Binomial theorem

Binomial theorem {displaystyle {begin{array}{c}1\1quad 1\1quad 2quad 1\1quad 3quad 3quad 1\1quad 4quad 6quad 4quad 1\1quad 5quad 10quad 10quad 5quad 1\1quad 6quad 15quad 20quad 15quad 6quad 1\1quad 7quad 21quad 35quad 35quad 21quad 7quad 1end{array}}} The binomial coefficient {displaystyle {tbinom {n}{k}}} appears as the kth entry in the nth row of Pascal's triangle (counting starts at 0). Each entry is the sum of the two above it.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example, for n = 4, {displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.} The coefficient a in the term of axbyc is known as the binomial coefficient {displaystyle {tbinom {n}{b}}} or {displaystyle {tbinom {n}{c}}} (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where {displaystyle {tbinom {n}{b}}} gives the number of different combinations of b elements that can be chosen from an n-element set. Therefore {displaystyle {tbinom {n}{b}}} is often pronounced as "n choose b".

Contents 1 History 2 Statement 3 Examples 3.1 Geometric explanation 4 Binomial coefficients 4.1 Formulas 4.2 Combinatorial interpretation 5 Proofs 5.1 Combinatorial proof 5.1.1 Example 5.1.2 General case 5.2 Inductive proof 6 Generalizations 6.1 Newton's generalized binomial theorem 6.2 Further generalizations 6.3 Multinomial theorem 6.4 Multi-binomial theorem 6.5 General Leibniz rule 7 Applications 7.1 Multiple-angle identities 7.2 Series for e 7.3 Probability 8 In abstract algebra 9 In popular culture 10 See also 11 Notes 12 References 13 Further reading 14 External links History Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned the special case of the binomial theorem for exponent 2.[1][2] There is evidence that the binomial theorem for cubes was known by the 6th century AD in India.[1][2] Binomial coefficients, as combinatorial quantities expressing the number of ways of selecting k objects out of n 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.[3]: 230  The commentator Halayudha from the 10th century AD explains this method using what is now known as Pascal's triangle.[3] By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient {textstyle {frac {n!}{(n-k)!k!}}} ,[4] and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara.[4] 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 in his "al-Bahir".[5][6][7] Al-Karaji described the triangular pattern of the binomial coefficients[8] and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction.[8] The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.[2] The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui[9] and also Chu Shih-Chieh.[2] Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.[3]: 142  In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express {displaystyle (1+a)^{n}} in terms of {displaystyle (1+a)^{n-1}} , via "Pascal's triangle".[10] Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.[11] However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.[10] Isaac Newton is generally credited with the generalized binomial theorem, valid for any rational exponent.[10][12] Statement According to the theorem, it is possible to expand any nonnegative integer power of x + y into a sum of the form {displaystyle (x+y)^{n}={n choose 0}x^{n}y^{0}+{n choose 1}x^{n-1}y^{1}+{n choose 2}x^{n-2}y^{2}+cdots +{n choose n-1}x^{1}y^{n-1}+{n choose n}x^{0}y^{n},} where {displaystyle ngeq 0} is an integer and each {displaystyle {tbinom {n}{k}}} 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 {textstyle {binom {n}{0}}x^{n}+cdots } .) This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written as {displaystyle (x+y)^{n}=sum _{k=0}^{n}{n choose k}x^{n-k}y^{k}=sum _{k=0}^{n}{n choose k}x^{k}y^{n-k}.} The final expression follows from the previous one by the symmetry of x and y 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 1 for y, so that it involves only a single variable. In this form, the formula reads {displaystyle (1+x)^{n}={n choose 0}x^{0}+{n choose 1}x^{1}+{n choose 2}x^{2}+cdots +{n choose {n-1}}x^{n-1}+{n choose n}x^{n},} or equivalently {displaystyle (1+x)^{n}=sum _{k=0}^{n}{n choose k}x^{k},} or more explicitly[13] {displaystyle (1+x)^{n}=1+nx+{frac {n(n-1)}{2!}}x^{2}+{frac {n(n-1)(n-2)}{3!}}x^{3}+cdots +nx^{n-1}+x^{n}.} Examples Here are the first few cases of the binomial theorem: {displaystyle {begin{aligned}(x+y)^{0}&=1,\[8pt](x+y)^{1}&=x+y,\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.end{aligned}}} In general, for the expansion of (x + y)n on the right side in the nth row (numbered so that the top row is the 0th row): the exponents of x in the terms are n, n − 1, ..., 2, 1, 0 (the last term implicitly contains x0 = 1); the exponents of y in the terms are 0, 1, 2, ..., n − 1, n (the first term implicitly contains y0 = 1); the coefficients form the nth row of Pascal's triangle; before combining like terms, there are 2n terms xiyj in the expansion (not shown); after combining like terms, there are n + 1 terms, and their coefficients sum to 2n.

