Quadratic irrationality

Quadratic irrationality is an irrational number that is the real root of some quadratic equation $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ $ax ^ {2} + bx + c = 0$ with rational coefficients $a, b, c$ ${\ displaystyle a, b, c}$ $a, b, c$ (or, equivalently, the real root of a polynomial of the 2nd degree with rational coefficients ^[1] $ax^{2}+bx+c$ ${\ displaystyle ax ^ {2} + bx + c}$ $ax ^ {2} + bx + c$ ) Some sources also include the complex roots of these equations in the number of quadratic irrationalities.

Irrationality of number $x$ ${\ displaystyle x}$ $x$ means that it cannot be represented as a rational number (fraction). It follows that the polynomial $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ $ax ^ {2} + bx + c = 0$ irreducible in the field of rational numbers $\mathbb {Q} ,$ ${\ displaystyle \ mathbb {Q},}$ ${\ displaystyle \ mathbb {Q},}$ that is, it does not decompose in this field into factors of the first degree ^[1] .

Content

Algebraic properties

The solution of the quadratic equation $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ gives the formula:

x_{1,2}={\frac {-b\pm {\sqrt {D}}}{2a}},

{\ displaystyle x_ {1,2} = {\ frac {-b \ pm {\ sqrt {D}}} {2a}},}

Where $D=b^{2}-4ac$ ${\ displaystyle D = b ^ {2} -4ac}$ ( discriminant of the equation). The materiality of the root means that $D\geqslant 0.$ ${\ displaystyle D \ geqslant 0.}$ Therefore, any quadratic irrationality has the form:

x=u+v{\sqrt {D}},

{\ displaystyle x = u + v {\ sqrt {D}},}

Where $u, v, D$ ${\ displaystyle u, v, D}$ Are rational numbers, moreover $v\neq 0$ ${\ displaystyle v \ neq 0}$ , and the root expression $D$ ${\ displaystyle D}$ non-negative and is not a full square of a rational number ^[2] .

Examples: $11-{\sqrt {2}};\quad {\frac {1+{\sqrt {5}}}{2}}$ ${\ displaystyle 11 - {\ sqrt {2}}; \ quad {\ frac {1 + {\ sqrt {5}}} {2}}}$

It follows from the definition that quadratic irrationalities are algebraic numbers of the second degree. Note that the inverse element for $x=u+v{\sqrt {D}}$ ${\ displaystyle x = u + v {\ sqrt {D}}}$ also is quadratic irrationality:

{1 \over u+v{\sqrt {D}}}={u-v{\sqrt {D}} \over u^{2}-v^{2}D}.

{\ displaystyle {1 \ over u + v {\ sqrt {D}}} = {uv {\ sqrt {D}} \ over u ^ {2} -v ^ {2} D}.}

Number $x'=u-v{\sqrt {D}}$ ${\ displaystyle x '= uv {\ sqrt {D}}}$ called conjugate for $x=u+v{\sqrt {D}}.$ ${\ displaystyle x = u + v {\ sqrt {D}}.}$ The following formulas hold:

(x+y)'=x'+y';\quad (xy)'=x'y';\quad \left({\frac {1}{x}}\right)'={\frac {1}{x'}}

{\ displaystyle (x + y) '= x' + y '; \ quad (xy)' = x'y '; \ quad \ left ({\ frac {1} {x}} \ right)' = {\ frac {1} {x '}}}

Canonical format

Without loss of generality, the equation can be simplified $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ in the following way.

The coefficients of the considered equation of the 2nd degree can be made integers , since it is easy to get rid of the denominators of fractions by multiplying both sides of the equation by the least common multiple of all denominators. Discriminant $D$ ${\ displaystyle D}$ then also becomes an integer.
If the senior coefficient $a<0,$ ${\ displaystyle a <0,}$ then we multiply the equation by $-1$ ${\ displaystyle -1}$ .
Finally, we divide the resulting equation $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ by the largest common divisor of gcd $(a,b,c)$ ${\ displaystyle (a, b, c)}$ .

As a result, we obtain the equation $ax^{2}+bx+c=0$ ${\ displaystyle ax ^ {2} + bx + c = 0}$ with integer mutually simple coefficients, and the senior coefficient is positive ^[3] . This equation is uniquely related to a pair of its roots, and many such equations are countable . Therefore, the set of quadratic irrationalities is also countable.

Often convenient in root expression $x=u+v{\sqrt {D}},$ ${\ displaystyle x = u + v {\ sqrt {D}},}$ perform one more modification: if in canonical decomposition $D$ ${\ displaystyle D}$ any squares come in, we put them outside the sign of the radical, so the remaining value $D$ ${\ displaystyle D}$ will be free from squares .

Quadratic Fields

Sum, difference and product of quadratic irrationalities with the same discriminant $D$ ${\ displaystyle D}$ either have the same format or are rational numbers, so together they form a field that is a normal extension of the second degree of the field of rational numbers ℚ . This field is indicated by $\mathbb {Q} ({\sqrt {D}})$ ${\ displaystyle \ mathbb {Q} ({\ sqrt {D}})}$ and is called a quadratic field . Any such extension $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ can be obtained by the described method. The Galois group of the extension, in addition to the identity automorphism , contains a map of an irrational number to its conjugate (in the above sense) ^[4] .

Assume that, as described above, $D$ ${\ displaystyle D}$ Is a square-free integer. Then for different values $D$ ${\ displaystyle D}$ different quadratic fields are obtained ^[5] .

