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Rational number

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,[4] or blackboard bold ${\displaystyle \mathbb {Q} .}$[5]

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...).[6] 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.[7] 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).[8]

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.[9]

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).

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Transcription

So let's talk a little bit about rational numbers. And the simple way to think about it is any number that can be represented as the ratio of two integers is a rational number. So for example, any integer is a rational number. 1 can be represented as 1/1 or as negative 2 over negative 2 or as 10,000/10,000. In all of these cases, these are all different representations of the number 1, ratio of two integers. And I obviously can have an infinite number of representations of 1 in this way, the same number over the same number. The number negative 7 could be represented as negative 7/1, or 7 over negative 1, or negative 14 over positive 2. And I could go on, and on, and on, and on. So negative 7 is definitely a rational number. It can be represented as the ratio of two integers. But what about things that are not integers? For example, let us imagine-- oh, I don't know-- 3.75. How can we represent that as the ratio of two integers? Well, 3.75, you could rewrite that as 375/100, which is the same thing as 750/200. Or you could say, hey, 3.75 is the same thing as 3 and 3/4-- so let me write it here-- which is the same thing as-- that's 15/4. 4 times 3 is 12, plus 3 is 15, so you could write this. This is the same thing as 15/4. Or we could write this as negative 30 over negative 8. I just multiplied the numerator and the denominator here by negative 2. But just to be clear, this is clearly rational. I'm giving you multiple examples of how this can be represented as the ratio of two integers. Now, what about repeating decimals? Well, let's take maybe the most famous of the repeating decimals. Let's say you have 0.333, just keeps going on and on forever, which we can denote by putting that little bar on top of the 3. This is 0.3 repeating. And we've seen-- and later we'll show how you can convert any repeating decimal as the ratio of two integers-- this is clearly 1/3. Or maybe you've seen things like 0.6 repeating, which is 2/3. And there's many, many, many other examples of this. And we'll see any repeating decimal, not just one digit repeating. Even if it has a million digits repeating, as long as the pattern starts to repeat itself over and over and over again, you can always represent that as the ratio of two integers. So I know what you're probably thinking. Hey, Sal, you've just included a lot. You've included all of the integers. You've included all of finite non-repeating decimals, and you've also included repeating decimals. What is left? Are there any numbers that are not rational? And you're probably guessing that there are, otherwise people wouldn't have taken the trouble of trying to label these as rational. And it turns out-- as you can imagine-- that actually some of the most famous numbers in all of mathematics are not rational. And we call these numbers irrational numbers. And I've listed there just a few of the most noteworthy examples. Pi-- the ratio of the circumference to the diameter of a circle-- is an irrational number. It never terminates. It goes on and on and on forever, and it never repeats. e, same thing-- never terminates, never repeats. It comes out of continuously compounding interest. It comes out of complex analysis. e shows up all over the place. Square root of 2, irrational number. Phi, the golden ratio, irrational number. So these things that really just pop out of nature, many of these numbers are irrational. Now, you might say, OK, are these irrational? These are just these special kind of numbers. But maybe most numbers are rational, and Sal's just picked out some special cases here. But the important thing to realize is they do seem exotic, and they are exotic in certain ways. But they aren't uncommon. It actually turns out that there is always an irrational number between any two rational numbers. Well, we could go on and on. There's actually an infinite number. But there's at least one, so that gives you an idea that you can't really say that there are fewer irrational numbers than rational numbers. And in a future video, we'll prove that you give me two rational numbers-- rational 1, rational 2-- there's going to be at least one irrational number between those, which is a neat result, because irrational numbers seem to be exotic. Another way to think about it-- I took the square root of 2, but you take the square root of any non-perfect square, you're going to end up with an irrational number. You take the sum of an irrational and a rational number-- and we'll see this later on. We'll prove it to ourselves. The sum of an irrational and a rational is going to be irrational. The product of an irrational and a rational is going to be irrational. So there's a lot, a lot, a lot of irrational numbers out there.

