Equidistribution theoremIn mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle , when a is an irrational number. It is a special case of the ergodic theorem where one takes the normalized angle measure . While this theorem was proved in 1909 and 1910 separately by Hermann Weyl, Wacław Sierpiński and Piers Bohl, variants of this theorem continue to be studied to this day. In 1916, Weyl proved that the sequence a, 22a, 32a, ...
Equidistributed sequenceIn mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration. A sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed on a non-degenerate interval [a, b] if for every subinterval [c, d ] of [a, b] we have (Here, the notation |{s1,.
Irrational numberIn mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Quadratic irrational numberIn mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients.
ConjectureIn mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Formal mathematics is based on provable truth.
Function field of an algebraic varietyIn algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions. In complex algebraic geometry the objects of study are complex analytic varieties, on which we have a local notion of complex analysis, through which we may define meromorphic functions.
PolynomialIn mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1. Polynomials appear in many areas of mathematics and science.
Proof that π is irrationalIn the 1760s, Johann Heinrich Lambert was the first to prove that the number pi is irrational, meaning it cannot be expressed as a fraction , where and are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction.
Algebraic expressionIn mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power 1/2, the following is also an algebraic expression: An algebraic equation is an equation involving only algebraic expressions.
Transcendental numberIn mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known – partly because it can be extremely difficult to show that a given number is transcendental – transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set.
Beal conjectureThe Beal conjecture is the following conjecture in number theory: If where A, B, C, x, y, and z are positive integers with x, y, z ≥ 3, then A, B, and C have a common prime factor. Equivalently, The equation has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 3. The conjecture was formulated in 1993 by Andrew Beal, a banker and amateur mathematician, while investigating generalizations of Fermat's Last Theorem. Since 1997, Beal has offered a monetary prize for a peer-reviewed proof of this conjecture or a counterexample.
Schanuel's conjectureIn mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. The conjecture is as follows: Given any n complex numbers z1, ..., zn that are linearly independent over the rational numbers , the field extension (z1, ..., zn, ez1, ..., ezn) has transcendence degree at least n over . The conjecture can be found in Lang (1966).
Polynomial ringIn mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers.
Square root of 2The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as or . It is an algebraic number, and therefore not a transcendental number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem.
Cramér's conjectureIn number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm.
Income statementAn income statement or profit and loss account (also referred to as a profit and loss statement (P&L), statement of profit or loss, revenue statement, statement of financial performance, earnings statement, statement of earnings, operating statement, or statement of operations) is one of the financial statements of a company and shows the company's revenues and expenses during a particular period. It indicates how the revenues (also known as the “top line”) are transformed into the net income or net profit (the result after all revenues and expenses have been accounted for).
Constant termIn mathematics, a constant term (sometimes referred to as a free term) is a term in an algebraic expression that does not contain any variables and therefore is constant. For example, in the quadratic polynomial the 3 is a constant term. After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common to speak of the quadratic polynomial where is the variable, as having a constant term of If the constant term is 0, then it will conventionally be omitted when the quadratic is written out.
Rational functionIn mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.
Algebraic numberAn algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x^2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x^4 + 4. All integers and rational numbers are algebraic, as are all roots of integers.
Transcendental number theoryTranscendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendental number The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers.
