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Number System. a 'number system' is a set of numbers , (in the broadest sense of the word), together with one or more operations, such as addition or multiplication .
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a 'number system' is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication. • Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers. • For a history of number systems, see number. For a history of the symbols used to represent numbers, see numeral system.
Natural Numbers • are the ordinary counting numbers 1, 2, 3, ... (sometimes zero is also included). Since the development of set theory by Georg Cantor, it has become customary to view such numbers as a set. There are two conventions for the set of natural numbers: it is either the set of positiveintegers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century.
Natural numbers have two main purposes: counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. (See English numerals.) A more recent notion is that of a nominal number, which is used only for naming. • Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitionenumeration, are studied in combinatorics.
Integers • The integers (from the Latininteger, literally "untouched", hence "whole": the word entire comes from the same origin, but via French) are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. The set of all integers is often denoted by a boldface Z (or blackboard bold , Unicode U+2124 ℤ), which stands for Zahlen (. • Integers can be thought of as discrete, equally spaced points on an infinitely long number line.
Rational Number • is any number that can be expressed as the quotient a/b of two integers, with the denominatorb not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode U+211a ℚ), which stands for quotient. • A real number that is not rational is called irrational. Irrational numbers include √2, π, and e. The decimal expansion of an irrational number continues forever without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost every real number is irrational.
The rational numbers can be formally defined as the equivalence classes of the quotient setZ × N / ~, where the cartesian productZ × N is the set of all ordered pairs (m,n) where m is integer and n is natural number (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 − m2n1 = 0. • Zero divided by any other integer equals zero, therefore zero is a rational number (although division by zero itself is undefined).
Algebraic Number • In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational (or equivalently, integer) coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental.
Real Numbers • In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an infinitely long number line. • A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero. Real numbers are used to measure continuous quantities.
Complex Number • is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication. • Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis.
Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra. • Complex numbers are plotted on the complex plane, on which the real part is on the horizontal axis, and the imaginary part on the vertical axis.
P-adic Number • p-adic number system for any prime numberp extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of absolute value. • First described by Kurt Hensel in 1897[, the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of p-adic analysis essentially provides an alternative form of calculus.
More formally, for a given prime p, the fieldQp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is itself derived from an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp. This is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure which gives the p-adic number systems their power and utility. • The p in p-adic is a variable and may be replaced with a constant (yielding, for instance, "the 2-adic numbers") or another placeholder variable (for expressions such as "the l-adic numbers").
Surreal Number • 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.[1] In a rigorous set theoretic sense, 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, are subfields of the surreals. The surreals also contain all transfiniteordinal numbers reachable in the set theory in which they are constructed.
The definition and construction of the surreals is due to John Horton Conway. They were introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had simply called numbers originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book On Numbers and Games.
Hyperreal Number • The system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. Such quantities had been in widespread use in various forms for several centuries prior to the introduction of hyperreal numbers. The hyperreals, or nonstandard reals, *R, are an extension of the real numbersR that contains numbers greater than anything of the form • The hyperreal numbers satisfy the transfer principle, which states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals.
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis; some find it more intuitive than standard real analysis.
Prepared by: Manilynn Lim III-B
