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Algebraic expansion of powers of a binomial
algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power
Binomial_theorem
Generalization of the binomial theorem to other polynomials
multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from
Multinomial_theorem
Family of polynomials
Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients
Gaussian_binomial_coefficient
Mathematical series
In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where α {\displaystyle
Binomial_series
Mathematical fallacy
known as freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or the Frobenius identity is the generally-false
Freshman's_dream
Number theory theorem
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ( m n ) {\displaystyle {\tbinom {m}{n}}} by a prime
Lucas's_theorem
Number of subsets of a given size
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is
Binomial_coefficient
Probability distribution
limit theorem. The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution
Binomial_distribution
Approximation of powers of some binomials
approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality, the left-hand side of the approximation
Binomial_approximation
Fictional book mentioned in stories of Sherlock Holmes
A Treatise on the Binomial Theorem is a fictional work of mathematics by the young Professor James Moriarty, the criminal mastermind and archenemy of the
A Treatise on the Binomial Theorem
A_Treatise_on_the_Binomial_Theorem
Mathematical identity involving sums of binomial coefficients
Abel's binomial theorem, named after Niels Henrik Abel, is a mathematical identity involving sums of binomial coefficients. It states the following: ∑
Abel's_binomial_theorem
Probability Theory
limit theorem states that the Poisson distribution may be used as an approximation to the binomial distribution, under certain conditions. The theorem was
Poisson_limit_theorem
Probability distribution
In probability theory and statistics, the negative binomial distribution, also called a Pascal distribution, is a discrete probability distribution that
Negative binomial distribution
Negative_binomial_distribution
Addition of several numbers or other values
{\displaystyle n^{k}=\sum _{i=0}^{n-1}\left((i+1)^{k}-i^{k}\right).} Using binomial theorem, this may be rewritten as: n k = ∑ i = 0 n − 1 ( ∑ j = 0 k − 1 ( k
Summation
Triangular array of the binomial coefficients
Several theorems related to the triangle were known, including the binomial theorem. Khayyam used a method of finding nth roots based on the binomial expansion
Pascal's_triangle
Persian polymath and poet (1048–1131)
the importance of a general binomial theorem. The argument supporting the claim that Khayyam had a general binomial theorem is based on his ability to
Omar_Khayyam
Mathematical set of all subsets of a set
numbers, in which case we cannot enumerate all irrational numbers. The binomial theorem is closely related to the power set. A k–elements combination from
Power_set
Q-analog of hypergeometric series
_{k=1}^{N}\left(1+yq^{k}\right)} of the q-binomial theorem (also sometimes known as the Cauchy binomial theorem). Here [ N n ] q {\displaystyle
Basic_hypergeometric_series
Topics referred to by the same term
of binomials Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition Binomial theorem, a theorem about powers of binomials Binomial type
Binomial
which is the statement of the theorem for a = k+1. ∎ In order to prove the lemma, we must introduce the binomial theorem, which states that for any positive
Proofs of Fermat's little theorem
Proofs_of_Fermat's_little_theorem
2.71828…, base of natural logarithms
characterizations using the limit and the infinite series can be proved via the binomial theorem. Jacob Bernoulli discovered this constant in 1683, while studying a
E_(mathematical_constant)
Generalization of the product rule in calculus
The Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking
General_Leibniz_rule
Mathematical expression with disputed status
= 1 is necessary for many polynomial identities. For example, the binomial theorem ( 1 + x ) n = ∑ k = 0 n ( n k ) x k {\textstyle (1+x)^{n}=\sum _{k=0}^{n}{\binom
Zero_to_the_power_of_zero
In mathematics, a polynomial with two terms
