some approximate formulas
{{Classes of natural numbers
Figurate numbers
Factorial and binomial topics
Integer sequences
Proof without words
Squares in number theory
Triangles
Simplex numbers