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,[10] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570.[11] 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)".[12][13]

This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".[14] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).[15]

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 ${\displaystyle {\tfrac {a}{b}},}$ 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 ${\displaystyle b=d}$[8]

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

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.[8]

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.[8][16]

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.[16]

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.[8][16]

Inverse

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

${\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}$

If ${\displaystyle {\tfrac {a}{b}}}$ is in canonical form, the same is true for its opposite.

A nonzero rational number ${\displaystyle {\tfrac {a}{b}}}$ has a multiplicative inverse, also called its reciprocal,

${\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}$

If ${\displaystyle {\tfrac {a}{b}}}$ is in canonical form, then the canonical form of its reciprocal is either ${\displaystyle {\tfrac {b}{a}}}$ or ${\displaystyle {\tfrac {-b}{-a}},}$ depending on the sign of a.

Division

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

${\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}$

Thus, dividing ${\displaystyle {\tfrac {a}{b}}}$ by ${\displaystyle {\tfrac {c}{d}}}$ is equivalent to multiplying ${\displaystyle {\tfrac {a}{b}}}$ by the reciprocal of ${\displaystyle {\tfrac {c}{d}}:}$[16]

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

Exponentiation to integer power

If n is a non-negative integer, then

${\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}$

The result is in canonical form if the same is true for ${\displaystyle {\tfrac {a}{b}}.}$ In particular,

${\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}$

If a ≠ 0, then

${\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}$

If ${\displaystyle {\tfrac {a}{b}}}$ is in canonical form, the canonical form of the result is ${\displaystyle {\tfrac {b^{n}}{a^{n}}}}$ if a > 0 or n is even. Otherwise, the canonical form of the result is ${\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}$

Continued fraction representation

A finite continued fraction is an expression such as

${\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}$

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

Other representations

• common fraction: ${\displaystyle {\tfrac {8}{3}}}$
• mixed numeral: ${\displaystyle 2{\tfrac {2}{3}}}$
• repeating decimal using a vinculum: ${\displaystyle 2.{\overline {6}}}$
• repeating decimal using parentheses: ${\displaystyle 2.(6)}$
• continued fraction using traditional typography: ${\displaystyle 2+{\tfrac {1}{1+{\tfrac {1}{2}}}}}$
• continued fraction in abbreviated notation: ${\displaystyle [2;1,2]}$
• Egyptian fraction: ${\displaystyle 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}$
• prime power decomposition: ${\displaystyle 2^{3}\times 3^{-1}}$
• quote notation: ${\displaystyle 3'6}$

are different ways to represent the same rational value.

Formal construction

The rational numbers may be built as equivalence classes of ordered pairs of integers.[8][16]

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

${\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}$[8][16]

Addition and multiplication can be defined by the following rules:

${\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}$
${\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}$[8]

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 ${\displaystyle \mathbb {Q} }$ is the defined as the quotient set by this equivalence relation, ${\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}$ 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.)[8]

The equivalence class of a pair (m, n) is denoted ${\displaystyle {\tfrac {m}{n}}.}$ Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if

${\displaystyle m_{1}n_{2}=m_{2}n_{1}.}$

This means that

${\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}$

if and only if[8][16]

${\displaystyle m_{1}n_{2}=m_{2}n_{1}.}$

Every equivalence class ${\displaystyle {\tfrac {m}{n}}}$ may be represented by infinitely many pairs, since

${\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}$

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 ${\displaystyle {\tfrac {n}{1}}.}$

A total order may be defined on the rational numbers, that extends the natural order of the integers. One has

${\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}$

If

{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}

Properties

The set ${\displaystyle \mathbb {Q} }$ of all rational numbers, together with the addition and multiplication operations shown above, forms a field.[8]

${\displaystyle \mathbb {Q} }$ has no field automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)

${\displaystyle \mathbb {Q} }$ is a prime field, which is a field that has no subfield other than itself.[17] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to ${\displaystyle \mathbb {Q} .}$

With the order defined above, ${\displaystyle \mathbb {Q} }$ is an ordered field[16] 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 ${\displaystyle \mathbb {Q} .}$

${\displaystyle \mathbb {Q} }$ is the field of fractions of the integers ${\displaystyle \mathbb {Z} .}$[18] The algebraic closure of ${\displaystyle \mathbb {Q} ,}$ i.e. the field of roots of rational polynomials, is the field of algebraic numbers.