An example illustrating the last two points: {displaystyle {begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{text{ terms}})\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{text{ terms}})end{aligned}}} with {displaystyle 1+3+3+1=2^{3}} .

A simple example with a specific positive value of y: {displaystyle {begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\&=x^{3}+6x^{2}+12x+8.end{aligned}}} A simple example with a specific negative value of y: {displaystyle {begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\&=x^{3}-6x^{2}+12x-8.end{aligned}}} Geometric explanation Visualisation of binomial expansion up to the 4th power For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side a + b can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

In calculus, this picture also gives a geometric proof of the derivative {displaystyle (x^{n})'=nx^{n-1}:} [14] if one sets {displaystyle a=x} and {displaystyle b=Delta x,} interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube, {displaystyle (x+Delta x)^{n},} where the coefficient of the linear term (in {displaystyle Delta x} ) is {displaystyle nx^{n-1},} the area of the n faces, each of dimension n − 1: {displaystyle (x+Delta x)^{n}=x^{n}+nx^{n-1}Delta x+{binom {n}{2}}x^{n-2}(Delta x)^{2}+cdots .} Substituting this into the definition of the derivative via a difference quotient and taking limits means that the higher order terms, {displaystyle (Delta x)^{2}} and higher, become negligible, and yields the formula {displaystyle (x^{n})'=nx^{n-1},} interpreted as "the infinitesimal rate of change in volume of an n-cube as side length varies is the area of n of its (n − 1)-dimensional faces".

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral {displaystyle textstyle {int x^{n-1},dx={tfrac {1}{n}}x^{n}}} – see proof of Cavalieri's quadrature formula for details.[14] Binomial coefficients Main article: Binomial coefficient The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written {displaystyle {tbinom {n}{k}},} and pronounced "n choose k".

Formulas The coefficient of xn−kyk is given by the formula {displaystyle {binom {n}{k}}={frac {n!}{k!;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written {displaystyle {binom {n}{k}}={frac {n(n-1)cdots (n-k+1)}{k(k-1)cdots 1}}=prod _{ell =1}^{k}{frac {n-ell +1}{ell }}=prod _{ell =0}^{k-1}{frac {n-ell }{k-ell }}} with k factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient {displaystyle {tbinom {n}{k}}} is actually an integer.

Combinatorial interpretation The binomial coefficient {displaystyle {tbinom {n}{k}}} can be interpreted as the number of ways to choose k elements from an n-element set. This is related to binomials for the following reason: if we write (x + y)n as a product {displaystyle (x+y)(x+y)(x+y)cdots (x+y),} then, according to the distributive law, there will be one term in the expansion for each choice of either x or y from each of the binomials of the product. For example, there will only be one term xn, corresponding to choosing x from each binomial. However, there will be several terms of the form xn−2y2, one for each way of choosing exactly two binomials to contribute a y. Therefore, after combining like terms, the coefficient of xn−2y2 will be equal to the number of ways to choose exactly 2 elements from an n-element set.

