In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients ( or equivalently — by clearing denominators — with integer coefficients ).

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only.

In mathematics, Horner's method ( also known as Horner scheme in the UK or Horner's rule in the U. S .) is either of two things: ( i ) an algorithm for calculating polynomials, which consists in transforming the monomial form into a computationally efficient form ; or ( ii ) a method for approximating the roots of a polynomial.

In mathematics, a polynomial is an expression of finite length constructed from variables ( also called indeterminates ) and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

In advanced mathematics, polynomials are used to construct polynomial rings, a central concept in abstract algebra and algebraic geometry.

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree.

In mathematics, a transcendental number is a ( possibly complex ) number that is not algebraic — that is, it is not a root of a non-constant polynomial equation with rational coefficients.

* Equally spaced polynomial in mathematics

In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half-plane of the complex plane, that is, the real part of every zero is negative.

* Reciprocal polynomial, in mathematics

" One such example of his impact on pure mathematics is his framework for understanding the Jones polynomial using Chern – Simons theory.

In mathematics, a Laurent polynomial ( named

In mathematics, factorization ( also factorisation in British English ) or factoring is the decomposition of an object ( for example, a number, a polynomial, or a matrix ) into a product of other objects, or factors, which when multiplied together give the original.

In mathematics, a Diophantine equation is an equation of the form P ( x < sub > 1 </ sub >, ..., x < sub > j </ sub >, y < sub > 1 </ sub >, ..., y < sub > k </ sub >)= 0 ( usually abbreviated P (,)= 0 ) where P (,) is a polynomial with integer coefficients.

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in ( D + 1 )- dimensional space defined as the locus of zeros of a quadratic polynomial.

In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g ( x ) with coefficients in K such that g ( a )= 0.

A quadratic function, in mathematics, is a polynomial function of the form

In mathematics, a polynomial is said to be irreducible if it cannot be factored into the product of two or more non-trivial polynomials whose coefficients are of a specified type.

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series ; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus ; in numerical analysis as Gaussian quadrature ; in finite element methods as Shape Functions for beams ; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.

In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial.

In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to ' prove ' them.

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

In mathematics, a Cauchy sequence ( pronounced ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence:

Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics.

The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.

* Homology ( mathematics ), a procedure to associate a sequence of abelian groups or modules with a given mathematical object

In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems.

In mathematics, given an infinite sequence of numbers

In mathematics, more specifically in general topology and related branches, a net or Moore – Smith sequence is a generalization of the notion of a sequence.

In mathematics, a sequence is an ordered list of objects ( or events ).

There are various and quite different notions of sequences in mathematics, some of which ( e. g., exact sequence ) are not covered by the notations introduced below.

In mathematics a topological space is called separable if it contains a countable dense subset ; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

* Series ( mathematics ), the sum of a sequence of terms

* Word ( mathematics ), an ordered sequence of symbols chosen from a predetermined set or alphabet

The Dirac delta function as the limit ( in the sense of distribution ( mathematics ) | distributions ) of the sequence of zero-centered normal distribution s as a → 0

In applied mathematics, the delta function is often manipulated as a kind of limit ( a weak limit ) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The anthropic principle is often criticized for lacking falsifiability and therefore critics of the anthropic principle may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, " conditions that are observed in the universe must allow the observer to exist ", is " easy " to support in mathematics and philosophy, i. e. it is a tautology or truism

In ordinary language, i. e. outside of contexts such as formal logic, mathematics and programming, " or " sometimes has the meaning of exclusive disjunction.

Reacting against authors such as J. S. Mill, Sigwart and his own former teacher Brentano, Husserl criticised their psychologism in mathematics and logic, i. e. their conception of these abstract and a-priori sciences as having an essentially empirical foundation and a prescriptive or descriptive nature.

He presented a paper which posed the question of correctly formed definitions in mathematics, i. e. " how do you define a definition ?".

Hutton's mother-Sarah Balfour-insisted on his education at the High School of Edinburgh where he was particularly interested in mathematics and chemistry, then when he was 14 he attended the University of Edinburgh as a " student of humanity " i. e. Classics ( Latin and Greek ).

This change from a quasi-intensional stance to a fully extensional stance also restricts predicate logic to the second order, i. e. functions of functions: " We can decide that mathematics is to confine itself to functions of functions which obey the above assumption " ( PM 2nd Edition p. 401, Appendix C ).

) In modern mathematics, this formula can be derived using integral calculus, i. e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from where the disk has radius r ( i. e. ) to where the disk has radius 0 ( i. e. ).

In mathematics, a self-similar object is exactly or approximately similar to a part of itself ( i. e. the whole has the same shape as one or more of the parts ).

The final passages argue that logic and mathematics express only tautologies and are transcendental, i. e. they lie outside of the metaphysical subject ’ s world.

The notation e < sub > i </ sub > is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering.

In mathematics, the dimension of a vector space V is the cardinality ( i. e. the number of vectors ) of a basis of V.

In mathematics, the limit inferior ( also called infimum limit, liminf, inferior limit, lower limit, or inner limit ) and limit superior ( also called supremum limit, limsup, superior limit, upper limit, or outer limit ) of a sequence can be thought of as limiting ( i. e., eventual and extreme ) bounds on the sequence.

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure ( i. e. the composition of morphisms ) of the categories involved.

In mathematics, an integer sequence is a sequence ( i. e., an ordered list ) of integers.

An interpretation ( i. e. a semantic explanation of the formal mathematics of quantum mechanics ) can be characterized by its treatment of certain matters addressed by Einstein, such as:

In mathematics, logic and computer science, a formal language is called recursively enumerable ( also recognizable, partially decidable or Turing-acceptable ) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i. e., if there exists a Turing machine which will enumerate all valid strings of the language.

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R ( where U is an open subset of R < sup > n </ sup >) which satisfies Laplace's equation, i. e.

In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator is a power of two, i. e., a number of the form a / 2 < sup > b </ sup > where a is an integer and b is a natural number ; for example, 1 / 2 or 3 / 8, but not 1 / 3.

In mathematics, the horizontal line test is a test used to determine whether a function is injective ( i. e., one-to-one ).

< i > al </ i >., ISBN 0-231-10939-3 .</ ref > Chinese text < i > Zhou bi </ i > 周髀 ( Cullen, 1996 )< ref > Astronomy and mathematics in ancient China: the Zhou bi suan jing, 周髀算經, 1996, pg.