(ax+b)(cx+d)=acx^{2}+(ad+bc)x+bd.} A binomial raised to the nth power, represented as (x + y)n can be expanded by means of the binomial theorem or, equivalently, using
Binomial_(polynomial)
Describes the highest power of primes dividing a binomial coefficient
mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other
Kummer's_theorem
English polymath (1642–1727)
He generalised the binomial theorem to any real number, introduced the Puiseux series, was the first to state Bézout's theorem, classified most of the
Isaac_Newton
Inequality about exponentiations of ''1+x''
again (4). One can prove Bernoulli's inequality for x ≥ 0 using the binomial theorem. It is true trivially for r = 0, so suppose r is a positive integer
Bernoulli's_inequality
Method for division with remainder
method can be used with factors that allow simplifications by the binomial theorem. Assume N / D {\displaystyle N/D} has been scaled by a power of
Division_algorithm
Result in number theory
parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by
Wolstenholme's_theorem
Branch of discrete mathematics
astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist
Combinatorics
Function in discrete mathematics
the star denotes complex conjugation. The Plancherel theorem is a special case of Parseval's theorem and states: ∑ n = 0 N − 1 | x n | 2 = 1 N ∑ k = 0 N
Discrete_Fourier_transform
filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient
List of factorial and binomial topics
List_of_factorial_and_binomial_topics
Fictional character from Sherlock Holmes stories
gains recognition at the age of 21 for writing "a treatise upon the Binomial Theorem", which leads to his being awarded the Mathematical Chair at one of
Professor_Moriarty
Result in enumerative combinatorics and linear algebra
[x_{1}^{k}x_{2}^{2n-k}](x_{1}-x_{2})^{2n},} which are computed from the binomial theorem. On the other hand, we can compute the determinant explicitly: det
MacMahon's_master_theorem
binomial theorem, which he had extended to include fractional and negative exponents. Newton succeeded in expanding the applicability of the binomial
History_of_calculus
Convergence in distribution of binomial to normal distribution
Moivre–Laplace theorem, which is a special case of the central limit theorem, states that the normal distribution may be used as an approximation to the binomial distribution
De_Moivre–Laplace_theorem
Mathematical connection between field theory and group theory
this case, may be replaced by formula manipulations involving the binomial theorem. One might object that A and B are related by the algebraic equation
Galois_theory
(graph theory) Abel's binomial theorem (combinatorics) Alspach's theorem (graph theory) Aztec diamond theorem (combinatorics) BEST theorem (graph theory) Baranyai's
List_of_theorems
Method of differentiating single-term polynomials
the terms cancel. This proof only works for natural numbers as the binomial theorem only works for natural numbers. Let y = x n {\displaystyle y=x^{n}}
Power_rule
Italian Renaissance polymath (1501–1576)
in the foundation of probability; he introduced the binomial coefficients and the binomial theorem in the Western world. He wrote more than 200 works on
Gerolamo_Cardano
Mathematical theorem on convolved binomial coefficients
ai = 0 for all integers i > m and bj = 0 for all integers j > n. By the binomial theorem, ( 1 + x ) m + n = ∑ r = 0 m + n ( m + n r ) x r . {\displaystyle (1+x)^{m+n}=\sum
Vandermonde's_identity
Norwegian mathematician (1802–1829)
work of a crank. As a 16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only
Niels_Henrik_Abel
Generalization of Pythagorean theorem
geometric proof than using the Pythagorean theorem alone. Algebraic manipulations (in particular the binomial theorem) are avoided. Case of acute angle γ, where
Law_of_cosines
_{k=0}^{\infty }{\alpha \choose k}z^{k},|z|<1} (see Binomial theorem § Newton's generalized binomial theorem) ∑ k = 0 ∞ ( α + k − 1 k ) z k = 1 ( 1 − z ) α
List_of_mathematical_series
Arithmetic operation, inverse of nth power
determine cube roots. In 1665, Isaac Newton discovered the general binomial theorem, which can convert an nth root into an infinite series. Based on approach
Nth_root
Mathematical result on arithmetic properties of binomial coefficients
The Star of David theorem is a mathematical result on arithmetic properties of binomial coefficients. It was discovered by Henry W. Gould in 1972. The