Proofs Combinatorial proof Example The coefficient of xy2 in {displaystyle {begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\&=xxx+xxy+xyx+{underline {xyy}}+yxx+{underline {yxy}}+{underline {yyx}}+yyy\&=x^{3}+3x^{2}y+{underline {3xy^{2}}}+y^{3}end{aligned}}} equals {displaystyle {tbinom {3}{2}}=3} because there are three x,y strings of length 3 with exactly two ys, namely, {displaystyle xyy,;yxy,;yyx,} corresponding to the three 2-element subsets of {1, 2, 3}, namely, {displaystyle {2,3},;{1,3},;{1,2},} where each subset specifies the positions of the y in a corresponding string.

General case Expanding (x + y)n yields the sum of the 2n products of the form e1e2 ... en where each ei is x or y. Rearranging factors shows that each product equals xn−kyk for some k between 0 and n. For a given k, the following are proved equal in succession: the number of copies of xn−kyk in the expansion the number of n-character x,y strings having y in exactly k positions the number of k-element subsets of {1, 2, ..., n} {displaystyle {tbinom {n}{k}},} either by definition, or by a short combinatorial argument if one is defining {displaystyle {tbinom {n}{k}}} as {displaystyle {tfrac {n!}{k!(n-k)!}}.} This proves the binomial theorem.

Inductive proof Induction yields another proof of the binomial theorem. When n = 0, both sides equal 1, since x0 = 1 and {displaystyle {tbinom {0}{0}}=1.} Now suppose that the equality holds for a given n; we will prove it for n + 1. For j, k ≥ 0, let [f(x, y)]j,k denote the coefficient of xjyk in the polynomial f(x, y). By the inductive hypothesis, (x + y)n is a polynomial in x and y such that [(x + y)n]j,k is {displaystyle {tbinom {n}{k}}} if j + k = n, and 0 otherwise. The identity {displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}} shows that (x + y)n+1 is also a polynomial in x and y, and {displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},} since if j + k = n + 1, then (j − 1) + k = n and j + (k − 1) = n. Now, the right hand side is {displaystyle {binom {n}{k}}+{binom {n}{k-1}}={binom {n+1}{k}},} by Pascal's identity.[15] On the other hand, if j + k ≠ n + 1, then (j – 1) + k ≠ n and j + (k – 1) ≠ n, so we get 0 + 0 = 0. Thus {displaystyle (x+y)^{n+1}=sum _{k=0}^{n+1}{binom {n+1}{k}}x^{n+1-k}y^{k},} which is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step.

Generalizations Newton's generalized binomial theorem Main article: Binomial series 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 r, one can define {displaystyle {r choose k}={frac {r(r-1)cdots (r-k+1)}{k!}}={frac {(r)_{k}}{k!}},} where {displaystyle (cdot )_{k}} is the Pochhammer symbol, here standing for a falling factorial. This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with |x| > |y|,[Note 1] and r is any complex number, one has {displaystyle {begin{aligned}(x+y)^{r}&=sum _{k=0}^{infty }{r choose k}x^{r-k}y^{k}\&=x^{r}+rx^{r-1}y+{frac {r(r-1)}{2!}}x^{r-2}y^{2}+{frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+cdots .end{aligned}}} When r is a nonnegative integer, the binomial coefficients for k > r are zero, so this equation reduces to the usual binomial theorem, and there are at most r + 1 nonzero terms. For other values of r, the series typically has infinitely many nonzero terms.

