Difference between revisions of "Real numbers"

Revision as of 22:53, April 11, 2007

Real numbers can be thought of as numbers which can be represented by some infinite or finite decimal representation, such as 0.707106781187...

In classical physics, measurements of things that can vary smoothly and continuously, like speed or temperature, are treated as real numbers.

In a number line representation, the real numbers correspond to all the points on a geometric line. The distance between any two points on a line is a real number.

In computer programming, some computer languages such as FORTRAN include a real data type that is intended to represent real numbers.^[1]

The real numbers include within them all of these other kinds of numbers:

The "natural numbers" or positive integers, 1, 2, 3, ...
Zero and the negative integers
Fractions, like 355/113
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)
Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractionsTemplate:Prove
Irrational numbers, like

↑ In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers

[1] In reality, the actual values the computer uses are very-high-precision fractions which can equal or approximate real numbers

[1]

@@ Line 14: / Line 14: @@
 *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)
 *Any decimal representation which repeats or recurs, like 1.86292929292929..., because these can be shown to be fractions{{prove}}
-*[[Irrational numbers]], like &sqrt;10 = 3.162277660168... &pi; = 3.1415926525..., whose decimal representations never repeat or terminate.{{prove}}
+*[[Irrational numbers]], like
-==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 cuts]] or [[Cauchy Sequences]], i.e. it is the compeletion of the [[metric space]] of rational numbers.
-==Infinity==
-The real numbers ''do not'' include <math>\infty</math> or <math>-\infty</math> (infinity and minus infinity).  However, there are non-standard models of real numbers which include <math>\infty</math> or include both <math>\infty</math> and <math>-\infty</math>.
-There is no largest real number, because you can always make a real number larger by adding 1 (or 137.035 or 6.023&middot;10<sup>23</sup>) to it, and no smallest real number, because you can always make a real number smaller by subtracting from it.
-Every real number is finite. One way to see this is to observe that you cannot subtract infinity from itself&mdash;the result is indeterminate&mdash;but, for any real number x, x - x = 0, exactly.
-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."
-==Notes and references==
-<references/>
-[[Category:Mathematics]]

Difference between revisions of "Real numbers"

Revision as of 22:53, April 11, 2007

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