Star_of_David_theorem
Type of polynomial used in Numerical Analysis
k}x^{k}(1-x)^{n-k}={x(1-x) \over n}.} ("variance") In fact, by the binomial theorem ( 1 + t ) n = ∑ k ( n k ) t k , {\displaystyle (1+t)^{n}=\sum _{k}{n
Bernstein_polynomial
Expression for sums of powers
nonzero elements of T ( h , d ) {\displaystyle T(h,d)} follow the binomial theorem, and that A {\displaystyle A} is just Pascal's triangle with each row's
Faulhaber's_formula
3rd–2nd century BC Indian mathematician and poet
Indian mathematicians History of the binomial theorem List of Indian mathematicians Amulya Kumar Bag, 'Binomial theorem in ancient India', Indian J. Hist
Pingala
and trigonometry. Topics in algebra include the binomial theorem, complex numbers, the Fundamental Theorem of Algebra, root extraction, polynomial long division
Mathematics education in the United States
Mathematics_education_in_the_United_States
numerals used throughout the world. Binomial theorem: The first formulation of the binomial theorem and the table of binomial coefficient can be found in a
List of inventions in the medieval Islamic world
List_of_inventions_in_the_medieval_Islamic_world
Mathematical operation on random variables
of Mathematics Florin Avram and Murad Taqqu, (1987) "Noncentral Limit Theorems and Appell Polynomials", Annals of Probability, volume 15, number 2, pages
Wick_product
Form of mathematical proof
around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle. Whilst the original work was lost
Mathematical_induction
Course designed to prepare students for calculus
exercised with trigonometric functions and trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences
Precalculus
Fundamental theorem in probability theory and statistics
of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem. Let ( X
Central_limit_theorem
Things confirmed about Sherlock Holmes in Sir Arthur Conan Doyle's stories
Club The Dynamics of an Asteroid Reichenbach Falls A Treatise on the Binomial Theorem Studies Sherlockian game Holmesian studies The New Annotated Sherlock
Canon_of_Sherlock_Holmes
In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special
Mahler's_theorem
combinatorics of meters corresponds to an elementary version of the binomial theorem. Pingala's work also contains the basic ideas of Fibonacci numbers
History_of_mathematics
Technique for proving sets have equal size
the pentagonal number theorem. Bijective proofs of the formula for the Catalan numbers. Binomial theorem Schröder–Bernstein theorem Double counting (proof
Bijective_proof
1887 detective novel by Arthur Conan Doyle
Club The Dynamics of an Asteroid Reichenbach Falls A Treatise on the Binomial Theorem Studies Sherlockian game Holmesian studies The New Annotated Sherlock
A_Study_in_Scarlet
Concept in mathematics
the Commens Dictionary of Peirce's Terms. Cut the knot: Sample corollaries of the Pythagorean theorem Geeks for geeks: Corollaries of binomial theorem
Corollary
sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context. 3rd century
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Type of polynomial sequence
the binomial theorem can be stated by saying that the sequence { x n : n = 0 , 1 , 2 , … } {\displaystyle \{x^{n}:n=0,1,2,\ldots \}} is of binomial type
Binomial_type
1902 crime detective novel by Arthur Conan Doyle
Club The Dynamics of an Asteroid Reichenbach Falls A Treatise on the Binomial Theorem Studies Sherlockian game Holmesian studies The New Annotated Sherlock
The_Hound_of_the_Baskervilles
Reasoning for mathematical statements
introduced in the Al-Fakhri (1000) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Modern proof theory treats proofs
Mathematical_proof
Gilbert & Sullivan song
mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news, With many cheerful facts about the
Major-General's_Song
which flourished for several hundred years. "He also discovered the binomial theorem for integer exponents, which "was a major factor in the development
Timeline_of_mathematics
Algorithm checking for prime numbers
This theorem is a generalization to polynomials of Fermat's little theorem. In one direction it can easily be proved using the binomial theorem together
AKS_primality_test
Annual entrance test held in India
