181
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − '''Real numbers''' include within them all of these other kinds of [[number]]s:+ − + − + − + − + − *Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions+ − *[[Irrational numbers]], like <math>\sqrt{10} = 3.162277660168...</math>π = 3.1415926535..., whose decimal representations never repeat or terminate.+ + + + + + + + + + Line 21:
Line 30: − − ==Real line== − − The real numbers can be thought of as a [[line]], called the '''real line'''. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a [[dense set]], so points accumulate around each other.<ref>http://abstractmath.org/MM/MMRealNumbers.htm</ref> − − The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[x-axis]] and [[y-axis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]]. − +
Fix many things; more to come
{{Math-m}}
*The positive [[integer]]s, 1, 2, 3, ...
The '''real numbers''' are a set of numbers with extremely important theoretical and practical properties. They can be considered the numbers used for ordinary measurement of physical things like length, area, weight, charge, etc. They are the 4<sup>th</sup> item in this hierarchy of types of [[number]]s:
*[[Zero]] and the negative integers
*[[Fraction]]s, like 355/113
*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
*Any decimal representation which terminates (comes to an end), like 6.023, because this is just a way of writing a fraction (in this case, 6023/1000)
*The "[[integer]]s"—positive, negative, and [[zero]]
*The "[[rational number]]s", or [[fraction]]s, like 355/113
*The "real numbers", including irrational numbers
*The "[[complex number]]s, which give solutions to polynomial equations
Real numbers are typically represented by a decimal (or any other base) representation, as in 3.1416. It can be shown that any decimal representation that either terminates or gets into an endless repeating pattern is rational. The other numbers are real numbers that are ''irrational''. Examples are <math>\sqrt{10} = 3.162277660168...\,</math> and <math>\pi = 3.1415926535...\,</math>. These decimal representations neither repeat nor terminate.
==Formal definition==
==Formal definition==
Formally, real numbers are defined as the unique [[Field (mathematics)|field]] which is [[ordered]], [[Complete (mathematics)|complete]], and [[Archimedean]]. The reals can be constructed from the [[rationals]] by means of [[Dedekind cut]]s or [[Cauchy sequence]]s, i.e. it is the completion of the [[metric space]] of rational numbers.
Formally, real numbers are defined as the unique [[Field (mathematics)|field]] which is [[ordered]], [[Complete (mathematics)|complete]], and [[Archimedean]]. The reals can be constructed from the [[rationals]] by means of [[Dedekind cut]]s or [[Cauchy sequence]]s, i.e. it is the completion of the [[metric space]] of rational numbers.
==Real line==
The real numbers can be thought of as a [[line]], called the '''real line'''. Each real number represents a point on the real line. However, it is a mistake to think of the real line as a row of individual points, like beads. There is no real number “just to the right” of a given real number. This is because the real numbers, like the rational numbers, are a [[dense set]], so points accumulate around each other.<ref>http://abstractmath.org/MM/MMRealNumbers.htm</ref>
The real line is useful as a [[coordinate system]] for [[graph]]ing [[functions]]. Thus, the [[x-axis]] and [[y-axis]] are both instances of the real line. The real line is the basis for geometric [[measurement]]s, and more generally for ideas in [[metric topology]].
==Infinity==
==Infinity==
It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."
It is sometimes convenient to have a set of numbers that ''does'' include infinity. For example, in computer programming, "real arithmetic" is often done by a specific system defined by standard IEEE 754-1985; this system is built in to modern processor chips. It provides for values which print out as INF and -INF and which participate in arithmetic as if they were numbers. Thus, division by zero, which was often an error that stopped calculation on older machines, can be a legal operation which simply produces a +INF or -INF result. The system of numbers implemented in IEEE 754 is known in mathematics as the "affinely extended real numbers."
==History==
==History==
The first serious consideration of the real numbers was by [[Archimedes]] and followed by other Greeks <!--Don't wikilink "Greek". The page refers to the language.-->such as [[Euclid]], [[Pappus]], and [[Zeno]]. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject subsumed all geometry known at the time, creating the field of [[analytic geometry]].
The ancient Greek <!--Don't wikilink "Greek". The page refers to the language.-->mathematicians ([[Archimedes]], [[Euclid]], [[Pappus]], [[Pythagoras]] and [[Zeno]]) are perhaps the first people to have created the abstract notion of a "number" (a real number; not an integer) to represent a geometrical measurement. They developed the correspondence between numbers and measurements such as distances, areas, and angles. To honor Archimedes' contribution, real analysts have named a property of the real numbers the [[Archimedean|Archimedean property]]. Real analysis remained in [[geometry]]'s shadow until the development of the subfield of [[calculus]]. This subject subsumed all geometry known at the time, creating the field of [[analytic geometry]].
==Notes and references==
==Notes and references==