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# Real number

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The term real number is a retronym (coined in response to imaginary number) that denotes the kind of number one might use to represent a continuous quantity (including 0 and negatives). The integers and rational numbers are both subsets of real numbers. Imaginary numbers and complex numbers are not real numbers, nor are things like vectors and tensors. [Floating point]? numbers are approximations of the real numbers that can be used by computers. What follows are more rigorous mathematical descriptions of the concept.

Real number axioms:

  Let R denote the set of real numbers, and w, x, y, z be elements of R. Then:

- The set R is a field (i.e., addition?, multiplication? and division? are defined and have the usual properties)
- The field R is ordered, i.e.
  - if x >= y then x + z >= y + z
  - if x >= 0 and y >= 0 then x * y >= 0
- Every non-empty subset S with an [upper bound]? in R has a [least upper bound]? in R.

The latter property is what differentiates the reals from the rationals. For example, the set of the rationals with square less than 2 has a rational upper bound (e.g. 1.5) but no least upperbound, because the [square root]? of 2 is not rational.

Every non negative real number has a square root. This shows that the order on R is determined by its algebraic structure.

The real numbers are the unique field satisfying these properties. There are various other properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily isomorphic to the reals.

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete?. This means the following:

A sequence {xn} of real numbers is called a Cauchy sequence if there exists an integer valued function N = N(ε) on the positive real (or rational) numbers such that for any ε > 0 the distance? |xn - xm| < ε provided that n, m > N(ε).

In other words, a sequence is a Cauchy sequence if its elements xn eventually remain arbitrarily close to each other.

A sequence {xn} has limit x if there exists an integer valued function N = N(ε) on the positive real (or rational) numbers such that for any ε > 0 the distance |xn - x | < ε provided that n > N(ε).

A sequence is convergent? if it has a limit.

It is easy to see that every convergent sequence is a Cauchy sequence. Now a space is complete if, conversely, every Cauchy sequence is convergent. Note that the rationals are not complete.

The existence of limits of Cauchy sequences is what makes analysis and the calculus work and is of great practical use. For example the standard series of the exponential function

  ex = Σn = 0∞ xn / n!

converges to a real number because for every x the tail sum

  Σn=NM xn / n!

can be made arbitrarily small by choosing N sufficiently large.

Also note that the standard numerical test if a sequence has a limit is to test that it is (or seems to be) a Cauchy sequence, as the limit is typically not known in advance.

If we have a space where Cauchy sequences are meaningful (a metric space, i.e. a space where distance is defined), a standard procedure to force all Cauchy sequences to converge is adding new points to the space (a process called completing?). When applied to the rational numbers, it gives the following useful [construction of the real numbers]?:

Let C be the set of Cauchy sequences of rational numbers. Two Cauchy sequences {xn} and {ym} are equivalent if there exists an integer valued function N = N(ε) on the positive reals (or rationals) such that for any ε > 0 the distance |xn - ym| < ε provided n, m > N(ε).

It is easy to see that this really defines an equivalence relation. Let R be the set of equivalence classes. Then R is a model for the real numbers.

A practical and concrete representative for an equivalence class representing a real number is provided by the representation to base b (usually 2 (binary), 10 (decimal) or 16 (hexadecimal)). For example the number π = 3.14159.... would correspond to the Cauchy sequence {3, 3.1, 3.14, 3.141, 3.1415, ...}. A real number can have two representations. For example: 1 = 0.99999.... .

The reals are uncountable?, that is, there are strictly more real numbers than integers. In fact, the cardinality? of the reals is 2(א0) (two to the aleph-null; see [transfinite numbers]?), i.e. the cardinality of the set of subsets of the integers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The nonexistence of a subset of the reals with cardinality strictly in between that of the integers and the reals is known as the continuum hypothesis. This can neither be proved nor be disproved, but is an independant axiom of set theory.

The real numbers have many useful topological properties. The reals are a contractible? (hence connected? and [simply connected]?), [locally compact]? separable? Hausdorff space, of dimension? 1, and are [everywhere dense]?. The real numbers are not compact.

RB: The dimension is actually difficult to define: the reals have dimension 1 for pretty much any sensible definition, but the best definition I know is that cohomology with compact support is non trivial in dimension 1 and vanishes above it.

The reals are one of the two [local fields]? of characteristic? 0 (the other one being the complex numbers).

A [number field]? is a [real field]? if every embedding in the complex numbers is real.

The reals carry a canonical measure?, the [Lebesgue measure]? which is the [Haar measure]? normalised such that the interval [0,1] has measure 1.

A real number is one that can be expressed in the form 'DDD.ddd'.

- DDD is zero or more decimal digits
- ddd is zero or more decimal digits

  Of course, DDD must be finite in length. This restriction does not apply to ddd.

Special cases:

- ddd is 0 - the format is 'DDD' and the [decimal point]? is not included
- DDD and ddd are 0 - this is represented by 0

Note: There are many subsets of the real numbers, including:

- natural numbers
- integers
- rational numbers

The rational numbers are the infinitely repeating decimal expansions (if you allow 12 = 11.9....; if you don't allow this, the set is base dependent, which is undesirable). This may be proved by summing as a [geometric progression]?, e.g.

  0.121212... = 12/100 + 12/1002 + 12/1003 + ...

Numbers that are not rational are called irrational.

See also:

- complex numbers
- countably infinite

/Talk

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