For a quadratic field, one can construct its ring of integers , that is, the set of roots of reduced polynomials with integer coefficients for which the leading coefficient is 1. Free from squares $D$ ${\ displaystyle D}$ cannot be divided by 4, therefore two cases are possible ^[4] depending on what residue gives $D$ ${\ displaystyle D}$ when divided by 4.

If a $D$ ${\ displaystyle D}$ has the form $4k+1,$ ${\ displaystyle 4k + 1,}$ then integer elements are numbers of the form $m+n\cdot {\tfrac {1+{\sqrt {D}}}{2}}$ ${\ displaystyle m + n \ cdot {\ tfrac {1 + {\ sqrt {D}}} {2}}}$ where $m, n$ ${\ displaystyle m, n}$ - integers.
If a $D$ ${\ displaystyle D}$ has the form $4k+2$ ${\ displaystyle 4k + 2}$ or $4k+3,$ ${\ displaystyle 4k + 3,}$ then integer elements are numbers of the form $m+n{\sqrt {D}}$ ${\ displaystyle m + n {\ sqrt {D}}}$ where $m, n$ ${\ displaystyle m, n}$ - integers.

Continuous Fraction Relationship

Real quadratic irrationality is connected with continued fractions by the Lagrange theorem (sometimes called the Euler – Lagrange theorem ) ^[6] :

A real number is quadratic irrationality if and only if it decomposes into an infinite periodic continued fraction.

Example:

{\sqrt {3}}=1.732\ldots =[1;1,2,1,2,1,2,\ldots ]

{\ displaystyle {\ sqrt {3}} = 1.732 \ ldots = [1; 1,2,1,2,1,2, \ ldots]}

A continued fraction in which a period begins from the first link is called purely periodic . The evolutionist Galois in 1828 proved: continued fraction for quadratic irrationality $x$ ${\ displaystyle x}$ will be purely periodic if and only if $x>1$ ${\ displaystyle x> 1}$ and conjugate irrationality $x^{'}$ ${\ displaystyle x '}$ lies in the range $(-1;0)$ ${\ displaystyle (-1; 0)}$ . He also proved that in the case of a purely periodic decomposition, conjugate quadratic irrationality has the same links, but in the opposite order ^[7] .

Summary

Quadratic irrationality is a special case of “irrationality” $n$ ${\ displaystyle n}$ degree ", which is the root of the irreducible in the field $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ polynomial $n$ ${\ displaystyle n}$ degree with integer coefficients. Rational numbers are obtained when $n=1,$ ${\ displaystyle n = 1,}$ and quadratic irrationalities correspond to the case $n=2.$ ${\ displaystyle n = 2.}$

Some sources also include the complex roots of quadratic equations (for example, Gaussian integers or Eisenstein numbers ) among the quadratic irrationalities.

G.F. Voronoi in his work “On algebraic integers depending on the root of a third-degree equation” (1894) extended the theory (including continued fractions) to the case of cubic irrationalities.

History

Theodore of Kirensky and his student Teetet of Athens (IV century BC) were the first to prove that if the number $N$ ${\ displaystyle N}$ does not constitute a full square then ${\sqrt {N}}$ ${\ displaystyle {\ sqrt {N}}}$ is not a rational number, that is, it cannot be accurately expressed as a fraction. This proof was based on the Euclidean lemma . Euclid devoted the tenth book of his Beginnings to these questions; he, like modern sources, used the basic theorem of arithmetic .

Notes

↑ ¹ ² Quadratic irrationality // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2. - S. 776.
↑ Galochkin A. I. Quadratic irrationality // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2. - S. 776.
↑ Nesterenko Yu.V., 2008 , p. 207.
↑ ¹ ² Ayerland K., Rosen M. A classic introduction to modern number theory. - M .: Mir, 1987 .-- S. 230-232. - 428 p.
↑ Bukhstab A.A., 2015 , p. 149-150.
↑ Nesterenko Yu.V., 2008 , p. 208-209.
↑ Davenport G. Higher Arithmetic. - M .: Nauka, 1965 .-- S. 100.

Literature

Buchshtab A. A. Quadratic irrationality and periodic continued fractions // Number theory. - 4th ed. - M .: Doe, 2015 .-- 384 p. - ISBN 978-5-8114-0847-4 .
Nesterenko Yu. V. Number theory: a textbook for students. higher textbook. institutions. - M .: Publishing Center "Academy", 2008. - 272 p. - ISBN 978-5-7695-4646-4 .
Khinchin A. Ya. Chain fractions . - M .: GIFFL, 1960.

Links

Weisstein, Eric W. Quadratic Surd on Wolfram MathWorld .
Continued fraction calculator for quadratic irrationals
Proof that e is not a quadratic irrational (English)

[ME-1] ¹ ² Quadratic irrationality // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2. - S. 776.

[2] Galochkin A. I. Quadratic irrationality // Mathematical Encyclopedia (in 5 volumes). - M .: Soviet Encyclopedia , 1979. - T. 2. - S. 776.

[_20678cdbaabb8afd-3] Nesterenko Yu.V., 2008 , p. 207.

[Ire-4] ¹ ² Ayerland K., Rosen M. A classic introduction to modern number theory. - M .: Mir, 1987 .-- S. 230-232. - 428 p.

[_9fbe07384f48d5e7-5] Bukhstab A.A., 2015 , p. 149-150.

[_bb25e0d06ca42cab-6] Nesterenko Yu.V., 2008 , p. 208-209.

[7] Davenport G. Higher Arithmetic. - M .: Nauka, 1965 .-- S. 100.