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There are at least two definitions of hypercomplex numbers. Clifford algebraists call their higher dimensional numbers hypercomplex, even though they do not share all the ...

Let c and d!=c be real numbers (usually taken as c=1 and d=0). The Dirichlet function is defined by D(x)={c for x rational; d for x irrational (1) and is discontinuous ...

A subfield which is strictly smaller than the field in which it is contained. The field of rationals Q is a proper subfield of the field of real numbers R which, in turn, is ...

A function f:X->R is measurable if, for every real number a, the set {x in X:f(x)>a} is measurable. When X=R with Lebesgue measure, or more generally any Borel measure, then ...

A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than ...

The only linear associative algebra in which the coordinates are real numbers and products vanish only if one factor is zero are the field of real numbers, the field of ...

A description of an object by properties that are different from those mentioned in its definition, but are equivalent to them. The following list gives a number of examples. ...

Let R be the class of expressions generated by 1. The rational numbers and the two real numbers pi and ln2, 2. The variable x, 3. The operations of addition, multiplication, ...

Let V be a real symmetric matrix of large order N having random elements v_(ij) that for i<=j are independently distributed with equal densities, equal second moments m^2, ...

Let [a_0;a_1,a_2,...] be the simple continued fraction of a "generic" real number, where the numbers a_i are the partial quotients. Then the Khinchin (or Khintchine) harmonic ...