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Encyclopedia > Binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version says Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ... Addition is one of the basic operations of arithmetic. ...

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

whenever n is any non-negative integer, the numbers

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

are the binomial coefficients, and n! denotes the factorial of n. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. ... For factorial rings in mathematics, see unique factorisation domain. ...

This formula and the triangular arrangement of the binomial coefficients are often attributed to Blaise Pascal, who described them in the 17th century. 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.^{[citation needed]} 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 The first six rows of Pascals triangle In mathematics, Pascals triangle is a geometric arrangement of the binomial coefficients in a triangle. ... John Blaise Pascal (pronounced ), (June 19, 1623â€“August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... Knowledge of Chinese mathematics before 100 BC is somewhat fragmentary, but there are elements that seem consistent. ... Yang Hui (æ¥Šè¼, c. ... For information about all peoples of Iran, see Demographics of Iran; for Central Asian Persians, see Tajiks. ... Islamic mathematics is the profession of Muslim Mathematicians. ... Omar KhayyÃ¡m, (Persian: Ø¹Ù…Ø± Ø®ÛŒØ§Ù…, born: May 31, 1048 in Nishapur, Iran (Persia) â€“ died: December 4, 1131), was a Persian poet, mathematician, philosopher and astronomer. ... The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE). ... Pingala (à¤ªà¤¿à¤™à¥à¤—à¤² ) is the supposed author of the Chandas shastra (, also Chandas sutra ), a Sanskrit treatise on prosody considered one of the Vedanga. ...

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

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

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

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

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

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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...

1 Newton's generalized binomial theorem
2 "Binomial type"
3 Proof
4 Binomial number
5 A quick way to expand binomials
6 Trivia
7 References
8 See also

Newton's generalized binomial theorem

Isaac Newton generalized the formula to other exponents by considering an infinite series: Sir Isaac Newton, (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... In mathematics, a series is a sum of a sequence of terms. ...

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

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 In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

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

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., do not appear. In mathematics, an empty product, or nullary product, is the result of multiplying no numbers. ...

Another way to express this quantity is

${r choose k}=frac{(-1)^k}{k!}(-r)_k,$

which is important when one is working with infinite series and would like to represent them in terms of generalized hypergeometric functions. The notation $(cdot)_k$ is the Pochhammer symbol. This form is vital 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. ... In mathematics, the Pochhammer symbol, introduced by Leo August Pochhammer, is used in the theory of special functions to represent the rising factorial or upper factorial and, confusingly, is used in combinatorics to represent the falling factorial or lower factorial To distinguish the two, the notations and are commonly used...

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 binomial theorem. ...

The sum in (2) converges and the equality is true whenever the real or complex numbers x and y are "close together" in the sense that the absolute value | x/y | is less than one. In mathematics, the absolute value (or modulus[1]) 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 y = 1 and r = −1. In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

Formula (2) is also valid for elements x and y of a Banach algebra as long as xy = yx, y is invertible and ||x/y|| < 1. In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ...

"Binomial type"

The binomial theorem can be stated by saying that the polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. ...

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

is of binomial type. Definition In mathematics, a polynomial sequence, i. ...

Proof

One way to prove the binomial theorem is with mathematical induction. When n = 0, we have Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

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

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.

as desired. In mathematics, Pascals rule is a combinatorial identity about binomial coefficients. ...

Binomial number

A binomial number is a number in the form of $x^n pm y^n$ . These binomial numbers can be factored algebraically.

$x^npm y^n=(xpm y)(x^{n-1} mp x^{n-2}y + cdots mp xy^{n-2} + y^{n-1})$

Examples:

x 2 - y 2 = (x - y)(x + y)

x 3 - y 3 = (x - y)(x 2 + x y + y 2)

x 3 + y 3 = (x + y)(x 2 - x y + y 2)

x 8 - y 8 = (x - y)(x + y)(x 2 + y 2)(x 4 + y 4)

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.

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, 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

$x^{10}+10x^9y+45x^8y^2+120x^7y^3+210x^6y^4+252x^5y^5+210x^4y^6+120x^3y^7+45x^2y^8+10xy^9+y^{10}.,$

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.

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.

Trivia

In the Sherlock Holmes books, the villain Professor Moriarty is the author of A Treatise on the Binomial Theorem.

The binomial theorem is mentioned in the Gilbert and Sullivan song "I am the Very Model of a Modern Major General".

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 Meaning of Life.

The binomial theorem is mentioned in the TV series NUMB3RS in episode #217 ("Mind Games") in Season 2.

The generalized binomial theorem is engraved on Isaac Newton's tomb in Westminster Abbey.

Professor Moriarty, illustration by Sidney Paget which accompanied the original publication of The Final Problem. Professor James Moriarty is a fictional character who is the best known antagonist (and archenemy) of the detective Sherlock Holmes. ... A Treatise on the Binomial Theorem is a fictional work by Professor James Moriarty, the implacible foe of Sherlock Holmes. ... W. S. Gilbert Arthur Sullivan Librettist William Schwenck Gilbert (1836â€“1911) and composer Arthur Seymour Sullivan (1842â€“1900) collaborated on a series of fourteen comic operas in Victorian England between 1871 and 1896. ... Henry Lytton as the Major-General The Major-Generals Song is a patter song from Gilbert and Sullivans 1879 comic opera The Pirates of Penzance. ... Monty Python, or The Pythons, is the collective name of the creators of Monty Pythonâ€™s Flying Circus, a British television comedy sketch show that first aired on the BBC on 5 October 1969. ... List of all 45 episodes from the television series Monty Pythons Flying Circus: // (aired October 5, 1969; recorded September 7, 1969) Its Wolfgang Amadeus Mozart Italian Lesson Whizzo Butter Its the Arts Arthur Two Sheds Jackson Picasso/Cycling Race The Funniest Joke in the World Trivia The... Cover of the VHS release of Monty Pythons Fliegender Zirkus. ... Monty Pythons The Meaning of Life is a comedy film/musical made in 1983 by Monty Python. ... NUMB3RS (Numbers) is an American television show that follows FBI Special Agent Don Eppes (Rob Morrow) and his mathematical genius brother, Charlie Eppes (David Krumholtz), who develops formulae to predict the actions of various criminals. ... Sir Isaac Newton, (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... The Collegiate Church of St Peter, Westminster, which is almost always referred to by its original name of Westminster Abbey, is a mainly Gothic church, on the scale of a cathedral (and indeed often mistaken for one), in Westminster, London, just to the west of the Palace of Westminster. ...

References

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

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