In
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
, the binomial theorem (or binomial expansion) describes the algebraic expansion of
powers of a
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
* Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
...
. According to the theorem, it is possible to expand the polynomial into a
sum involving terms of the form , where the exponents and are
nonnegative integers with , and the
coefficient of each term is a specific
positive integer depending on and . For example, for ,
The coefficient in the term of is known as the
binomial coefficient or
(the two have the same value). These coefficients for varying and can be arranged to form
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
. These numbers also occur in
combinatorics, where
gives the number of different
combinations
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
of
elements that can be chosen from an -element
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
. Therefore
is often pronounced as " choose ".
History
Special cases of the binomial theorem were known since at least the 4th century BC when
Greek mathematician Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
mentioned the special case of the binomial theorem for exponent .
There is evidence that the binomial theorem for cubes was known by the 6th century AD in India.
Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting objects out of without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the ''Chandaḥśāstra'' by the Indian lyricist
Pingala
Acharya Pingala ('; c. 3rd2nd century BCE) was an ancient Indian poet and mathematician, and the author of the ' (also called the ''Pingala-sutras''), the earliest known treatise on Sanskrit prosody.
The ' is a work of eight chapters in the la ...
(c. 200 BC), which contains a method for its solution.
The commentator
Halayudha
Halayudha (Sanskrit: हलायुध) was a 10th-century Indian mathematician who wrote the ',Maurice Winternitz, ''History of Indian Literature'', Vol. III a commentary on Pingala's ''Chandaḥśāstra''. The latter contains a clear descri ...
from the 10th century AD explains this method using what is now known as
Pascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...
.
By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient
,
and a clear statement of this rule can be found in the 12th century text ''Lilavati'' by
Bhaskara.
The first formulation of the binomial theorem and the table of binomial coefficients, to our knowledge, can be found in a work by
Al-Karaji, quoted by
Al-Samaw'al
Al-Samawʾal ibn Yaḥyā al-Maghribī ( ar, السموأل بن يحيى المغربي, ; c. 1130 – c. 1180), commonly known as Samau'al al-Maghribi, was a mathematician, Islamic astronomy, astronomer and Islamic medicine, physician. Born to ...
in his "al-Bahir".
Al-Karaji described the triangular pattern of the binomial coefficients
and also provided a
mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
of both the binomial theorem and Pascal's triangle, using an early form of
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 ...
.
The Persian poet and mathematician
Omar Khayyam
Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...
was probably familiar with the formula to higher orders, although many of his mathematical works are lost.
The binomial expansions of small degrees were known in the 13th century mathematical works of
Yang Hui
Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theo ...
and also
Chu Shih-Chieh
Zhu Shijie (, 1249–1314), courtesy name Hanqing (), pseudonym Songting (), was a Chinese mathematician and writer. He was a Chinese mathematician during the Yuan Dynasty. Zhu was born close to today's Beijing. Two of his mathematical works ha ...
.
Yang Hui attributes the method to a much earlier 11th century text of
Jia Xian
Jia Xian (; ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty.
Biography
According to the history of the Song dynasty, Jia was a palace eunuch of the Left Duty Group. He studied under the mathematician Chu Yan, and ...
, although those writings are now also lost.
In 1544,
Michael Stifel
Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena Universi ...
introduced the term "binomial coefficient" and showed how to use them to express
in terms of
, via "Pascal's triangle".
Blaise Pascal studied the eponymous triangle comprehensively in his ''Traité du triangle arithmétique''. However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel,
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republi ...
, and
Simon Stevin
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
.
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
is generally credited with the generalized binomial theorem, valid for any rational exponent.
Statement
According to the theorem, it is possible to expand any nonnegative integer power of into a sum of the form
where
is an integer and each
is a positive integer known as a
binomial coefficient. (When an exponent is zero, the corresponding power expression is taken to be 1 and this multiplicative factor is often omitted from the term. Hence one often sees the right hand side written as
.) This formula is also referred to as the binomial formula or the binomial identity. Using
summation notation, it can be written as
The final expression follows from the previous one by the symmetry of and in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical. A simple variant of the binomial formula is obtained by
substituting for , so that it involves only a single
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
. In this form, the formula reads
or equivalently
or more explicitly
Examples
Here are the first few cases of the binomial theorem:
In general, for the expansion of on the right side in the th row (numbered so that the top row is the 0th row):
* the exponents of in the terms are (the last term implicitly contains );
* the exponents of in the terms are (the first term implicitly contains );
* the coefficients form the th row of Pascal's triangle;
* before combining like terms, there are terms in the expansion (not shown);
* after combining like terms, there are terms, and their coefficients sum to .
An example illustrating the last two points:
with
.
