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real number

 Definition/Summary The real numbers are most commonly encountered number system, familiar to the layman via the number line, and as the number system lying behind decimal notation. Because the real numbers have many nice arithmetic and geometric properties, they feature prominently in many fields of mathematics.

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 Extended explanation This is a preliminary article. More information will be added in the future. A Brief History The real numbers originate from a need to quantify the geometric notion of 'length'. It was known in ancient times that the rational numbers are not adequate, culminating with Eudoxus's 'method of exhaustion'. Formal Definition With the development of modern calculus, it became increasingly clear that a rigorous definition of the real numbers was required. Cantor provided the first definition, by identifying a real number with the set of Cauchy sequences of rational that ought to converge to that real number. Since then, many other equivalent definitions have been proveded. The following definition is the one traditionally used today, except that the Dedekind completeness axiom has been replaced with an equivalent axiom: Language The real numbers consist of: A set $\mathbb{R}$ whose elements are called real numbers (also written R) A distinguished real number $0$ (zero) A distinguished real number $1$ (one) A binary operation $+$ (addition) A binary operation $\cdot$ (multiplication) A unary operation $-$ (negation) A unary partial operation ${}^{-1}$ (reciprocal) A relation $\leq$ (less than or equal to) In what follows, the symbols $a, b, c$ denote real numbers, $d$ denotes a nonzero real number (meaning $d \neq 0$). Field Axioms $a+b$ is a real number $a \cdot b$ is a real number $-a$ is a real number $d^{-1}$ is a real number $a+b = b+a$ (commutativity of addition) $a \cdot b = b \cdot a$ (commutativity of multiplication) $a+(b+c) = (a+b)+c$ (associativity of addition) $a\cdot (b\cdot c) = (a\cdot b)\cdot c$ (associativity of multiplication) $a\cdot (b+c) = (a\cdot b) + (a \cdot c)$ (distributivity of multiplication over addition) $a + 0 = a$ (0 is the additive identity) $a \cdot 1 = a$ (1 is the multiplicative identity) $a + (-a) = 0$ (additive inverses) $d \cdot (d^{-1}) = 1$ (multiplicative inverses) Ordering axioms If $a \leq b$ and $b \leq a$ then $a = b$ If $a \leq b$ and $b \leq c$ then $a \leq c$ $a \leq b$ or $b \leq a$ If $a \leq b$ then $a + c \leq b + c$ If $0 \leq c$ and $a \leq b$ then $a \cdot c \leq b \cdot c$ Completeness axiom The completeness axiom is significantly more complicated. One way to state it is via the least upper bound property of the calculus of sequences. Let $\{ f_n \}$ be a sequence of real numbers If $f_n \leq f_{n+1} \leq a$ for every natural number $n$, then $\lim_{n \rightarrow +\infty} f_n$ exists Other Operations and Identities These axioms are sufficient to derive all of the familiar properties of the real numbers. Some examples: Subtraction is defined by $a - b = a + (-b)$ Division is defined by $a / d = a \cdot (d^{-1})$ Common Errors The word "real" in "the real numbers" is often mistaken for the ordinary English word. In actuality, it is simply a name, and is used to distinguish the number system from other familiar systems like "the integers" or "the rational numbers" or "the Galois field of 49 elements".

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pierrecurie @ 07:24 AM Sep21-08
 Any real number is uniquely defined as being < all the rational numbers > itself, and > all the rational numbers < itself.
This sounds either very cyclic, or obvious, given the definition of ordering.
Limits of cauchy sequences is somewhat related, and much clearer.

 matt grime @ 10:22 AM Jun8-08 I realize this is preliminary, so here are some comments for whoever writes the final version. The summary states that the real numbers are 'generated by the rationals'. In what sense is this meant? It is never explained in the main body of the text. The same goes for 'constructive numbers'. There are no references. Where did Cantor define the real numbers? The axioms of a field and ordered field merit their own entries. All of the section on "Language" applies equally to the rationals, or indeed any subfield of R. And I'd suggest that the real numbers should be in analysis, not algebra, since most questions about them, and their very definition, are analytic.

 mathwonk @ 10:23 AM May12-08 Need for real numbers: real numbers are needed in order to assign a length to every line segment. if only rational numbers are used, the Greeks knew that the diagonal of a unit square would not have an assigned length. the essence of the axioms above is that real numbers are those numbers which can be approximated by rational numbers to any degree of accuracy. It turns out then that not only does X^2 - 2 = 0 have a real solution, solving the problem of the Greeks, but every odd degree rational polynomial has a real solution. A disadvantage of real numbers is there are so many of them they usually cannot be expressed using only a finite number of integers. I.e. it usually takes an infinite decimal expansion to write one real number. Thus real numbers are usually represented as limits, and to understand them well, a working knowledge of limits is very helpful. Nonetheless, the question of when a given limiting expression represents a rational number is still very challenging. E.g. I believe it is still unknown, when k is fixed, whether the limit of the sum, for all n > 0, of -kth powers 1/n^k, is ever rational.