Binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information Its simplest version says

$(x+y)^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^{k}\quad\quad\quad(1)$

whenever n is any non-negative integer, the number

${n \choose k}=\frac{n!}{k!\,(n-k)!}$

is the binomial coefficient (using the choose function), and n! denotes the factorial of n. A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x k term in the Polynomial In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x k term in the Polynomial Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century. \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix Blaise Pascal (blɛz paskal (June 19 1623 &ndash August 19 1662 was a French Mathematician, Physicist, and religious Philosopher However, it was known to many mathematicians who preceded him; 13th-century Chinese mathematician Yang Hui, 11th-century Persian mathematician Omar Khayyám, and 3rd-century BC Indian mathematician Pingala all derived similar results. Mathematics in China emerged independently by the 11th century BC Yang Hui ( ca 1238–1298 Courtesy name Qianguang (谦光 was a Chinese Mathematician from Qiantang (modern Hangzhou layout and formatting it should ensure no clashes with the top of the infobox For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra ^[1]

For example, here are the cases where 2 ≤ n ≤ 5:

$(x + y)^2 = x^2 + 2xy + y^2\,$

$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\,$

$(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\,$

$(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 +5xy^4 + y^5\,$

Formula (1) is valid for all real or complex numbers x and y, and more generally for any elements x and y of a semiring as long as xy = yx (the theorem is true even more generally: note that associativity is not required, just alternativity). In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, associativity is a property that a Binary operation can have In Abstract algebra, a magma G is said to be left alternative if ( xx) y = x ( xy) for all x and y

1 Simple derivation
2 Newton's generalized binomial theorem
3 "Binomial type"
4 Proof
5 Binomial number
6 A quick way to expand binomials
7 The binomial theorem in popular culture
8 References
9 See also
10 External links

Simple derivation

Consider $a = (x + y) n$ . a can be written as a product of sums, $a=s_1s_2 \cdots s_n$ , where each $s i = x + y$ . The expansion of a is the sum of all products involving one term—either x or y—from each $s i$ . For example, the term $x n$ in the expansion of a is had by picking x in each $s i$ .

The coefficient of each term in the expansion of a is determined by how many different ways there are to pick terms from the $s i$ such that their product is of the same form as the term (excluding the coefficient). Consider $t = x n - 1 y$ . t can be formed from a by picking y from one of the $s i$ and x in the rest of them. There are n ways to pick a $s i$ to provide the y; t is thus formed in n different ways in the expansion of a, making its coefficient n. In general, for $t = x n - k y k$ , there are

${n \choose k}$

different ways to pick the $s i$ that provide the ys (since k ys are picked from the n $s i$ ), and thus this must be the coefficient for t. The binomial theorem follows naturally from here.

Newton's generalized binomial theorem

Isaac Newton generalized the formula to other exponents by considering an infinite series:

${(x+y)^r=\sum_{k=0}^\infty {r \choose k} x^{r-k} y^k \quad\quad\quad(2)}$

where r can be any complex number (in particular r can be any real number, not necessarily positive and not necessarily an integer), and the coefficients are given by

$\begin{align} {r \choose k} &{}= {1 \over k!}\prod_{n=0}^{k-1}(r-n)=\frac{r(r-1)(r-2)\cdots(r-(k-1))}{k!}. \end{align}$

In case k = 0, this is a product of no numbers at all and therefore equal to 1, and in case k = 1 it is equal to r, as the additional factors (r − 1), etc. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers , do not appear.

Another way to express this quantity is

${r \choose k}=\frac{(r)_k}{k!},$

which is important when one is working with infinite series and would like to represent them in terms of generalized hypergeometric functions. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function The notation $(\cdot)_k$ is the Pochhammer symbol. In Mathematics, the Pochhammer symbol (x_{n}\ introduced by Leo August Pochhammer, represents either the rising or the falling This form is used in applied mathematics, for example, when evaluating the formulas that model the statistical properties of the phase-front curvature of a light wave as it propagates through optical atmospheric turbulence.

A particularly handy but non-obvious form holds for the reciprocal power:

$\frac{1}{(1-x)^r}=\sum_{k=0}^\infty {r+k-1 \choose k} x^k \equiv \sum_{k=0}^\infty {r+k-1 \choose r-1} x^k.$

For a more extensive account of Newton's generalized binomial theorem, see binomial series. In Mathematics, the binomial series generalizes the purely algebraic formula of the Binomial theorem to complex values of α

The sum in (2) converges and the equality is true whenever x is nonzero and the real or complex numbers x and y are "close together" in the sense that the absolute value | y/x | is less than one. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

The geometric series is a special case of (2) where we choose x = 1 and r = − 1. In Mathematics, a geometric series is a series with a constant ratio between successive terms.

Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, x is invertible and ||y/x|| < 1. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the

"Binomial type"

The binomial theorem can be stated by saying that the polynomial sequence

$\left\{\,x^k:k=0,1,2,\dots\,\right\}\,$

is of binomial type. In Mathematics, a polynomial sequence is a Sequence of Polynomials indexed by the nonnegative integers 0 1 2 3. In Mathematics, a Polynomial sequence, ie a sequence of Polynomials indexed by { 0 1 2 3.