Quadratic Equations Linear Inequalities Permutations and Combinations Binomial Theorem Sequences and Series Straight Lines Conic Sections Three Dimensional
IISER_Aptitude_Test
Topics referred to by the same term
Binomial identity may refer to: Binomial theorem Binomial type Binomial (disambiguation) This disambiguation page lists articles associated with the title
Binomial_identity
Counting technique in combinatorics
{t}{1}}-{\binom {t}{2}}+\cdots +(-1)^{t+1}{\binom {t}{t}}.\end{aligned}}} By the binomial theorem, 0 = ( 1 − 1 ) t = ( t 0 ) − ( t 1 ) + ( t 2 ) − ⋯ + ( − 1 ) t ( t
Inclusion–exclusion_principle
Function defined by a hypergeometric series
or equal to 1. This can be proved by expanding (1 − zx)−a using the binomial theorem and then integrating term by term for z with absolute value smaller
Hypergeometric_function
Sigmoid shape special function
{\displaystyle Q^{n}(x)} for positive integers n {\displaystyle n} via the binomial theorem, suggesting potential adaptability for powers of erfc ( x ) {\displaystyle
Error_function
Graphical aid for deriving some concepts in combinatorics
n = 10 and k = 4, the theorem gives the number of solutions to x1 + x2 + x3 + x4 = 10 (with x1, x2, x3, x4 > 0) as the binomial coefficient ( n − 1 k
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
(Mathematical) decomposition into a product
) {\displaystyle x^{4}+x^{2}+1=(x^{2}+x+1)(x^{2}-x+1)} Binomial expansions The binomial theorem supplies patterns that can easily be recognized from the
Factorization
Statistical confidence interval for success counts
Moivre–Laplace theorem (the original, binomial-only version of the central limit theorem) and becomes unreliable when it violates the theorems' premises,
Binomial proportion confidence interval
Binomial_proportion_confidence_interval
Chinese mathematician and writer (c. 1238–1298)
Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui's triangle
Yang_Hui
Mathematical rule for inverting probabilities
Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes (/beɪz/), gives a mathematical rule for inverting conditional probabilities
Bayes'_theorem
Curve used in computer graphics and related fields
of less straightforward Bernstein polynomials. Application of the binomial theorem to the definition of the curve followed by some rearrangement will
Bézier_curve
Concept in combinatorics (part of mathematics)
}{\frac {x^{n}}{(q;q)_{n}}},} which are both special cases of the q-binomial theorem: ( a x ; q ) ∞ ( x ; q ) ∞ = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n x n
Q-Pochhammer_symbol
Type of proof technique
{n}{k}}={\binom {n}{n-k}}.} In fact, one can use similar reasoning to prove the binomial theorem. One example of the double counting method counts the number of ways
Double counting (proof technique)
Double_counting_(proof_technique)
Algebraic structure
characteristic p {\displaystyle p} . This follows from the binomial theorem, as each binomial coefficient of the expansion of ( x + y ) p {\displaystyle
Finite_field
Identity expressing an integral as a sum
The correct result in a general commutative context is given by the binomial theorem. All the terms vanish at 0 because lim x → 0 + x m ( log x ) n =
Sophomore's_dream
Describes approximate behavior of a function
M_{b}\cdot x.} This particular statement follows from the general binomial theorem. Another example, common in the theory of Taylor series, is e x = 1
Big_O_notation
Mathematical concept
_{k=0}^{n}{\frac {x^{k}}{k!}},\qquad e^{x}=\lim _{n\to \infty }s_{n}.} By the binomial theorem, t n = ∑ k = 0 n ( n k ) x k n k = 1 + x + ∑ k = 2 n n ( n − 1 ) (
Characterizations of the exponential function
Characterizations_of_the_exponential_function
) {\textstyle \tau ^{p}{\pmod {p}}} using the binomial theorem. Because the cross terms in the binomial expansion all contain factors of p, we find that
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Mathematical notation
_{|\alpha |=k}{\binom {k}{\alpha }}\,x^{\alpha }} Multi-binomial theorem ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha
Multi-index_notation
Theorem in calculus
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through
Divergence_theorem
Result from multiplying no factors
and implies that x0 = 1 for all x), Stirling number, König's theorem, binomial type, binomial series, difference operator and Pochhammer symbol. Since logarithms