A simple example with a specific positive value of :
A simple example with a specific negative value of :
Geometric explanation
For positive values of and , the binomial theorem with is the geometrically evident fact that a square of side can be cut into a square of side , a square of side , and two rectangles with sides and . With , the theorem states that a cube of side can be cut into a cube of side , a cube of side , three rectangular boxes, and three rectangular boxes.
In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, this picture also gives a geometric proof of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
if one sets
and
interpreting as an
infinitesimal change in , then this picture shows the infinitesimal change in the volume of an -dimensional
hypercube,
where the coefficient of the linear term (in
) is
the area of the faces, each of dimension :
Substituting this into the
definition of the derivative via a
difference quotient
In single-variable calculus, the difference quotient is usually the name for the expression
: \frac
which when taken to the limit as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact ...
and taking limits means that the higher order terms,
and higher, become negligible, and yields the formula
interpreted as
:"the infinitesimal rate of change in volume of an -cube as side length varies is the area of of its -dimensional faces".
If one integrates this picture, which corresponds to applying the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, one obtains
Cavalieri's quadrature formula, the integral
– see
proof of Cavalieri's quadrature formula for details.
Binomial coefficients
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written
and pronounced " choose ".
Formulas
The coefficient of is given by the formula
which is defined in terms of the
factorial function . Equivalently, this formula can be written
with factors in both the numerator and denominator of the
fraction. Although this formula involves a fraction, the binomial coefficient
is actually an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
.
Combinatorial interpretation
The binomial coefficient
can be interpreted as the number of ways to choose elements from an -element set. This is related to binomials for the following reason: if we write as a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
then, according to the
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
, there will be one term in the expansion for each choice of either or from each of the binomials of the product. For example, there will only be one term , corresponding to choosing from each binomial. However, there will be several terms of the form , one for each way of choosing exactly two binomials to contribute a . Therefore, after
combining like terms, the coefficient of will be equal to the number of ways to choose exactly elements from an -element set.
Proofs
Combinatorial proof
Example
The coefficient of in
equals
because there are three strings of length 3 with exactly two s, namely,
corresponding to the three 2-element subsets of , namely,
where each subset specifies the positions of the in a corresponding string.
General case
Expanding yields the sum of the products of the form where each is or . Rearranging factors shows that each product equals for some between and . For a given , the following are proved equal in succession:
* the number of copies of in the expansion
* the number of -character strings having in exactly positions
* the number of -element subsets of
*
either by definition, or by a short combinatorial argument if one is defining
as
This proves the binomial theorem.
Inductive proof
Induction
Induction, Inducible or Inductive may refer to:
Biology and medicine
* Labor induction (birth/pregnancy)
* Induction chemotherapy, in medicine
* Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...
yields another proof of the binomial theorem. When , both sides equal , since and
Now suppose that the equality holds for a given ; we will prove it for . For , let denote the coefficient of in the polynomial . By the inductive hypothesis, is a polynomial in and such that is
if , and otherwise. The identity
shows that is also a polynomial in and , and
since if , then and . Now, the right hand side is
by
Pascal's identity
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers ''n'' and ''k'',
+ = ,
where \tbinom is a binomial coefficient; one interpretation of ...
. On the other hand, if , then and , so we get . Thus
which is the inductive hypothesis with substituted for and so completes the inductive step.
Generalizations
Newton's generalized binomial theorem
Around 1665,
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
exponents.) In this generalization, the finite sum is replaced by an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
. In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number , one can define
where
is the
Pochhammer symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
, here standing for a
falling factorial. This agrees with the usual definitions when is a nonnegative integer. Then, if and are real numbers with ,
[This is to guarantee convergence. Depending on , the series may also converge sometimes when .] and is any complex number, one has
When is a nonnegative integer, the binomial coefficients for are zero, so this equation reduces to the usual binomial theorem, and there are at most nonzero terms. For other values of , the series typically has infinitely many nonzero terms.
For example, gives the following series for the square root:
Taking , the generalized binomial series gives the
geometric series formula, valid for :
More generally, with , we have for :
So, for instance, when ,
Replacing with yields:
So, for instance, when , we have for :
Further generalizations
The generalized binomial theorem can be extended to the case where and are complex numbers. For this version, one should again assume
and define the powers of and using a
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
branch of log defined on an open disk of radius centered at . The generalized binomial theorem is valid also for elements and of a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
as long as , and is invertible, and .
A version of the binomial theorem is valid for the following
Pochhammer symbol
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
:\begin
(x)_n = x^\underline &= \overbrace^ \\
&= \prod_^n(x-k+1) = \prod_^(x-k) \,.
\e ...
-like family of polynomials: for a given real constant , define
and