Real number

A positive number, a negative number or zero. The concept of a real number arose by a generalization of the concept of a rational number. Such a generalization was rendered necessary both by practical applications of mathematics — viz., the expression of the value of a given magnitude by a definite number — and by the internal development of mathematics itself; in particular, by the desire to extend the domain of applicability of certain operations on numbers (root extraction, computation of logarithms, solution of equations, etc.). The general concept of real number was already studied by Greek mathematicians of Antiquity in their theory of non-commensurable segments, but it was formulated as an independent concept only in the 17th century by I. Newton, in his Arithmetica Universalis as follows: "A number is not so much the totality of several units, as an abstract ratio between one magnitude and another, of the same kind, and which is accepted as a unit" . Rigorous theories of real numbers were constructed at the end of the 19th century by K. Weierstrass, G. Cantor and R. Dedekind.

Real numbers form a non-empty totality of elements which contains more than one element and displays the following properties.

I) The property of being ordered. Any two numbers $ a $ and $ b $ have a definite order relation, i.e. one and only one of the following relations will be true: $ a < b $, $ a = b $ or $ a > b $; also, if $ a < b $ and $ b < c $, then $ a < c $( transitivity of the order).

II) The property of an addition operation. For any ordered pair of numbers $ a $ and $ b $ there is a unique number, known as their sum and denoted by $ a + b $, such that the following properties hold: $ \textrm{ II } _ {1} $) $ a +b = b + a $( commutativity); $ \textrm{ II } _ {2} $) for any numbers $ a $, $ b $ and $ c $ one has $ a + (b+c) = (a+b) +c $( associativity); $ \textrm{ II } _ {3} $) there exists a number, called zero and denoted by $ 0 $, such that $ a + 0 = a $ for any $ a $; $ \textrm{ II } _ {4} $) for any number $ a $ there exists a number, called the opposite of $ a $ and denoted by $ -a $, such that $ a + (-a) = 0 $; $ \textrm{ II } _ {5} $) if $ a < b $, then $ a + c < b + c $ for any $ c $.

The zero is unique, and the number opposite to any given number is unique. For any ordered pair of numbers $ a $ and $ b $ the number $ a + (-b) $ is called the difference between the numbers $ a $ and $ b $ and is denoted by $ a -b $.

III) The property of a multiplication operation. For any ordered pair of numbers $ a $ and $ b $ there exists a unique number, known as their product and denoted by $ ab $, such that: $ \textrm{ III } _ {1} $) $ ab = ba $( commutativity); $ \textrm{ III } _ {2} $) $ a (bc) = (ab) c $ for any numbers $ a , b , c $( associativity); $ \textrm{ III } _ {3} $) there exists a number, known as the unit and denoted by $ 1 $, such that $ a1 = a $ for any number $ a $; $ \textrm{ III } _ {4} $) for any non-zero number $ a $ there exists a number, known as its reciprocal and denoted by $ 1 / a $, such that $ a(1/a) = 1 $; $ \textrm{ III } _ {5} $) if $ a < b $ and $ c > 0 $, then $ ac < bc $.

These properties ensure that the unit and the reciprocal of each element are unique. For each ordered pair of numbers $ a $ and $ b $, $ b \neq 0 $, the number $ a (1/b) $ is known as the quotient obtained by dividing $ a $ by $ b $; it is denoted by $ a/b $.

The number $ 1 + 1 $ is denoted by $ 2 $, the number $ 2 + 1 $ is denoted by $ 3 $, etc. The numbers $ 1, 2, 3 \dots $ are known as the natural numbers (cf. Natural number). Numbers larger than zero are said to be positive, while numbers smaller than zero are said to be negative. The numbers $ 0, \pm1, \pm2 \dots $ are called integers (it is assumed that $ +a = a $; cf. Integer). Numbers of the type $ m/n $, where $ m $ is an integer, while $ n $ is a natural number, are known as rational numbers or fractions. They include all integers. The number $ (a)1 $ is identified with $ a $. Real numbers which are not rational are also called irrational numbers.

IV) The property of distributivity of multiplication with respect to addition. For any three numbers $ a $, $ b $ and $ c $, $ (a+b) c = ac +bc $.

A non-empty totality of elements which has all the above properties forms a totally ordered field (cf. Totally ordered set; Field). Real numbers also have two other important properties.

V) The Archimedean property. For any number $ a $ there exists an integer $ n $ such that $ n > a $. A totality of elements having properties I–V forms an Archimedean ordered field. Examples are not only the field of real numbers, but also the field of rational numbers.

