JH WEEKLIES ISSUE #13 2011
... Prime Numbers A Prime number is a Whole number that has only two divisors (or factors)—1 and itself. Euclid proved in approximately 300 B.C. that there are infinitely-many Prime numbers. As of August 2011, the largest known Prime has nearly 13 million digits (243112609 - 1). It is greatly beneficial ...
... Prime Numbers A Prime number is a Whole number that has only two divisors (or factors)—1 and itself. Euclid proved in approximately 300 B.C. that there are infinitely-many Prime numbers. As of August 2011, the largest known Prime has nearly 13 million digits (243112609 - 1). It is greatly beneficial ...
An Investigation Relating Square and Triangular Numbers
... In yet another marginal note penned in his copy of Diophantus’ Arithmetica, Fermat wrote, in 1636, speaking of the positive integers, that … every number is either a triangular number or the sum of two or three triangular numbers; every number is a square or the sum of two, three, or four squares; e ...
... In yet another marginal note penned in his copy of Diophantus’ Arithmetica, Fermat wrote, in 1636, speaking of the positive integers, that … every number is either a triangular number or the sum of two or three triangular numbers; every number is a square or the sum of two, three, or four squares; e ...
Compare and Order Rational Numbers
... Replace the with <, >, or = to make a true statement. e. -3_ -3.625 ...
... Replace the with <, >, or = to make a true statement. e. -3_ -3.625 ...
The Number System: Operations to Add, Subtract, Multiply and
... negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, pq=p+(-q ...
... negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, pq=p+(-q ...
p. 1 Math 490 Notes 4 We continue our examination of well
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
... empty set φ is a well-ordered set (vacuously), and the ordinal containing φ is naturally denoted 0 (zero). Now consider all well-ordered sets with exactly n elements for some n ∈ N. It should be easy to see that all such well-ordered sets are similar to each other, and thus they all belong to the sa ...
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.