For example, r = 1/2 gives the following series for the square root: {displaystyle {sqrt {1+x}}=1+{frac {1}{2}}x-{frac {1}{8}}x^{2}+{frac {1}{16}}x^{3}-{frac {5}{128}}x^{4}+{frac {7}{256}}x^{5}-cdots } Taking r = −1, the generalized binomial series gives the geometric series formula, valid for |x| < 1: {displaystyle (1+x)^{-1}={frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+cdots } More generally, with s = −r: {displaystyle {frac {1}{(1-x)^{s}}}=sum _{k=0}^{infty }{s+k-1 choose k}x^{k}.} So, for instance, when s = 1/2, {displaystyle {frac {1}{sqrt {1+x}}}=1-{frac {1}{2}}x+{frac {3}{8}}x^{2}-{frac {5}{16}}x^{3}+{frac {35}{128}}x^{4}-{frac {63}{256}}x^{5}+cdots } Further generalizations The generalized binomial theorem can be extended to the case where x and y are complex numbers. For this version, one should again assume |x| > |y|[Note 1] and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius |x| centered at x. The generalized binomial theorem is valid also for elements x and y of a Banach algebra as long as xy = yx, and x is invertible, and ||y/x|| < 1. A version of the binomial theorem is valid for the following Pochhammer symbol-like family of polynomials: for a given real constant c, define {displaystyle x^{(0)}=1} and {displaystyle x^{(n)}=prod _{k=1}^{n}[x+(k-1)c]} for {displaystyle n>0.} Then[16] {displaystyle (a+b)^{(n)}=sum _{k=0}^{n}{binom {n}{k}}a^{(n-k)}b^{(k)}.} The case c = 0 recovers the usual binomial theorem.

More generally, a sequence {displaystyle {p_{n}}_{n=0}^{infty }} of polynomials is said to be of binomial type if {displaystyle deg p_{n}=n} for all {displaystyle n} , {displaystyle p_{0}(0)=1} , and {displaystyle p_{n}(x+y)=sum _{k=0}^{n}{binom {n}{k}}p_{k}(x)p_{n-k}(y)} for all {displaystyle x} , {displaystyle y} , and {displaystyle n} .