An important property of real numbers is their continuity; rational numbers do not have this property.

VI) The property of continuity. For any system of nested segments

$$ \{ [ a _ {n} , b _ {n} ] \} ,\ a _ {n} \leq a _ {n+1} \leq \dots \leq b _ {n+1} \leq b _ {n} ,\ n= 1 , 2 \dots $$

there exists at least one number which belongs to all the segments of the system. This property is also known as Cantor's principle of nested segments. If the lengths $ b _ {n} - a _ {n} $ of the nested segments tend to zero as $ n \rightarrow \infty $, there exists a unique point which belongs to all these segments.

The properties of real numbers listed above entail many others; thus, it follows from the properties I to V that $ 1 > 0 $; there also follow the rules of operations on rational fractions, the sign rules to be observed when multiplying and dividing real numbers, the properties of the absolute value of a real number, the rules governing transformations of equalities and inequalities, etc. Properties I to VI are a complete description of the properties of the field of real numbers and only of this field; in other words, if these properties are taken as axioms, it follows that the real numbers form the unique totality of elements satisfying them. This means that properties I to VI define the set of real numbers up to an isomorphism: If there are two sets $ X $ and $ Y $ satisfying the properties I to VI, there always exists a mapping of $ X $ onto $ Y $, isomorphic with respect to the order and to the operations of addition and multiplication, i.e. this mapping (denoted $ x \rightarrow y $, where $ y \in Y $ is the element corresponding to the element $ x \in X $) maps $ X $ onto $ Y $ in a one-to-one correspondence so that if

$$ x _ {1} < x _ {2} ,\ x _ {1} , x _ {2} \in X,\ \textrm{ and } \ \ x _ {1} \rightarrow y _ {1} ,\ x _ {2} \rightarrow y _ {2} , $$

then

$$ y _ {1} < y _ {2} ,\ x _ {1} + x _ {2} \rightarrow y _ {1} + y _ {2} ,\ \ x _ {1} x _ {2} \rightarrow y _ {1} y _ {2} . $$

A consequence of this is that the field of real numbers (as distinct, for example, from the field of rational numbers) cannot be extended while preserving the properties I to V, i.e. there is no field with the property of being ordered and with addition and multiplication operations in accordance with properties I to V, which would contain a subset isomorphic to the field of real numbers without being identical with it.

There are many more real numbers than rational numbers; in fact, the rational numbers form a countable subset of the set of real numbers, which is itself uncountable (cf. Cardinality). Both the rational and the irrational numbers are dense in the set of all real numbers (cf. Dense set): For any two real numbers $ a $ and $ b $, $ a < b $, it is possible to find a rational number $ r $ such that $ a < r < b $ and an irrational number $ \xi $ such that $ a < \xi < b $.

The property of continuity of real numbers is closely connected with the property of their completeness, to wit, that any Cauchy sequence of real numbers is convergent. It should be noted that the field of rational numbers only is no longer complete: It contains Cauchy sequences which do not converge to any rational number. The continuity (or completeness) of the set of real numbers is closely connected with their utilization in measuring several kinds of continuous magnitudes, e.g. in determining the length of geometrical segments; if a unique unit segment is chosen, then, in view of the continuity of the set of real numbers, it is possible to bring any segment into correspondence with a positive real number — its length. The continuity of the set of real numbers may be described, in an illustrative manner, by saying that it contains no "empty spaces" . A consequence of the continuity of the set of real numbers is the fact that it is possible to extract the $ n $- th root of any positive number (where $ n $ is a natural number), and the fact that any positive number has a logarithm to any base $ a $, $ a > 0 $, $ a \neq 1 $.

The property of continuity of real numbers may also be formulated in a different manner.

VI') Any non-empty set bounded from above has a least upper bound (cf. Upper and lower bounds).

The concept of a cut $ A \mid B $ in the domain of real numbers (cf. Dedekind cut) may also be employed. One says that the cut $ A \mid B $ is effected by the number $ \alpha $ if $ a \leq \alpha \leq b $ for all $ a \in A $, $ b \in B $( here, either $ \alpha \in A $ or $ \alpha \in B $). Any number effects a cut.

The property of continuity, known as the Dedekind continuity of the real numbers, consists in the validity of the converse postulate.

VI) Any cut of real numbers is effected by some number. Such a number is unique, and is either the highest in the lower class or the lowest in the higher class.

