The binomial theorem is a proposition of mathematics which , in its simplest form, allows the powers of a binomial , i.e. an expression of the form
as a polynomial of the -th degree in the variables and .
In algebra , the binomial theorem indicates how to multiply an expression of the form .
Binomial theorem for natural exponents
For all elements and a commutative unitary ring and for all natural numbers the equation applies:
In particular, this is true for real or complex numbers and (by convention ).
The coefficients of this polynomial expression are the binomial coefficients
which got their name because of their appearance in the binomial theorem. With which this Faculty of designated.
The terms are to be understood as a scalar multiplication of the whole number to the ring element , i.e. H. Here, the ring than in his capacity - module used.
The binomial theorem for the case is called the first binomial formula .
- The binomial theorem also applies to elements and in arbitrary unitary rings , provided that only these elements commute with one another, i.e. H. applies.
- The existence of the one in the ring is also dispensable, provided that the proposition is paraphrased in the following form:
The proof for any natural number can be given by complete induction . This formula can also be obtained for each concrete by multiplying it out.
, where is the imaginary unit .
Binomial series, theorem for complex exponents
A generalization of the theorem to any real exponent by means of infinite series is thanks to Isaac Newton . But the same statement is also valid if is any complex number .
The binomial theorem is in its general form:
This series is called the binomial series and converges for all with and .
In the special case , equation (2) changes into (1) and is then even valid for all , since the series then breaks off.
The generalized binomial coefficients used here are defined as
In this case , an empty product is created, the value of which is defined as 1.
For and , the geometric series results from (2) as a special case .
↑ Wikibook's evidence archive: Algebra: Rings: Binomial Theorem