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Real number

Related subjects: Mathematics

Background to the schools Wikipedia


In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339.... The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and the square root of 2, and can be represented as points along an infinitely long number line.

A more rigorous definition of the real numbers was one of the most important developments of 19th century mathematics. Popular definitions in use today include equivalence classes of Cauchy sequences of rational numbers, Dedekind cuts, a more sophisticated version of "decimal representation", and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field.

The name real numbers arose to distinguish them from what were then called imaginary numbers (and now complex numbers).

Basic properties


Real numbers measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247... The ellipsis (three dots) indicate that there would still be more digits to come.

More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of −1 to the real numbers, obtaining the complex numbers, the result is algebraically closed.




Computers can only approximate most real numbers. Most commonly, they can represent a certain subset of the rationals exactly, via either floating point numbers or fixed-point numbers, and these rationals are used as an approximation for other nearby real values. Arbitrary-precision arithmetic is a method to represent arbitrary rational numbers, limited only by available memory, but more commonly one uses a fixed number of bits of precision determined by the size of the processor registers. In addition to these rational values, computer algebra systems are able to treat many (countable) irrational numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their rational approximation. Note that a few programming languages use "real" to describe their main numeric data type, such as AppleScript.

Mathematicians use the symbol R (or alternatively,  \Bbb{R} , the letter " R" in blackboard bold, Unicode ℝ) to represent the set of all real numbers. The notation Rn refers to an n- dimensional space with real coordinates; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra. As a substantive, the term is used almost strictly in reference to the real numbers, themselves (e.g., The "set of all reals").


Vulgar fractions had been used by the Egyptians around 1000 BC; the Vedic " Sulba Sutras" ("rule of chords" in Sanskrit), ca. 600 BC, include what may be the first 'use' of irrational numbers.






Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,...} converges to a unique real number. For details and other constructions of real numbers, see construction of real numbers.

Axiomatic approach

Let R denote the set of all real numbers. Then:

  • The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
  • The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
    • if xy then x + zy + z;
    • if x ≥ 0 and y ≥ 0 then xy ≥ 0.
  • The order is Dedekind-complete; that is, every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.







A sequence (xn) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |xn − xm| is less than ε for all n and m that are both greater than N. In other words, a sequence is a Cauchy sequence if its elements xn eventually come and remain arbitrarily close to each other.



Every Cauchy sequence of real numbers is convergent.

That is, the reals are complete.



For example, the standard series of the exponential function

\mathrm{e}^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}

converges to a real number because for every x the sums

\sum_{n=N}^{M} \frac{x^n}{n!}


"The complete ordered field"






Advanced properties






Generalizations and extensions


  • The complex numbers contain solutions to all polynomial equations and hence are an algebraically closed field unlike the real numbers. However, the complex numbers are not an ordered field.
  • The affinely extended real number system adds two elements +∞ and −∞. It is a compact space. It is no longer a field, not even an additive group; it still has a total order; moreover, it is a complete lattice.
  • The real projective line adds only one value ∞. It is also a compact space. Again, it is no longer a field, not even an additive group. However, it allows division of a non-zero element by zero. It is not ordered anymore.
  • The long real line pastes together ℵ1* + ℵ1 copies of the real line plus a single point (here ℵ1* denotes the reversed ordering of ℵ1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ1 in the long real line but not in the real numbers. The long real line is the largest ordered set that is complete and locally archimedean. As with the previous two examples, this set is no longer a field or additive group.

"Reals" in set theory

In set theory, specifically descriptive set theory the Baire space is used as a surrogate for the real numbers since the latter have some topological properties (connectedness) that are a technical inconvenience. Elements of Baire space are referred to as "reals".

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