Each one of the postulates VI, VI' and VI is equivalent to each one of the others, in the sense that if any one of them, as well as the remaining properties I to V, is taken as axiom, the other two will follow. Moreover, both property VI' and property VI (in conjunction with properties I–IV) entail not merely VI, but also the Archimedean property V. The definition of the set real numbers as the non-empty totality of elements with properties I–VI is an axiomatic construction of the theory of real numbers. Several methods of constructing this theory on the base of rational numbers are available.

The first such theory was constructed by Dedekind on the basis of the concept of a cut $ R _ {1} \mid R _ {2} $ in the domain of rational numbers. If, for a given cut $ R _ {1} \mid R _ {2} $ there exists a largest rational number in $ R _ {1} $ or a smallest rational number in $ R _ {2} $, one says that the cut $ R _ {1} \mid R _ {2} $ is effected by this number. Any rational number effects a cut. A cut for which there is no largest number in the lower class, and no smallest number in the upper class, is said to be an irrational number. Rational and irrational numbers are called real numbers; here, for the sake of uniformity, rational numbers are considered as the cuts which they effect.

Let $ x = R _ {1} \mid R _ {2} $ and $ x ^ \prime = R _ {1} ^ \prime \mid R _ {2} ^ \prime $. The real number $ x $ is said to be smaller than the real number $ x ^ \prime $( or, which is the same thing, $ x ^ \prime $ is said to be larger than $ x $) if $ R _ {1} \subset R _ {1} ^ \prime $, $ R _ {1} \neq R _ {1} ^ \prime $. The concepts of positive and negative real numbers (see above) and of the absolute value $ | x | $ of a real number $ x $ are introduced in the usual way. The sum of the real numbers $ x $ and $ x ^ \prime $ is defined to be the number $ x + x ^ \prime $ such that for all $ r _ {1} \in R _ {1} $, $ r _ {1} ^ \prime \in R _ {1} ^ \prime $, $ r _ {2} \in R _ {2} $, $ r _ {2} ^ \prime \in R _ {2} ^ \prime $ the inequalities

$$ r _ {1} + r _ {1} ^ \prime \leq x + x ^ \prime \leq r _ {2} + r _ {2} ^ \prime $$

are valid. The product of two positive real numbers $ x $ and $ x ^ \prime $ is the number $ x x ^ \prime $ such that for all positive $ r _ {1} , r _ {1} ^ \prime , r _ {2} , r _ {2} ^ \prime $ the inequalities $ r _ {1} r _ {1} ^ \prime \leq x x ^ \prime \leq r _ {2} r _ {2} ^ \prime $ are satisfied. The product of two non-zero real numbers $ x $ and $ x ^ \prime $ is defined as the real number whose absolute value is $ | x | | x ^ \prime | $, and which is positive if $ x $ and $ x ^ \prime $ have the same sign, and negative if they have opposite signs. Finally, for any real number $ x $ it is assumed that $ 0x = x0 = 0 $.

The sum and product of real numbers always exist, are unique, and the totality of real numbers thus defined, together with the introduced order and operations of addition and multiplication, displays the properties I–VI.

