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## Real numberIn mathematics, thereal numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number".Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247... (where the three dots express that there would still be more digits to come, no matter how many more might be added at the end). Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent expresses the sense of an underlying real number. It is as if one says "I'm writing down only the part of the number that I know; it's infinitely long, and my stopping after a finite number of digits echoes the fact that I'm stopping short of doing more and more refined experiments forever, and getting further along in the infinite series of digits, which would be the only way to avoid an approximate final result." The real numbers are the central object of study in real analysis.
A real number is said to be Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed point numbers; see Real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation.
Mathematicians use the symbol In mathematics, the term "real XXX" means that the underlying number field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.
## HistoryFractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians lead by Pythagoras realized the need for irrational numbers. Negative numbers began to be generally accepted in the 1600s and were invented by Muslim mathematicians. The development of the calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871. ## Definition## Construction from the rational numbersReal numbers could be constructed as the topological completion of rational numbers. For details and other construction of real numbers, see Construction of real numbers## Axiomatic approach
Let - The set
**R**is a field, i.e., addition, subtraction, multiplication and division are defined and have the usual properties. - The field
**R**is ordered, i.e., there is a total order ≥ such that, for all real numbers*x*,*y*and*z*:- if
*x*≥*y*then*x*+*z*≥*y*+*z*; - if
*x*≥ 0 and*y*≥ 0 then*x**y*≥ 0.
- if
- The order is Dedekind-complete, i.e., every non-empty subset
*S*of**R**with an upper bound in**R**has a least upper bound (also called supremum) in**R**.
The real numbers are uniquely specified by the above properties.
More precisely, given any two Dedekind complete ordered fields ## Properties## CompletenessThe main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following:
A sequence (
A sequence ( It is easy to see that every convergent sequence is a Cauchy sequence. Now the important fact about the real numbers is that the converse is true: **Every Cauchy sequence of real numbers is convergent.**
Note that the rationals are not complete. For example, the sequence (1,1.4,1.41,1.414,1.4142,1.41421,...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example the standard series of the exponential function
x the sums
N sufficiently large.
This proves that the sequence is Cauchy, so we know that the sequence converges even if we don't know ahead of time what the limit is.## "The complete ordered field"
First, an order can be lattice complete.
It's easy to see that no ordered field can be lattice complete, because it can have no largest element (given any element
Additionally, an order can be Dedekind-complete, as defined in the section
These two notions of completeness ignore the field structure.
However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section
But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it.
He meant that the real numbers form the ## Advanced properties
The reals are uncountable, that is, there are strictly more real numbers than natural numbers (even though both sets are infinite).
This is proved with Cantor's diagonal argument.
In fact, the cardinality of the reals is 2
The real numbers form a metric space: the distance between
Every nonnegative real number has a square root in The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1.
The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement.
It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves.
The set of hyperreal numbers is much bigger than ## Generalizations and ExtensionsThe real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, formal elements +∞ and -∞ are added to the reals to form the extended real number line, a compact space which is not a field anymore but retains many of the properties of the real numbers. Self-adjoint operatorss on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers.
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