An operator {displaystyle Q} on the space of polynomials is said to be the basis operator of the sequence {displaystyle {p_{n}}_{n=0}^{infty }} if {displaystyle Qp_{0}=0} and {displaystyle Qp_{n}=np_{n-1}} for all {displaystyle ngeqslant 1} . A sequence {displaystyle {p_{n}}_{n=0}^{infty }} is binomial if and only if its basis operator is a Delta operator.[17] Writing {displaystyle E^{a}} for the shift by {displaystyle a} operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference {displaystyle I-E^{-c}} for {displaystyle c>0} , the ordinary derivative for {displaystyle c=0} , and the forward difference {displaystyle E^{-c}-I} for {displaystyle c<0} . Multinomial theorem Main article: Multinomial theorem The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is {displaystyle (x_{1}+x_{2}+cdots +x_{m})^{n}=sum _{k_{1}+k_{2}+cdots +k_{m}=n}{binom {n}{k_{1},k_{2},ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}cdots x_{m}^{k_{m}},} where the summation is taken over all sequences of nonnegative integer indices k1 through km such that the sum of all ki is n. (For each term in the expansion, the exponents must add up to n). The coefficients {displaystyle {tbinom {n}{k_{1},cdots ,k_{m}}}} are known as multinomial coefficients, and can be computed by the formula {displaystyle {binom {n}{k_{1},k_{2},ldots ,k_{m}}}={frac {n!}{k_{1}!cdot k_{2}!cdots k_{m}!}}.} Combinatorially, the multinomial coefficient {displaystyle {tbinom {n}{k_{1},cdots ,k_{m}}}} counts the number of different ways to partition an n-element set into disjoint subsets of sizes k1, ..., km. Multi-binomial theorem When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to {displaystyle (x_{1}+y_{1})^{n_{1}}dotsm (x_{d}+y_{d})^{n_{d}}=sum _{k_{1}=0}^{n_{1}}dotsm sum _{k_{d}=0}^{n_{d}}{binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}dotsc {binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.} This may be written more concisely, by multi-index notation, as {displaystyle (x+y)^{alpha }=sum _{nu leq alpha }{binom {alpha }{nu }}x^{nu }y^{alpha -nu }.} General Leibniz rule Main article: General Leibniz rule The general Leibniz rule gives the nth derivative of a product of two functions in a form similar to that of the binomial theorem:[18] {displaystyle (fg)^{(n)}(x)=sum _{k=0}^{n}{binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).} Here, the superscript (n) indicates the nth derivative of a function. If one sets f(x) = eax and g(x) = ebx, and then cancels the common factor of e(a + b)x from both sides of the result, the ordinary binomial theorem is recovered.[19] Applications Multiple-angle identities For the complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine. According to De Moivre's formula, {displaystyle cos left(nxright)+isin left(nxright)=left(cos x+isin xright)^{n}.} Using the binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos(nx) and sin(nx). For example, since {displaystyle left(cos x+isin xright)^{2}=cos ^{2}x+2icos xsin x-sin ^{2}x,} De Moivre's formula tells us that {displaystyle cos(2x)=cos ^{2}x-sin ^{2}xquad {text{and}}quad sin(2x)=2cos xsin x,} which are the usual double-angle identities. Similarly, since {displaystyle left(cos x+isin xright)^{3}=cos ^{3}x+3icos ^{2}xsin x-3cos xsin ^{2}x-isin ^{3}x,} De Moivre's formula yields {displaystyle cos(3x)=cos ^{3}x-3cos xsin ^{2}xquad {text{and}}quad sin(3x)=3cos ^{2}xsin x-sin ^{3}x.} In general, {displaystyle cos(nx)=sum _{k{text{ even}}}(-1)^{k/2}{n choose k}cos ^{n-k}xsin ^{k}x} and {displaystyle sin(nx)=sum _{k{text{ odd}}}(-1)^{(k-1)/2}{n choose k}cos ^{n-k}xsin ^{k}x.} Series for e The number e is often defined by the formula {displaystyle e=lim _{nto infty }left(1+{frac {1}{n}}right)^{n}.} Applying the binomial theorem to this expression yields the usual infinite series for e. In particular: {displaystyle left(1+{frac {1}{n}}right)^{n}=1+{n choose 1}{frac {1}{n}}+{n choose 2}{frac {1}{n^{2}}}+{n choose 3}{frac {1}{n^{3}}}+cdots +{n choose n}{frac {1}{n^{n}}}.} The kth term of this sum is {displaystyle {n choose k}{frac {1}{n^{k}}}={frac {1}{k!}}cdot {frac {n(n-1)(n-2)cdots (n-k+1)}{n^{k}}}} As n → ∞, the rational expression on the right approaches 1, and therefore {displaystyle lim _{nto infty }{n choose k}{frac {1}{n^{k}}}={frac {1}{k!}}.} This indicates that e can be written as a series: {displaystyle e=sum _{k=0}^{infty }{frac {1}{k!}}={frac {1}{0!}}+{frac {1}{1!}}+{frac {1}{2!}}+{frac {1}{3!}}+cdots .} Indeed, since each term of the binomial expansion is an increasing function of n, it follows from the monotone convergence theorem for series that the sum of this infinite series is equal to e. Probability The binomial theorem is closely related to the probability mass function of the negative binomial distribution. The probability of a (countable) collection of independent Bernoulli trials {displaystyle {X_{t}}_{tin S}} with probability of success {displaystyle pin [0,1]} all not happening is {displaystyle Pleft(bigcap _{tin S}X_{t}^{C}right)=(1-p)^{|S|}=sum _{n=0}^{|S|}{|S| choose n}(-p)^{n}.