Another theory was proposed by G. Cantor. It is based on the concept of a Cauchy sequence of rational numbers, i.e. a sequence $ \{ r _ {n} \} $ of rational numbers such that for any rational number $ \epsilon > 0 $ there exists a number $ n _ \epsilon $ such that for all $ n \geq n _ \epsilon $ and $ m \geq n _ \epsilon $ the inequality $ | r _ {n} - r _ {m} | < \epsilon $ is valid. A sequence of rational numbers $ \{ r _ {n} \} $ is said to be a zero-sequence if for any rational number $ \epsilon > 0 $ there exists a number $ n _ \epsilon $ such that for all $ n \geq n _ \epsilon $ the inequality $ | r _ {n} | < \epsilon $ is valid. Two Cauchy sequences of rational numbers $ \{ r _ {n} \} $ and $ \{ r _ {n} ^ \prime \} $ are said to be equivalent if the sequence $ \{ r _ {n} - r _ {n} ^ \prime \} $ is a zero-sequence. This definition of equivalence displays the properties of reflexivity, symmetry and transitivity, and this is the reason why the whole set of Cauchy sequences of rational numbers splits into equivalence classes. The totality of all these equivalence classes is also known in this case as the set of real numbers. By virtue of this definition, any real number represents an equivalence class of Cauchy sequences of rational numbers. Each such sequence is said to be a representative of the given real number. A Cauchy sequence of rational numbers $ \{ r _ {n} \} $ is said to be positive (negative) if there exists a rational number $ r > 0 $ $ (r < 0 ) $ such that all terms of this sequence, beginning with some term, are larger than $ r $( smaller than $ -r $). Any Cauchy sequence of rational numbers is either a zero-sequence, a positive sequence or a negative sequence. If a Cauchy sequence of rational numbers is positive (negative), then any Cauchy sequence of rational numbers equivalent to it will also be positive (negative). A real number is said to be positive (negative) if some one (and hence any one) of its representatives is positive (negative). A real number is said to be zero if some one (and hence any one) of its representatives is a zero-sequence. Any real number is either positive, negative or zero. In order to add or to multiply two real numbers $ x $ and $ x ^ \prime $ one has to add (respectively, multiply) any two of their representatives $ \{ r _ {n} \} \in x $, $ \{ r _ {n} ^ \prime \} \in x ^ \prime $; this again yields Cauchy sequences of rational numbers, $ \{ r _ {n} + r _ {n} ^ \prime \} $ and $ \{ r _ {n} r _ {n} ^ \prime \} $. The equivalence classes which they represent are known in this case as the sum $ x + x ^ \prime $ and the product $ x x ^ \prime $ of these numbers. These operations are unambiguously defined, i.e. they do not depend on the choice of representatives of these numbers. Subtraction and division of real numbers are defined as the operations inverse to addition and multiplication, respectively. If, for two real numbers $ x $ and $ y $, one has $ x - y > 0 $, the real number $ x $ is said to be larger than the real number $ y $. The totality of real numbers thus defined, together with the property of ordering described above and the operations of addition and multiplication, again displays the properties I–VI.

Still another theory, based on infinite decimal expansions, was developed by Weierstrass. According to this theory, a real number is any infinite decimal expansion with a plus or a minus sign:

$$ \pm \alpha _ {0} . \alpha _ {1} \alpha _ {2} \dots \alpha _ {n} \dots , $$

where $ \alpha _ {0} $ is a non-negative integer (integers are assumed to be given) while each $ \alpha _ {n} $, $ n = 1 , 2 \dots $ is one of the digits $ 0 , 1 \dots 9 $. Here, an infinite decimal expansion which after some time consists of 9's only (i.e. which has a period consisting of 9:

$$ \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} (9) ,\ \ a _ {n} \neq 9 , $$

is considered equal to the infinite decimal expansion

$$ \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n-1} ( \alpha _ {n} + 1 ) 00 \dots 0 \dots $$

(if $ n = 0 $, it is equal to the infinite decimal expansion $ ( \alpha _ {0} + 1 ) . 00 \dots 0 \dots $). This expansion may also be written as the finite decimal expansion

$$ \alpha _ {0} . \alpha _ {1} \dots ( \alpha _ {n} + 1 ) , $$

and one says that it has $ n $ significant figures after the decimal point. An infinite decimal expansion without the period

is said to be an allowable infinite decimal expansion. Clearly, any real number can be uniquely (re)written as an allowable infinite decimal expansion. If a real number $ x $ is rewritten as an allowable infinite decimal expansion with a plus (minus) sign, and if the digits $ \alpha _ {n} $ contain at least one non-zero digit, $ x $ is said to be positive (negative), written as $ x > 0 $( $ x < 0 $). If all $ \alpha _ {n} = 0 $, $ n = 0 , 1 \dots $ it is said to be zero: $ x = 0 $. For the number

$$ x = \pm \alpha _ {0} . \alpha _ {1} \dots a _ {n} \dots , $$

the number $ \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} \dots $ is said to be its absolute value and is denoted by $ | x | $. The number with the plus sign (the minus sign) replaced by the minus sign (the plus sign) is said to be opposite to the given number and is denoted by $ -x $. If

$$ x = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} \dots $$

is an allowable infinite decimal expansion, then the numbers

$$ \underline{x _ {n} } = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} $$

and

$$ \overline{ {x _ {n} }}\; = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} + 10 ^ {-n} , $$

are said to be, respectively, the lower (higher) decimal approximation of order $ n $ of the number $ x $. Let $ x $ and $ y $ be two positive numbers, written as allowable infinite decimal expansions

$$ x = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} \dots $$

and

$$ y = \beta _ {0} . \beta _ {1} \dots \beta _ {n} \dots . $$

By definition, $ x < y $ if either $ \alpha _ {0} < \beta _ {0} $ or if there exists a number $ n _ {0} $, $ n _ {0} =0 , 1 \dots $ such that $ \alpha _ {k} = \beta _ {k} $, $ k = 0 \dots n _ {0} $, but $ \alpha _ {n _ {0} + 1 } < \beta _ {n _ {0} + 1 } $. Every negative number and zero are considered to be smaller than every positive number. If $ x $ and $ y $ are both negative and $ | y | < | x | $, then $ x < y $.

