In
elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of
powers
Powers may refer to:
Arts and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
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
Sum most commonly means the total of two or more numbers added together; see addition.
Sum can also refer to:
Mathematics
* Sum (category theory), the generic concept of summation in mathematics
* Sum, the result of summation, the additio ...
involving terms of the form , where the exponents and are
nonnegative integer
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 with , and the
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
of each term is a specific
positive integer
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 ...
depending on and . For example, for ,
The coefficient in the term of is known as the
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 ...
or
(the two have the same value). These coefficients for varying and can be arranged to form
Pascal's triangle. These numbers also occur in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, where
gives the number of different
combinations of
elements
Element or elements may refer to:
Science
* Chemical element, a pure substance of one type of atom
* Heating element, a device that generates heat by electrical resistance
* Orbital elements, parameters required to identify a specific orbit of ...
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
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathem ...
Euclid 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 (c. 200 BC), which contains a method for its solution.
The commentator
Halayudha from the 10th century AD explains this method using what is now known as
Pascal's triangle.
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
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal sur ...
, quoted by
Al-Samaw'al in his "al-Bahir".
Al-Karaji
( fa, ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a 10th-century Persian people, Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal sur ...
described the triangular pattern of the binomial coefficients
and also provided a
mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of
mathematical induction.
The Persian poet and mathematician
Omar Khayyam 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 theor ...
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, 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 Universit ...
introduced the term "binomial coefficient" and showed how to use them to express
in terms of
, via "Pascal's triangle".
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
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, 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 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
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 ...
. (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, this picture also gives a geometric proof of the
derivative if one sets
and
interpreting as an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
change in , then this picture shows the infinitesimal change in the volume of an -dimensional
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
,
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
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. F ...
via a
difference quotient 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, 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
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 . Equivalently, this formula can be written
with factors in both the numerator and denominator of the
fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. Although this formula involves a fraction, the binomial coefficient
is actually an
integer.
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
then, according to the
distributive law, 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
In mathematics, like terms are summands in a sum that differ only by a numerical factor. Like terms can be regrouped by adding their coefficients.
Typically, in a polynomial expression, like terms are those that contain the same variables to the ...
, 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 generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to
complex exponents.) In this generalization, the finite sum is replaced by an
infinite series. 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, here standing for a
falling factorial
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 ...
. 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 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 as long as , and is invertible, and .
A version of the binomial theorem is valid for the following
Pochhammer symbol-like family of polynomials: for a given real constant , define
and