Proof

One way to prove the binomial theorem (1) is with mathematical induction. Mathematical induction is a method of Mathematical proof typically used to establish that a given statement is true of all Natural numbers It is done by proving that When n = 0, we have

$(a+b)^0 = 1 = \sum_{k=0}^0 { 0 \choose k } a^{0-k}b^k.$

For the inductive step, assume the theorem holds when the exponent is m. Then for n = m + 1

$(a+b)^{m+1} = a(a+b)^m + b(a+b)^m \,$

$= a \sum_{k=0}^m { m \choose k } a^{m-k} b^k + b \sum_{j=0}^m { m \choose j } a^{m-j} b^j$

by the inductive hypothesis

$= \sum_{k=0}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1}$

by multiplying through by a and b

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{j=0}^m { m \choose j } a^{m-j} b^{j+1}$

by pulling out the k = 0 term

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1} b^k + \sum_{k=1}^{m+1} { m \choose k-1 }a^{m-k+1}b^{k}$

by letting j = k − 1

$= a^{m+1} + \sum_{k=1}^m { m \choose k } a^{m-k+1}b^k + \sum_{k=1}^{m} { m \choose k-1 }a^{m+1-k}b^{k} + b^{m+1}$

by pulling out the k = m + 1 term from the right hand side

$= a^{m+1} + b^{m+1} + \sum_{k=1}^m \left[ { m \choose k } + { m \choose k-1 } \right] a^{m+1-k}b^k$

by combining the sums

$= a^{m+1} + b^{m+1} + \sum_{k=1}^m { m+1 \choose k } a^{m+1-k}b^k$

from Pascal's rule

$= \sum_{k=0}^{m+1} { m+1 \choose k } a^{m+1-k}b^k$

by adding in the m + 1 terms. In Mathematics, Pascal's rule is a combinatorial identity about Binomial coefficients It states that for any Natural number n

Binomial number

A binomial number is a number in the form of $\scriptstyle x^n \,\pm\, y^n$ (for n at least 2). When the sign is minus or n is odd these binomial numbers can be factored algebraically:

$x^n\pm y^n=(x\pm y)(x^{n-1} \mp x^{n-2}y + \cdots \mp xy^{n-2} + y^{n-1}).\,$

Examples:

$x^2-y^2=(x-y)(x+y)\,$

$x^3-y^3=(x-y)(x^2+xy+y^2)\,$

$x^3+y^3=(x+y)(x^2-xy+y^2)\,$

$x^8-y^8=(x-y)(x+y)(x^2+y^2)(x^4+y^4)\,$

To factorise $\scriptstyle x^n\,-\,y^n$ simply, use

$x^n-y^n=(x-y) \left( \sum_{k=0}^{n-1}x^ky^{n-1-k} \right).$

A quick way to expand binomials

To quickly expand binomials of the form

$(x+y)^n \,$

The first term is

$x^n \,$

(this follows directly from the generalized binomial theorem) and the coefficient of each subsequent term is the current coefficient multiplied by the current exponent of x, divided by the current term number. Exponents of x decrease each term, while exponents of y increase each term (from 0 in the first term) until the exponent of x is 0 and that of y is n.

Example:

$(x+y)^{10} \,$

The first term is

$x^{10} \,$

To find the coefficient of the second term, multiply 1 (the current coefficient) by 10 (the current exponent of x), and divide by the current term number (1, since this is the first term) to get 10. The exponent of x decrements, and the exponent of y increments. The next term is therefore

$10x^9y \,$

Similarly, the next coefficient is 10×9/2×1, which gives 45. After that, it is (10×9×8)/(3×2×1). This continues until (10×9×8×7×6)/(5×4×3×2×1), after which, the coefficients are symmetrical. The whole thing is

x 10 + 10 x 9 y + 45 x 8 y 2 + 120 x 7 y 3 + 210 x 6 y 4 + 252 x 5 y 5 + 210 x 4 y 6 + 120 x 3 y 7 + 45 x 2 y 8 + 10 x y 9 + y 10 .

Notice that the coefficients are perfectly symmetrical. This will happen when the coefficients of x and y within the parentheses of the original expression are the same. Recognizing this can save even more time.

More formally, given a term

$kx^my^n \,$

The next term in the binomial is

$\frac{km}{n+1}x^{m-1}y^{n+1}=\frac{d}{dx}\left( \int kx^my^n\,dy\right)$

If the original expression instead was

$(2x+y)^{10} \,$

then the resulting expansion would be the same, except with (2x) in place of x in every place. The factor of 2 must get raised to the power of x in each term. The same holds true if either x or y is raised to a power inside the parentheses of the original expression.

