Implementing real numbers with RZ
... convergence which usually results in estimates that overshoot the precision. We end up computing much more than is needed. Our implementation via the interval domain does not have this drawback. ...
... convergence which usually results in estimates that overshoot the precision. We end up computing much more than is needed. Our implementation via the interval domain does not have this drawback. ...
Multiplying and Dividing Rational Numbers 2.4
... ACTIVITY: Multiplying by −1 Work with a partner. a. Graph each number below on three different number lines. Then multiply each number by −1 and graph the product on the appropriate number line. ...
... ACTIVITY: Multiplying by −1 Work with a partner. a. Graph each number below on three different number lines. Then multiply each number by −1 and graph the product on the appropriate number line. ...
1.2 The Integers and Rational Numbers
... Recap: Rational numbers are equivalence classes of integer fractions, and they have a very satisfactory arithmetic, with additive inverses and multiplicative inverses (of everything except 0) allowing us to define subtraction and exact division (by anything except 0). On the other hand, from the poi ...
... Recap: Rational numbers are equivalence classes of integer fractions, and they have a very satisfactory arithmetic, with additive inverses and multiplicative inverses (of everything except 0) allowing us to define subtraction and exact division (by anything except 0). On the other hand, from the poi ...
Problems for Chapter 1
... Try to find a number in the form q whose decimal representation consists of a very long repeating part. Is there any limit to how long this repeating part can be? Prove that the sum of two rational numbers is rational. Prove that the product of two rational numbers is rational Prove that the sum of ...
... Try to find a number in the form q whose decimal representation consists of a very long repeating part. Is there any limit to how long this repeating part can be? Prove that the sum of two rational numbers is rational. Prove that the product of two rational numbers is rational Prove that the sum of ...
Completed versus Incomplete Infinity in Arithmetic
... a non-zero number y 0 that is divisible by all non-zero numbers z with z ≤ x. By induction, we need only prove the inductive step, that Sx has the property p; that is, that there exists a non-zero number y that is divisible by every non-zero number z with z ≤ Sx. But this is true: let y = y 0 · Sx a ...
... a non-zero number y 0 that is divisible by all non-zero numbers z with z ≤ x. By induction, we need only prove the inductive step, that Sx has the property p; that is, that there exists a non-zero number y that is divisible by every non-zero number z with z ≤ Sx. But this is true: let y = y 0 · Sx a ...
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations.In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.Research on the go endgame by John Horton Conway led to a simpler definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.