A sequence of integers $ n _ {k} $, $ k = 1 , 2 \dots $ is said to be stabilizing to a number $ m $ if there exists a number $ k _ {0} $ such that $ n _ {k} = m $ for all $ k \geq k _ {0} $. A sequence of infinite decimal expansions

$$ x ^ {(k)} = \alpha _ {0} ^ {(k)} . \ \alpha _ {1} ^ {(k)} \dots \alpha _ {n} ^ {(k)} \dots $$

is said to be stabilizing to a number

$$ x = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} \dots , $$

if the $ i $- th column of the infinite matrix $ \| \alpha _ {i} ^ {(k)} \| $, where $ i $ is the column index and $ k $ is the row index, stabilizes to the number $ \alpha _ {i} $ for any $ i = 0, 1 ,\dots $. If $ x > 0 $ and $ y > 0 $, the finite decimal expansions

$$ \underline{x _ {n} } + \underline{y _ {n} } ,\ \ \underline{x _ {n} } - \underline{y _ {n} } ,\ \ \underline{\left ( \underline{x _ {n} } \underline{y _ {n} } \right ) _ {n} } \ \textrm{ and } \ \ \underline{\left ( \frac{\underline{x _ {n} } }{\underline{y _ {n} } } \right ) _ {n} } $$

have $ n $ significant figures to the right of the decimal point and form sequences stabilizing to certain numbers. These numbers are known, respectively, as the sum $ x + y $, the difference $ x - y $, the product $ xy $, and the quotient $ x/y $ of $ x $ and $ y $. These definitions are extended to real numbers of arbitrary sign. For instance, if $ x \leq 0 $ and $ y \leq 0 $, then $ x + y = - ( | x | + | y | ) $; if the signs of $ x $ and $ y $ are different, then $ x + y = \pm | | x | - | y | | $, the sign of this result being identical with the sign of that number $ x $ or $ y $ which has the larger absolute value. For any numbers $ x $ and $ y $ it is assumed that $ x - y = x + (-y) $( if $ x > 0 $, $ y > 0 $, this definition is identical with that given above), etc. The totality of allowable infinite decimal expansions with the order relation and with the operations of addition, subtraction, multiplication, and division thus defined, satisfies the axioms I–VI.

In constructing the theory of real numbers it is also possible to use non-decimal computation systems, i.e. systems to the base two, three, etc. It is important to note that none of the constructions of the theory of real numbers given above (axiomatic, based on cuts of rational numbers, based on Cauchy sequences of rational numbers or on infinite decimal expansions) is a proof of the existence (self-consistency) of the set of real numbers. From this point of view all these methods are equivalent.

Geometrically, the set of real numbers can be represented by an oriented (directed) straight line, while the individual numbers are represented by points on that line. Accordingly, the totality of real numbers is often called the number axis, while the individual numbers are called points. When such a representation of real numbers is employed, instead of saying that $ a $ is smaller than $ b $( respectively, that $ b $ is larger than $ a $) one says that the point $ a $ lies to the left of the point $ b $( respectively, $ b $ lies to the right of $ a $). There is an order-preserving one-to-one correspondence between the points on a Euclidean straight line ordered in accordance with their locations on it and the elements of the number axis. This is a justification for representing the set of real numbers as a straight line.

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The most important theory of proportions in Antiquity has been given by Eudoxus of Cnidus (ca. 400 B.C.– 347 B.C.). One can find this theory in Euclid's Elements, book V. See also [a2], [a3] and $ Elements $ of Euclid.

Irrational numbers can be divided into two different kinds: algebraic numbers and transcendental numbers. An algebraic number is a root of an algebraic equation with (rational) integers as coefficients. A transcendental number is not the root of any algebraic equation with (rational) integral coefficients. The usual notation for the field of rational (respectively, real) numbers is $ \mathbf Q $( respectively, $ \mathbf R $).

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