Empty_product
Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not
History_of_trigonometry
Fundamental study of potential theory
laborious way of achieving the same result is by using the generalized binomial theorem. The resulting series is the generating function for the Legendre polynomials:
Gravitational_potential
average rate of change binomial coefficient Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly
Glossary_of_calculus
Rational number sequence
(B+1)^{m}-B_{m}=0,} where the power is expanded formally using the binomial theorem and B k {\displaystyle B^{k}} is replaced by B k {\displaystyle B_{k}}
Bernoulli_number
Algorithms for calculating square roots
the work of François Viète, published c. 1600., and is based on the binomial theorem and is essentially an inverse algorithm solving ( x + y ) 2 = x 2 +
Square_root_algorithms
1890 detective novel by Arthur Conan Doyle
Club The Dynamics of an Asteroid Reichenbach Falls A Treatise on the Binomial Theorem Studies Sherlockian game Holmesian studies The New Annotated Sherlock
The_Sign_of_the_Four
Mnemonic for finding the product of two binomial functions
algebra, FOIL is a mnemonic for the standard method of multiplying two binomials—hence the method may be referred to as the FOIL method. The word FOIL
FOIL_method
Number with an integer power equal to 1
{\displaystyle {\frac {(z+1)^{n}-1}{(z+1)-1}},} and expanding via the binomial theorem. Every nth root of unity is a primitive dth root of unity for exactly
Root_of_unity
Latin phrase referring to several years during which events of major importance occurred
University by an outbreak of the plague. He stated and proved the binomial theorem, discovered calculus, formulated the universal law of gravitation,
Annus_mirabilis
Identity in mathematical combinatorics
proofs of the q-Vandermonde identity. The following proof uses the q-binomial theorem. One standard proof of the Chu–Vandermonde identity is to expand the
Q-Vandermonde_identity
Sum of inverse squares of natural numbers
x}{\sin x}}\right)^{n}\\[4pt]&=(\cot x+i)^{n}.\end{aligned}}} From the binomial theorem, we have ( cot x + i ) n = ( n 0 ) cot n x + ( n 1 ) ( cot n −
Basel_problem
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
Female
English
Variant spelling of English Star, STARR means "star."
Boy/Male
Muslim
Lightning
Boy/Male
Latin American English French
Hammer.
Boy/Male
Arabic, Muslim
Decorated Throne
Girl/Female
Tamil
Heart
Girl/Female
Arabic
Sheep; Goat; Name of a Valley Between Makkah and Taif; Shinning Star; Garden of Roses
Boy/Male
French American
Rejoicing.
Surname or Lastname
English
English : habitational name from either of two places called Butterley, in Derbyshire and Herefordshire, or from Butterleigh in Devon. All are named with Old English butere ‘butter’ + lēah ‘pasture’.
Boy/Male
Tamil
Soumyajyoti | ஸோஉஂமà¯à®¯à®¾à®œà¯à®¯à¯‹à®¤à¯€
Girl/Female
Hindu, Indian
Beautiful Moodek
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
n.
A quantity consisting of three terms, connected by the sign + or -; as, x + y + z, or ax + 2b - c2.
n.
An expression of the condition of equality between two algebraic quantities or sets of quantities, the sign = being placed between them; as, a binomial equation; a quadratic equation; an algebraic equation; a transcendental equation; an exponential equation; a logarithmic equation; a differential equation, etc.
n.
One who constructs theorems.
a.
Consisting of two terms; pertaining to binomials; as, a binomial root.
v. t.
To formulate into a theorem.
a.
Binominal.
n.
A name or term.
a.
Consisting of three terms; of or pertaining to trinomials; as, a trinomial root.
n.
An expression consisting of two terms connected by the sign plus (+) or minus (-); as, a + b, or 7 - 3.
a.
Having two names; -- used of the system by which every animal and plant receives two names, the one indicating the genus, the other the species, to which it belongs.
n.
A monomial.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
n.
A numerical coefficient in any particular case of the binomial theorem.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
a.
Consisting of but a single term or expression.
a.
Alt. of Theorematical
n. & a.
Trinomial.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
a.
Of or pertaining to two names; binomial.
a.
Theorematic.