Positive real numbers  In mathematics, the set of positive real numbers, formula_1, is the subset of those real numbers that are greater than zero. The non-negative real numbers, formula_2, also include zero. Although the symbols formula_3 and formula_4 are ambiguously used for either of these, the notation formula_3 or formula_4 for formula_7 and formula_8 or formula_9 for formula_10 has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, formula_11 is identified with the positive real axis and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers formula_12 with argument formula_13. The set formula_11 is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or of an additive topological semigroup. For a given positive real number "x", the sequence {"x""n"} of its integral powers has three different fates: When "x" ∈ (0, 1) the limit is zero and when "x" ∈ (1, ∞) the limit is infinity, while the sequence is constant for "x" = 1. The case "x" > 1 thus leads to an unbounded sequence.  formula_15 and the multiplicative inverse function exchanges the intervals. The functions floor, formula_16, and excess, formula_17, have been used to describe an element formula_18 as a continued fraction formula_19 which is a sequence of integers obtained from the floor function after the excess has been reciprocated. For formula_20 the sequence terminates with an exact fractional expression of formula_21, and for quadratic irrational formula_21 the sequence becomes a periodic continued fraction. In the study of classical groups, for every formula_23, the determinant gives a map from matrices over the reals to the real numbers formula_24 Restricting to invertible matrices gives a map from the general linear group to non-zero real numbers: formula_25. Restricting to matrices with a positive determinant gives the map formula_26; interpreting the image as a quotient group by the normal subgroup relation expresses the positive reals as a Lie group.