Binomial theorem

Summary

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,

The binomial coefficient appears as the kth entry in the nth row of Pascal's triangle (where the top is the 0th row ). Each entry is the sum of the two above it.

The coefficient a in the term of axbyc is known as the binomial coefficient or (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 gives the number of different combinations (i.e. subsets) of b elements that can be chosen from an n-element set. Therefore is usually pronounced as "n choose b".

History

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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  .[1] Greek mathematician Diophantus cubed various binomials, including  .[1] Indian mathematician Aryabhata's method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent  .[1]

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.[2]: 230  The commentator Halayudha from the 10th century AD explains this method.[2]: 230  By the 6th century AD, the Indian mathematicians probably knew how to express this as a quotient  ,[3] and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara.[3]

The first known formulation of the binomial theorem and the table of binomial coefficients appears in a work by Al-Karaji, quoted by Al-Samaw'al in his "al-Bahir".[4][5][6] Al-Karaji described the triangular pattern of the binomial coefficients[7] and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction.[7] The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost.[1] The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui[8] and also Chu Shih-Chieh.[1] Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.[2]: 142 

In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express   in terms of  , via "Pascal's triangle".[9] Blaise Pascal studied the eponymous triangle comprehensively in his Traité du triangle arithmétique.[10] However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia, and Simon Stevin.[9]

Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665.[9][11] It was discovered independently in 1670 by James Gregory.[12]

Statement

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According to the theorem, the expansion of any nonnegative integer power n of the binomial x + y is a sum of the form

 
where each   is a positive integer known as a binomial coefficient, defined as

 

This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written more concisely as

 

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

 

Examples

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Here are the first few cases of the binomial theorem:

 
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:

 
with  .

A simple example with a specific positive value of y:

 

A simple example with a specific negative value of y:

 

Geometric explanation

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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  [13] if one sets   and   interpreting b as an infinitesimal change in a, then this picture shows the infinitesimal change in the volume of an n-dimensional hypercube,   where the coefficient of the linear term (in  ) is   the area of the n faces, each of dimension n − 1:

 
Substituting this into the definition of the derivative 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 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   – see proof of Cavalieri's quadrature formula for details.[13]

Binomial coefficients

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The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written   and pronounced "n choose k".

Formulas

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The coefficient of xnkyk is given by the formula

 
which is defined in terms of the factorial function n!. Equivalently, this formula can be written
 
with k factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient   is actually an integer.

Combinatorial interpretation

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The binomial coefficient   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

 
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

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Combinatorial proof

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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 xnkyk for some k between 0 and n. For a given k, the following are proved equal in succession:

  • the number of terms equal to xnkyk 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}
  •   either by definition, or by a short combinatorial argument if one is defining   as  

This proves the binomial theorem.

Example

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The coefficient of xy2 in

 
equals   because there are three x,y strings of length 3 with exactly two y's, namely,
 
corresponding to the three 2-element subsets of {1, 2, 3}, namely,
 
where each subset specifies the positions of the y in a corresponding string.

Inductive proof

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Induction yields another proof of the binomial theorem. When n = 0, both sides equal 1, since x0 = 1 and   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   if j + k = n, and 0 otherwise. The identity

 
shows that (x + y)n+1 is also a polynomial in x and y, and
 
since if j + k = n + 1, then (j − 1) + k = n and j + (k − 1) = n. Now, the right hand side is
 
by Pascal's identity.[14] On the other hand, if j + kn + 1, then (j – 1) + kn and j + (k – 1) ≠ n, so we get 0 + 0 = 0. Thus
 
which is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step.

Generalizations

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Newton's generalized binomial theorem

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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

 
where   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
 

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:

 

Taking r = −1, the generalized binomial series gives the geometric series formula, valid for |x| < 1:

 

More generally, with r = −s, we have for |x| < 1:[15]

 

So, for instance, when s = 1/2,

 

Replacing x with -x yields:

 

So, for instance, when s = 1/2, we have for |x| < 1:

 

Further generalizations

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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   and

 
for   Then[16]
 
The case c = 0 recovers the usual binomial theorem.

More generally, a sequence   of polynomials is said to be of binomial type if

  •   for all  ,
  •  , and
  •   for all  ,  , and  .

An operator   on the space of polynomials is said to be the basis operator of the sequence   if   and   for all  . A sequence   is binomial if and only if its basis operator is a Delta operator.[17] Writing   for the shift by   operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference   for  , the ordinary derivative for  , and the forward difference   for  .

Multinomial theorem

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The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is

 

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   are known as multinomial coefficients, and can be computed by the formula

 

Combinatorially, the multinomial coefficient   counts the number of different ways to partition an n-element set into disjoint subsets of sizes k1, ..., km.

