Rational number

Last updated
The rational numbers (Q) are included in the real numbers (R), while themselves including the integers (Z), which in turn include the natural numbers (N) Number-systems.svg
The rational numbers (ℚ) are included in the real numbers (ℝ), while themselves including the integers (ℤ), which in turn include the natural numbers (ℕ)

In mathematics, a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. [1] For example, −3/7 is a rational number, as is every integer (e.g. 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", [2] the field of rationals [3] or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode 𝐐/ℚ); [4] [5] it was thus denoted in 1895 by Giuseppe Peano after quoziente , Italian for "quotient".

Contents

The decimal expansion of a rational number either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...). [6] Conversely, any repeating or terminating decimal represents a rational number. These statements are true in base 10, and in every other integer base (for example, binary or hexadecimal).

A real number that is not rational is called irrational. [7] Irrational numbers include 2 , π, e , and φ . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. [1]

Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows:

The fraction p/q then denotes the equivalence class of (p, q).

Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. [8]

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (for more, see Construction of the real numbers).

Terminology

The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.

Etymology

Although nowadays rational numbers are defined in terms of ratios, the term rational is not a derivation of ratio. On the opposite, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660, [9] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570. [10] This meaning of rational came from the mathematical meaning of irrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of ἄλογος)". [11] [12]

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers". [13] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek). [14]

This etymology is similar to that of imaginary numbers and real numbers.

Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers and b > 0. This is often called the canonical form of the rational number.

Starting from a rational number a/b, its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.

Embedding of integers

Any integer n can be expressed as the rational number n/1, which is its canonical form as a rational number.

Equality

if and only if

If both fractions are in canonical form, then:

if and only if and

Ordering

If both denominators are positive (particularly if both fractions are in canonical form):

if and only if

On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator.

Addition

Two fractions are added as follows:

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.

Subtraction

If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.

Multiplication

The rule for multiplication is:

where the result may be a reducible fraction—even if both original fractions are in canonical form.

Inverse

Every rational number a/b has an additive inverse, often called its opposite,

If a/b is in canonical form, the same is true for its opposite.

A nonzero rational number a/b has a multiplicative inverse, also called its reciprocal,

If a/b is in canonical form, then the canonical form of its reciprocal is either b/a or b/a, depending on the sign of a.

Division

If b, c, and d are nonzero, the division rule is

Thus, dividing a/b by c/d is equivalent to multiplying a/b by the reciprocal of c/d:

Exponentiation to integer power

If n is a non-negative integer, then

The result is in canonical form if the same is true for a/b. In particular,

If a ≠ 0, then

If a/b is in canonical form, the canonical form of the result is bn/an if a > 0 or n is even. Otherwise, the canonical form of the result is bn/an.

Continued fraction representation

A finite continued fraction is an expression such as

where an are integers. Every rational number a/b can be represented as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a, b).

Other representations

are different ways to represent the same rational value.

Formal construction

A diagram showing a representation of the equivalent classes of pairs of integers Rational Representation.svg
A diagram showing a representation of the equivalent classes of pairs of integers

The rational numbers may be built as equivalence classes of ordered pairs of integers.

More precisely, let (Z × (Z \ {0})) be the set of the pairs (m, n) of integers such n ≠ 0. An equivalence relation is defined on this set by

Addition and multiplication can be defined by the following rules:

This equivalence relation is a congruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers Q is the defined as the quotient set by this equivalence relation, (Z × (Z \ {0})) / ~, equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain and produces its field of fractions.)

The equivalence class of a pair (m, n) is denoted m/n. Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if m1n2 = m2n1. This means that m1/n1 = m2/n2 if and only m1n2 = m2n1.

Every equivalence class m/n may be represented by infinitely many pairs, since

Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class such that m and n are coprime, and n > 0. It is called the representation in lowest terms of the rational number.

The integers may be considered to be rational numbers identifying the integer n with the rational number n/1.

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has m1/n1m2/n2 if

Properties

A diagram illustrating the countability of the positive rationals Diagonal argument.svg
A diagram illustrating the countability of the positive rationals

The set Q of all rational numbers, together with the addition and multiplication operations shown above, forms a field.

Q has no field automorphism other than the identity.

With the order defined above, Q is an ordered field that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic to Q.

