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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
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
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
mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian numbers, Gaussian polynomials, or q-binomial coefficients) are q-analogs
Gaussian_binomial_coefficient
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
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 distribution
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes
Binomial_distribution
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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)
(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
binomial theorem, which he had extended to include fractional and negative exponents. Newton succeeded in expanding the applicability of the binomial
History_of_calculus
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
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
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
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
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
_{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
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
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
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
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
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
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
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
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
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
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
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
Theorem in p-adic analysis
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
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
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
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
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
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
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
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
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
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
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
function Octic function Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root (radical)
List_of_polynomial_topics
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
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
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
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
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
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
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
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
(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
Effects of pore diffusion on Rate of heterogeneous chemical reaction
dr} for small values of beta this can be approximated using the binomial theorem: η = 1 − n β 4 {\displaystyle \eta =1-{\dfrac {n\beta }{4}}} Assuming
Weisz–Prater_criterion
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)
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
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
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)
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
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
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
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
genetics Beta-galactosidase Bimodal distribution Binary fission Binomial expansion Binomial theorem Biochemical genetics Bioinformatics Biolistic Bioremediation
Index_of_genetics_articles
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
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
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
) {\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
Recurrence relations of binomial coefficients in Pascal's triangle
{X^{r}-X^{n+1}}{1-X}}={\frac {X^{n+1}-X^{r}}{x}}} . Further, by the binomial theorem, we also find that X r + k = ( 1 + x ) r + k = ∑ i = 0 r + k ( r +
Hockey-stick_identity
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
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
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
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
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
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
1915 Sherlock Holmes 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_Valley_of_Fear
12th-century Muslim mathematician, astronomer and physician
without stating them explicitly. He used this to extend results for the binomial theorem up to n=12 and Pascal's triangle previously given by al-Karaji. Arabic
Al-Samawal_al-Maghribi
Ethnic group native to Italy
established the foundation of probability and introduced the binomial coefficients and the binomial theorem; he also invented several mechanical devices.[citation
Italians
Solved prime-number problem
{\displaystyle {\frac {4^{n}}{2n}}\leq {\binom {2n}{n}}.} Proof: Applying the binomial theorem, 4 n = ( 1 + 1 ) 2 n = ∑ k = 0 2 n ( 2 n k ) = 2 + ∑ k = 1 2 n − 1
Proof_of_Bertrand's_postulate
Relativistic equation relating total energy to mass and momentum
{1+\left({\frac {p}{mc}}\right)^{2}}}\,,} and expanding into power series by the binomial theorem (or a Taylor series) around ( p m c ) 2 = 0 {\displaystyle \left({\frac
Energy–momentum_relation
Product of numbers from 1 to n
{\displaystyle n!/e} . In algebra, the factorials arise through the binomial theorem, which uses binomial coefficients to expand powers of sums. They also occur in
Factorial
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
Girl/Female
Assamese, Bengali, Celebrity, Gujarati, Hindu, Indian, Jain, Kannada, Malayalam, Marathi, Mythological, Oriya, Sanskrit, Sindhi, Tamil, Telugu, Traditional
Worship; Prayer; Idol Worship; Derived from Pooja; Celebration of Worship
Male
English
Anglicized form of Hebrew Abiyram, ABIRAM means "my father is exalted." In the bible, this is the name of the eldest son of Hiel the Bethelite, and the name of a son of Eliab who joined Korah in his rebellion against Moses.
Male
Slovene
Slovene pet form of Alojz, LOJZE means "famous warrior."
Surname or Lastname
English
English : unexplained; possibly a variant of Burdett.Robert Burdick was a freeman of Newport, RI, in 1655.
Boy/Male
Muslim
Eminent, Exalted, High, Sublime, Similar
Biblical
to whisper; an embalmer; a village of the tribe of David
Boy/Male
Indian, Punjabi, Sikh
Love for the Proximity of God
Boy/Male
Tamil
Bal Bhadra | பலபதà¯à®°
(Brother of Krishna)
Boy/Male
Tamil
Lord Vishnu and Lord Shiva
Boy/Male
Arabic, Muslim
Sword of the Faith
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
BINOMIAL THEOREM
a.
Consisting of three terms; of or pertaining to trinomials; as, a trinomial root.
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.
a.
Consisting of but a single term or expression.
n.
A rule or principle expressed in algebraic language; as, the binominal formula.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
v. t.
To formulate into a theorem.
a.
Consisting of two terms; pertaining to binomials; as, a binomial root.
a.
Binominal.
n.
One who constructs theorems.
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.
a.
Of or pertaining to two names; binomial.
n.
A monomial.
n.
A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality.
n.
An expression consisting of two terms connected by the sign plus (+) or minus (-); as, a + b, or 7 - 3.
a.
Theorematic.
n.
A name or term.
n.
A quantity consisting of three terms, connected by the sign + or -; as, x + y + z, or ax + 2b - c2.
n. & a.
Trinomial.
a.
Alt. of Theorematical