The binomial theorem in popular culture

In the Sherlock Holmes books, the villain Professor Moriarty is the author of A Treatise on the Binomial Theorem. Professor James Moriarty is a Fictional character who is the best known Antagonist (and Nemesis) of the detective Sherlock Holmes. A Treatise on the Binomial Theorem is a brilliant work of mathematics by the young James Moriarty, the evil Archenemy of the detective Sherlock Holmes
The binomial theorem is mentioned in the Gilbert and Sullivan song "I am the Very Model of a Modern Major General". Gilbert and Sullivan refers to the Victorian era partnership of Librettist W The Major-General's Song is a Patter song from Gilbert and Sullivan 's 1879 Comic opera The Pirates of Penzance.
The binomial theorem appears in at least three different works by Monty Python - Coal Mine in Llandarogh Carmarthen, The Tale of Happy Valley, and the film Monty Python's The Meaning of Life. Monty Python (sometimes known as The Pythons) is the collective name of the six creators of Monty Python's Flying Circus, a British Television This is a list of all 45 episodes from the television series Monty Python's Flying Circus: Series 1 (Oct Monty Python's Fliegender Zirkus consisted of two 45-minute Monty Python specials produced by WDR for West German television in 1971 Monty Python's The Meaning of Life is a 1983 musical Comedy film by the Monty Python comedy team
The binomial theorem is mentioned in the TV series NUMB3RS in episode #217 ("Mind Games") in Season 2. NUMB3RS (pronounced Numbers) is an American Television show produced by brothers Ridley and Tony Scott.
Contrary to popular belief, the generalized binomial theorem is not engraved on Isaac Newton's tomb in Westminster Abbey. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The Collegiate Church of St Peter at Westminster, which is almost always referred to by its original name of Westminster Abbey, is a large mainly Gothic church ^[2]
In chapter 18 of Mikhail Bulgakov's "The Master and Margarita," the black magic practitioner Woland says, "But by Newton's binomial theorem, I predict that he will die in nine month's time. Mikhail Afanasievich Bulgakov (Михаил Афанасьевич Булгаков, Kiev &ndash March 10, 1940, Moscow) was a Russian . . " From this, "it's hardly Newton's binomial theorem" became a popular Russian expression.
There is a short poem by Álvaro de Campos, heteronym of the Portuguese writer Fernando Pessoa about the binomial theorem that roughly translates into: "Newton's binomial is as beautiful as the Venus de Milo. Álvaro de Campos was one of Fernando Pessoa 's various heteronyms widely known by his powerful and wraithful writing style The literary concept of heteronym, invented by Portuguese poet Fernando Pessoa, refers to one or more imaginary character(s created by a poet to write Portuguese ( or língua portuguesa) is a Romance language that originated in what is now Galicia (Spain and northern Portugal. Fernando António Nogueira Pessoa (fɨɾˈnɐ̃du pɨˈsoɐ (b The Aphrodite of Milos (Greek "Αφροδίτη της Μήλου" better known as the Venus de Milo is an ancient Greek statue The truth is few people notice it. "
In record 5 of Yevgeny Zamyatin's "We", the protagonist D-503 says, ". . . to me it's nothing more than the four equal angles, but for you that might be, I don't know, as tough as Newton's binomial theorem. "

References

Amulya Kumar Bag. Binomial Theorem in Ancient India. Indian J. History Sci. ,1:68-74,1966.

^ Landau, James A (1999-05-08). Year 1999 ( MCMXCIX) was a Common year starting on Friday (link will display full 1999 Gregorian calendar) Events 589 - Reccared summons the Third Council of Toledo 1450 - Jack Cade's Rebellion: Kentishmen Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle (mailing list email). Archives of Historia Matematica. Retrieved on 2007-04-13. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1111 - Henry V is crowned Holy Roman Emperor. 1204 - The Fourth Crusade sacks Constantinople
^ Cajori, Florian (1985). A History of Mathematics. New York: Chelsea Publishing Company, 205. ISBN 0-8284-1303-X.

External links

Binomial Theorem by Stephen Wolfram, and "Binomial Theorem (Step-by-Step)" by Bruce Colletti and Jeff Bryant, The Wolfram Demonstrations Project, 2007. WikipediaWikiProject Probability#Standards for a discussion of standards used for probability distribution articles such as this one In Mathematics, the binomial inverse theorem is useful for expressing matrix inverses in different ways In Mathematics, the binomial series generalizes the purely algebraic formula of the Binomial theorem to complex values of α In Mathematics, the multinomial theorem says how to write a power of a sum in terms of powers of the terms in that sum In Probability and Statistics the negative binomial distribution is a Discrete probability distribution. \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix Yang Hui ( ca 1238–1298 Courtesy name Qianguang (谦光 was a Chinese Mathematician from Qiantang (modern Hangzhou Stephen Wolfram (born August 29, 1959 in London) is a British Physicist, Mathematician and Businessman known for his

This article incorporates material from inductive proof of binomial theorem on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(mathematics) A formula giving the expansion of an algebraic sum (e.g. (a + b)) raised to any positive integer power

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