Q is a prime field, which is a field that has no subfield other than itself. [15] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to Q.

Q is the field of fractions of the integers Z. [16] The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

The set of all rational numbers is countable, while the set of all real numbers (as well as the set of irrational numbers) is uncountable. Being countable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that

(where are positive), we have

Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.

Real numbers and topological properties

The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.

By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x, y) = |xy|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x, y) = |xy| above.

p-adic numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:

Let p be a prime number and for any non-zero integer a, let |a|p = pn, where pn is the highest power of p dividing a.

In addition set |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p/|b|p.

Then dp(x, y) = |xy|p defines a metric on Q.

The metric space (Q, dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

See also

Related Research Articles

An irreducible fraction is a fraction in which the numerator and denominator are integers that have no other common divisors than 1. In other words, a fraction ab is irreducible if and only if a and b are coprime, that is, if a and b have a greatest common divisor of 1. In higher mathematics, "irreducible fraction" may also refer to rational fractions such that the numerator and the denominator are coprime polynomials. Every positive rational number can be represented as an irreducible fraction in exactly one way.

Field of fractions

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.

Square root Number whose square is a given number

In mathematics, a square root of a number x is a number y such that y2 = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x. For example, 4 and −4 are square roots of 16, because 42 = (−4)2 = 16. Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by where the symbol is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by because 32 = 3 ⋅ 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case 9.

In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers with q > 1 such that

Dyadic rational

In mathematics, a dyadic rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures and in musical time signatures.

In mathematics, an nth root of a number x is a number r which, when raised to the power n, yields x:

Diophantine approximation approximating real numbers with rational numbers

In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

In mathematics, a quadratic irrational number is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

Square root of 2 Unique positive real number which when multiplied by itself gives 2

The square root of 2, or the one-half power of 2, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it must be called the principal square root of 2, to distinguish it from the negative number with the same property.

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.

In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.

Pell number

In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.

In mathematics, the mediant of two fractions, generally made up of four positive integers

Fraction Mathematical representation of a portion of a whole

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of a numerator displayed above a line, and a non-zero denominator, displayed below that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

Stern–Brocot tree

In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree.

Square root of 3 Unique positive real number which when multiplied by itself gives 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.

In the metrical theory of regular continued fractions, the kth complete quotient ζk is obtained by ignoring the first k partial denominators ai. For example, if a regular continued fraction is given by

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

In number theory, specifically in Diophantine approximation theory, the Markov constant of an irrational number is the factor for which Dirichlet's approximation theorem can be improved for .

References

  1. 1 2 Rosen, Kenneth. Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN   978-0-07-288008-3.
  2. Lass, Harry (2009). Elements of Pure and Applied Mathematics (illustrated ed.). Courier Corporation. p. 382. ISBN   978-0-486-47186-0. Extract of page 382
  3. Robinson, Julia (1996). The Collected Works of Julia Robinson. American Mathematical Soc. p. 104. ISBN   978-0-8218-0575-6. Extract of page 104
  4. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-11.
  5. Rouse, Margaret. "Mathematical Symbols" . Retrieved 1 April 2015.
  6. "Rational number". Encyclopedia Britannica. Retrieved 2020-08-11.
  7. Weisstein, Eric W. "Rational Number". mathworld.wolfram.com. Retrieved 2020-08-11.
  8. Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN   0-534-40264-X.
  9. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry ratio, n., sense 2.a.
  10. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry rational, a. (adv.) and n.1, sense 5.a.
  11. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry irrational, a. and n., sense 3.
  12. Shor, Peter (2017-05-09). "Does rational come from ratio or ratio come from rational". Stack Exchange. Retrieved 2021-03-19.
  13. Coolman, Robert (2016-01-29). "How a Mathematical Superstition Stultified Algebra for Over a Thousand Years" . Retrieved 2021-03-20.
  14. Kramer, Edna (1983). The Nature and Growth of Modern Mathematics. Princeton University Press. p. 28.
  15. Sūgakkai, Nihon (1993). Encyclopedic Dictionary of Mathematics, Volume 1. London, England: MIT Press. p. 578. ISBN   0-2625-9020-4.
  16. Bourbaki, N. (2003). Algebra II: Chapters 4 - 7. Springer Science & Business Media. p. A.VII.5.