The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.[8] For example, for any two fractions such that

${\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}$

(where ${\displaystyle b,d}$ are positive), we have

${\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}$

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.[19]

Countability

The set of all rational numbers is countable, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a square lattice as in a Cartesian coordinate system, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any non-zero number divided by itself will always equal one.

It is possible to generate all of the rational numbers without such redundancies: examples include the Calkin–Wilf tree and Stern–Brocot tree.

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

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.[8] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.[20]

In the usual topology of the real numbers, the rationals are neither an open set nor a closed set.[21]

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 ${\displaystyle d(x,y)=|x-y|,}$ and this yields a third topology on ${\displaystyle \mathbb {Q} .}$ 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, and the real numbers are the completion of ${\displaystyle \mathbb {Q} }$ under the metric ${\displaystyle d(x,y)=|x-y|}$ above.[16]

In addition to the absolute value metric mentioned above, there are other metrics which turn ${\displaystyle \mathbb {Q} }$ into a topological field:

Let p be a prime number and for any non-zero integer a, let ${\displaystyle |a|_{p}=p_{-n},}$ where pn is the highest power of p dividing a.

In addition set ${\displaystyle |0|_{p}=0.}$ For any rational number ${\displaystyle {\frac {a}{b}},}$ we set

${\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}$

Then

${\displaystyle d_{p}(x,y)=|x-y|_{p}}$

defines a metric on ${\displaystyle \mathbb {Q} .}$[22]

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

Number systems
Complex ${\displaystyle :\;\mathbb {C} }$
Real ${\displaystyle :\;\mathbb {R} }$
Rational ${\displaystyle :\;\mathbb {Q} }$
Integer ${\displaystyle :\;\mathbb {Z} }$
Natural ${\displaystyle :\;\mathbb {N} }$
 Zero: 0 One: 1 Prime numbers Composite numbers
Negative integers
Imaginary

References

1. ^ a b Rosen, Kenneth (2007). 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. ^ It was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient",[citation needed]
5. ^ It first appeared in Bourbaki's Algèbre.
6. ^ "Rational number". Encyclopedia Britannica. Retrieved 2020-08-11.
7. ^ Weisstein, Eric W. "Rational Number". mathworld.wolfram.com. Retrieved 2020-08-11.
8. Biggs, Norman L. (2002). Discrete Mathematics. India: Oxford University Press. pp. 75–78. ISBN 978-0-19-871369-2.
9. ^ Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0-534-40264-X.
10. ^ Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry ratio, n., sense 2.a.
11. ^ Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry rational, a. (adv.) and n.1, sense 5.a.
12. ^ Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry irrational, a. and n., sense 3.
13. ^ Shor, Peter (2017-05-09). "Does rational come from ratio or ratio come from rational". Stack Exchange. Retrieved 2021-03-19.
14. ^ Coolman, Robert (2016-01-29). "How a Mathematical Superstition Stultified Algebra for Over a Thousand Years". Retrieved 2021-03-20.
15. ^ Kramer, Edna (1983). The Nature and Growth of Modern Mathematics. Princeton University Press. p. 28.
16. "Fraction - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-17.
17. ^ Sūgakkai, Nihon (1993). Encyclopedic Dictionary of Mathematics, Volume 1. London, England: MIT Press. p. 578. ISBN 0-2625-9020-4.
18. ^ Bourbaki, N. (2003). Algebra II: Chapters 4 - 7. Springer Science & Business Media. p. A.VII.5.
19. ^ Giese, Martin; Schönegge, Arno (December 1995). Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV (PDF) (Technical report). Retrieved 17 August 2021.
20. ^ Anthony Vazzana; David Garth (2015). Introduction to Number Theory (2nd, revised ed.). CRC Press. p. 1. ISBN 978-1-4987-1752-6. Extract of page 1
21. ^ Richard A. Holmgren (2012). A First Course in Discrete Dynamical Systems (2nd, illustrated ed.). Springer Science & Business Media. p. 26. ISBN 978-1-4419-8732-7. Extract of page 26
22. ^ Weisstein, Eric W. "p-adic Number". mathworld.wolfram.com. Retrieved 2021-08-17.
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