} An upper bound for this quantity is {displaystyle e^{-p|S|}.} [20] In abstract algebra The binomial theorem is valid more generally for two elements x and y in a ring, or even a semiring, provided that xy = yx. For example, it holds for two n × n matrices, provided that those matrices commute; this is useful in computing powers of a matrix.[21] The binomial theorem can be stated by saying that the polynomial sequence {1, x, x2, x3, ...} is of binomial type. In popular culture The binomial theorem is mentioned in the Major-General's Song in the comic opera The Pirates of Penzance. Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem. The Portuguese poet Fernando Pessoa, using the heteronym Álvaro de Campos, wrote that "Newton's Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it."[22] In the 2014 film The Imitation Game, Alan Turing makes reference to Isaac Newton's work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park. See also Mathematics portal Binomial approximation Binomial distribution Binomial inverse theorem Stirling's approximation Tannery's theorem Notes ^ Jump up to: a b This is to guarantee convergence. Depending on r, the series may also converge sometimes when |x| = |y|. References ^ Jump up to: a b Weisstein, Eric W. "Binomial Theorem". Wolfram MathWorld. ^ Jump up to: a b c d Coolidge, J. L. (1949). "The Story of the Binomial Theorem". The American Mathematical Monthly. 56 (3): 147–157. doi:10.2307/2305028. JSTOR 2305028. ^ Jump up to: a b c Jean-Claude Martzloff; S.S. Wilson; J. Gernet; J. Dhombres (1987). A history of Chinese mathematics. Springer. ^ Jump up to: a b Biggs, N. L. (1979). "The roots of combinatorics". Historia Math. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0. ^ "THE BINOMIAL THEOREM: A WIDESPREAD CONCEPT IN MEDIEVAL ISLAMIC MATHEMATICS" (PDF). core.ac.uk. p. 401. Retrieved 2019-01-08. ^ "Taming the unknown. A history of algebra from antiquity to the early twentieth century" (PDF). Bulletin of the American Mathematical Society: 727. However, algebra advanced in other respects. Around 1000, al-Karaji stated the binomial theorem ^ Rashed, R. (1994-06-30). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Springer Science & Business Media. p. 63. ISBN 9780792325659. ^ Jump up to: a b O'Connor, John J.; Robertson, Edmund F., "Abu Bekr ibn Muhammad ibn al-Husayn Al-Karaji", MacTutor History of Mathematics archive, University of St Andrews ^ Landau, James A. (1999-05-08). "Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle" (mailing list email). Archives of Historia Matematica. Retrieved 2007-04-13. ^ Jump up to: a b c Kline, Morris (1972). History of mathematical thought. Oxford University Press. p. 273. ^ Katz, Victor (2009). "14.3: Elementary Probability". A History of Mathematics: An Introduction. Addison-Wesley. p. 491. ISBN 978-0-321-38700-4. ^ Bourbaki, N. (18 November 1998). Elements of the History of Mathematics Paperback. J. Meldrum (Translator). ISBN 978-3-540-64767-6. ^ Mathematical Methods for Physicists. 2013. p. 34. doi:10.1016/c2009-0-30629-7. ISBN 9780123846549. ^ Jump up to: a b Barth, Nils R. (2004). "Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube". The American Mathematical Monthly. 111 (9): 811–813. doi:10.2307/4145193. ISSN 0002-9890. JSTOR 4145193. ^ Binomial theorem – inductive proofs Archived February 24, 2015, at the Wayback Machine ^ Sokolowsky, Dan; Rennie, Basil C. (February 1979). "Problem 352". Crux Mathematicorum. 5 (2): 55–56. ^ Aigner, Martin (1997) [Reprint of the 1979 Edition]. Combinatorial Theory. Springer. p. 105. ISBN 3-540-61787-6. ^ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006. ^ Spivey, Michael Z. (2019). The Art of Proving Binomial Identities. CRC Press. p. 71. ISBN 978-1351215800. ^ Cover, Thomas M.; Thomas, Joy A. (2001-01-01). Data Compression. John Wiley & Sons, Inc. p. 320. doi:10.1002/0471200611.ch5. ISBN 9780471200611. ^ Artin, Algebra, 2nd edition, Pearson, 2018, equation (4.7.11). ^ "Arquivo Pessoa: Obra Édita - O binómio de Newton é tão belo como a Vénus de Milo". arquivopessoa.net. Further reading Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci. 1 (1): 68–74. Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153–256. ISBN 978-0-201-55802-9. OCLC 17649857. External links The Wikibook Combinatorics has a page on the topic of: The Binomial Theorem Solomentsev, E.D. (2001) [1994], "Newton binomial", Encyclopedia of Mathematics, EMS Press Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, Wolfram Demonstrations Project, 2007. 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