Multi-binomial theorem

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When working in more dimensions, it is often useful to deal with products of binomial expressions. By the binomial theorem this is equal to

 

This may be written more concisely, by multi-index notation, as

 

General Leibniz rule

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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]

 

Here, the superscript (n) indicates the nth derivative of a function,  . If one sets f(x) = eax and g(x) = ebx, cancelling the common factor of e(a + b)x from each term gives the ordinary binomial theorem.[19]

Applications

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Multiple-angle identities

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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,

 

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

 
But De Moivre's formula identifies the left side with  , so
 
which are the usual double-angle identities. Similarly, since
 
De Moivre's formula yields
 
In general,
 
and
 
There are also similar formulas using Chebyshev polynomials.

Series for e

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The number e is often defined by the formula

 

Applying the binomial theorem to this expression yields the usual infinite series for e. In particular:

 

The kth term of this sum is

 

As n → ∞, the rational expression on the right approaches 1, and therefore

 

This indicates that e can be written as a series:

 

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

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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   with probability of success   all not happening is

 

An upper bound for this quantity is  [20]

In abstract algebra

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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.

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See also

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Notes

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  1. ^ a b This is to guarantee convergence. Depending on r, the series may also converge sometimes when |x| = |y|.

References

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  1. ^ a b c d e Coolidge, J. L. (1949). "The Story of the Binomial Theorem". The American Mathematical Monthly. 56 (3): 147–157. doi:10.2307/2305028. JSTOR 2305028.
  2. ^ a b c Jean-Claude Martzloff; S.S. Wilson; J. Gernet; J. Dhombres (1987). A history of Chinese mathematics. Springer.
  3. ^ a b Biggs, N. L. (1979). "The roots of combinatorics". Historia Math. 6 (2): 109–136. doi:10.1016/0315-0860(79)90074-0.
  4. ^ Yadegari, Mohammad (1980). "The Binomial Theorem: A Widespread Concept in Medieval Islamic Mathematics". Historia Mathematica. 7 (4): 401–406. doi:10.1016/0315-0860(80)90004-X.
  5. ^ Stillwell, John (2015). "Taming the unknown. A history of algebra ... by Victor J. Katz and Karen Hunger Parshall". Bulletin of the American Mathematical Society (Book review). 52 (4): 725–731. doi:10.1090/S0273-0979-2015-01491-6. p. 727: However, algebra advanced in other respects. Around 1000, al-Karaji stated the binomial theorem
  6. ^ Rashed, Roshdi (1994). The Development of Arabic Mathematics: Between Arithmetic and Algebra. Kluwer. p. 63. ISBN 0-7923-2565-6.
  7. ^ 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
  8. ^ Landau, James A. (1999-05-08). "Historia Matematica Mailing List Archive: Re: [HM] Pascal's Triangle". Archives of Historia Matematica. Archived from the original (mailing list email) on 2021-02-24. Retrieved 2007-04-13.
  9. ^ a b c Kline, Morris (1972). History of mathematical thought. Oxford University Press. p. 273.
  10. ^ Katz, Victor (2009). "14.3: Elementary Probability". A History of Mathematics: An Introduction. Addison-Wesley. p. 491. ISBN 978-0-321-38700-4.
  11. ^ Bourbaki, N. (18 November 1998). Elements of the History of Mathematics Paperback. Translated by J. Meldrum. ISBN 978-3-540-64767-6.
  12. ^ Stillwell, John (2010). Mathematics and its history (third ed.). Springer. p. 186. ISBN 978-1-4419-6052-8.
  13. ^ 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.
  14. ^ Binomial theorem – inductive proofs Archived February 24, 2015, at the Wayback Machine
  15. ^ Weisstein, Eric W. "Negative Binomial Series". Wolfram MathWorld.
  16. ^ Sokolowsky, Dan; Rennie, Basil C. (February 1979). "Problem 352". Crux Mathematicorum. 5 (2): 55–56.
  17. ^ Aigner, Martin (1997) [Reprint of the 1979 Edition]. Combinatorial Theory. Springer. p. 105. ISBN 3-540-61787-6.
  18. ^ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006.
  19. ^ Spivey, Michael Z. (2019). The Art of Proving Binomial Identities. CRC Press. p. 71. ISBN 978-1351215800.
  20. ^ Cover, Thomas M.; Thomas, Joy A. (2001-01-01). Data Compression. John Wiley & Sons, Inc. p. 320. doi:10.1002/0471200611.ch5. ISBN 9780471200611.
  21. ^ Artin, Algebra, 2nd edition, Pearson, 2018, equation (4.7.11).
  22. ^ "Arquivo Pessoa: Obra Édita - O binómio de Newton é tão belo como a Vénus de Milo". arquivopessoa.net.

Further reading

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  • 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.
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