In mathematics, a rational number is a number that can be expressed as the quotient or fraction ${\displaystyle {\tfrac {p}{q}}}$ of two integers, a numerator p and a non-zero denominator q.^[1] For example, ${\displaystyle {\tfrac {3}{7}}}$ is a rational number, as is every integer (e.g., ${\displaystyle -5={\tfrac {-5}{1}}}$). 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 boldface Q, or blackboard bold ${\displaystyle \mathbb {Q} .}$

A rational number is a real number. The real numbers that are rational are those whose decimal expansion 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...).^[4] This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).

A real number that is not rational is called irrational.^[5] Irrational numbers include the square root of 2 (${\displaystyle {\sqrt {2}}}$), π, e, and the golden ratio (φ). 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:

${\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}$

The fraction ${\displaystyle {\tfrac {p}{q}}}$ then denotes the equivalence class of (p, q).^[6]

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 ${\displaystyle \mathbb {Q} }$ are called algebraic number fields, and the algebraic closure of ${\displaystyle \mathbb {Q} }$ is the field of algebraic numbers.^[7]

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 (see Construction of the real numbers).

Terminology

The term rational in reference to the set ${\displaystyle \mathbb {Q} }$ 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 contrary, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660,^[8] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570.^[9] 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)".^[10][11]

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those lengths as numbers".^[12] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).^[13]

Arithmetic

Irreducible fraction

Every rational number may be expressed in a unique way as an irreducible fraction ${\displaystyle {\tfrac {a}{b}},}$ 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 ${\displaystyle {\tfrac {a}{b}},}$ 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 ${\displaystyle {\tfrac {n}{1}},}$ which is its canonical form as a rational number.

Equality

${\displaystyle {\frac {a}{b}}={\frac {c}{d}}}$ if and only if ${\displaystyle ad=bc}$

If both fractions are in canonical form, then:

${\displaystyle {\frac {a}{b}}={\frac {c}{d}}}$ if and only if ${\displaystyle a=c}$ and ${\displaystyle b=d}$^[6]

Ordering

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

${\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}$ if and only if ${\displaystyle ad<bc.}$

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.^[6]

Addition

Two fractions are added as follows:

${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}$

If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.^[6][14]

Subtraction

${\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}$

If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.^[14]

Multiplication

The rule for multiplication is:

${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}$

where the result may be a reducible fraction—even if both original fractions are in canonical form.^[6][14]

Inverse

Every rational number ${\displaystyle {\tfrac {a}{b}}}$ has an additive inverse, often called its opposite,

${\displaystyle -<!--\; -\; -->\left(\frac{a}{b}\right)=\frac{-<!--\; -\; -->a}{b}}$Zdroj:https://en.wikipedia.org?pojem=Rational_number
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---