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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
American Airlines is a major airline in the United States headquartered in Fort Worth, Texas, within the Dallas–Fort Worth metroplex. It is the largest airline in the world when measured by scheduled passengers carried, revenue passenger mile. American, together with its regional partners and affiliates, operates an extensive international ...
Binomial distribution for = with n and k as in Pascal's triangleThe probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is /. In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no ...
- History
- Statement
- Examples
- Binomial Coefficients
- Proofs
- Generalizations
- Applications
- In Abstract Algebra
- In Popular Culture
- See Also
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 n = 2 {\displaystyle n=2} . Greek mathematician Diophantus cubed various binomials, including x − 1 {\displaystyle x-1} . Indian mathematician Aryabhata's method for findi...
According to the theorem, the expansion of any nonnegative integer power n of the binomial x + yis a sum of the form 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 fir...
Here are the first few cases of the binomial theorem: 1. the exponents of x in the terms are n, n − 1, ..., 2, 1, 0 (the last term implicitly contains x0 = 1); 2. the exponents of y in the terms are 0, 1, 2, ..., n − 1, n (the first term implicitly contains y0 = 1); 3. the coefficients form the nth row of Pascal's triangle; 4. before combining like...
The coefficients that appear in the binomial expansion are called binomial coefficients. These are usually written ( n k ) , {\displaystyle {\tbinom {n}{k}},} and pronounced "n choose k".
Combinatorial proof
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 xn−kyk for some k between 0 and n. For a given k, the following are proved equal in succession: 1. the number of terms equal to xn−kykin the expansion 2. the number of n-character x,y strings having y in exactly kpositions 3. the number of k-element subsets of {1, 2, ..., n} 4. ( n k ) , {\displaystyle {\tbinom {n}{k}},} either by definition,...
Inductive proof
Induction yields another proof of the binomial theorem. When n = 0, both sides equal 1, since x0 = 1 and ( 0 0 ) = 1. {\displaystyle {\tbinom {0}{0}}=1.} 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 ( n k ) {\displaystyle {\tbinom {n}{k}}} if j + k = n, and 0otherwise. The identity
Newton's generalized binomial theorem
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 When r is a nonnegative intege...
Further generalizations
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 s...
Multinomial theorem
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 ( n k 1 , ⋯ , k m ) {\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}} are known as multinomial coefficients, and can be computed by the formula Combinatorially, the multin...
Multiple-angle identities
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
Series for e
The number eis 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 ecan 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
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 { X t } t ∈ S {\displaystyle \{X_{t}\}_{t\in S}} with probability of success p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} all not happening is 1. P ( ⋂ t ∈ S X t C ) = ( 1 − p ) | S | = ∑ n = 0 | S | ( | S | n ) ( − p ) n . {\displaystyle P\left(\bigcap _{t\in S}X_{t}^{C}\right)=(1-p)^{|S|}=\sum _{n=0}^{|S|}{|S|...
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 × nmatrices, provided that those matrices commute; this is useful in computing powers of a matrix. The binomial theorem can be stated by saying that the polynomial sequence {1, x, x2, x3, ...} i...
The binomial theorem is mentioned in the Major-General's Song in the comic opera The Pirates of Penzance.Professor Moriarty is described by Sherlock Holmes as having written a treatise on the binomial theorem.The Portuguese poet Fernando Pessoa, using the heteronym Álvaro de Campos, wrote that "Newton's Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it."In the 2014 film The Imitation Game, Alan Turing makes reference to Isaac Newton's work on the binomial theorem during his first meeting with Commander Denniston at Bletchley Park.Gross domestic product (GDP) is the market value of all final goods and services from a nation in a given year. [2] Countries are sorted by nominal GDP estimates from financial and statistical institutions, which are calculated at market or government official exchange rates. Nominal GDP does not take into account differences in the ...
The Northrop B-2 Spirit, also known as the Stealth Bomber, is an American heavy strategic bomber, featuring low-observable stealth technology designed to penetrate dense anti-aircraft defenses.
Premise Set thousands of years before the events of the novels The Hobbit and The Lord of the Rings by J. R. R. Tolkien, the series is based on the author's history of Middle-earth.It begins during a time of relative peace and covers the major events of Middle-earth's Second Age: the forging of the Rings of Power, the rise of the Dark Lord Sauron, the fall of the island kingdom of Númenor ...
