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| In [[mathematics]], an '''algebraic number''' is a number that is a [[root of a function|root]] of a non-zero [[polynomial]] in one variable with [[rational number|rational]] coefficients (or equivalently—by clearing [[denominator]]s—with [[integer]] coefficients). Numbers such as ''[[Pi|{{pi}}]]'' that are not algebraic are said to be [[transcendental number|transcendental]]; [[almost all]] [[real number|real]] and [[complex number]]s are transcendental. (Here "almost all" has the sense "all but a [[countable set]]"; see Properties below.)
| | <br><br>It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you have any kind of inquiries pertaining to where and the best ways to make use of [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist], you can call us at the site. |
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| ==Examples==
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| *The [[rational number]]s, expressed as the quotient of two [[integer]]s ''a'' and ''b'', ''b'' not equal to zero, satisfy the above definition because <math>x = a/b</math> is the root of <math>bx-a</math>.<ref>Some of the following examples come from Hardy and Wright 1972:159–160 and pp. 178–179</ref>
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| *The [[quadratic surd]]s (irrational roots of a quadratic polynomial <math>ax^2 + bx + c</math> with integer coefficients <math>a</math>, <math>b</math>, and <math>c</math>) are algebraic numbers. If the quadratic polynomial is monic <math>(a = 1)</math> then the roots are [[quadratic integer]]s.
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| *The [[constructible number]]s are those numbers that can be constructed from a given unit length using straightedge and compass and their opposites. These include all quadratic surds, all rational numbers, and all numbers that can be formed from these using the [[Arithmetic#Arithmetic operations|basic arithmetic operations]] and the extraction of square roots. (Note that by designating cardinal directions for 1, −1, <math>i</math>, and <math>-i</math>, complex numbers such as <math>3+\sqrt{2}i</math> are considered constructible.)
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| *Any expression formed using any combination of the basic arithmetic operations and extraction of [[nth root|''n''th roots]] gives an algebraic number.
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| *Polynomial roots that ''cannot'' be expressed in terms of the basic arithmetic operations and extraction of ''n''th roots (such as the roots of <math>x^5 - x + 1 </math>). This [[Abel–Ruffini theorem|happens with many]], but not all, polynomials of degree 5 or higher.
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| *[[Gaussian integer]]s: those complex numbers <math>a+bi</math> where both <math>a</math> and <math>b</math> are integers are also quadratic integers.
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| *[[Trigonometric functions]] of [[rational number|rational]] multiples of <math>\pi</math> (except when undefined). For example, each of <math>\cos(\pi/7)</math>, <math>\cos(3\pi/7)</math>, <math>\cos(5\pi/7)</math> satisfies <math>8x^3 - 4x^2 - 4x + 1 = 0</math>. This polynomial is [[irreducible polynomial|irreducible]] over the rationals, and so these three cosines are ''conjugate'' algebraic numbers. Likewise, <math>\tan(3\pi/16)</math>, <math>\tan(7\pi/16)</math>, <math>\tan(11\pi/16)</math>, <math>\tan(15\pi/16)</math> all satisfy the irreducible polynomial <math>x^4 - 4x^3 - 6x^2 + 4x + 1</math>, and so are conjugate [[algebraic integers]].
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| *Some [[irrational number]]s are algebraic and some are not:
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| **The numbers <math>\sqrt{2}</math> and <math>\sqrt[3]{3}/2</math> are algebraic since they are roots of polynomials <math>x^2 - 2</math> and <math>8x^3 - 3</math>, respectively.
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| **The [[golden ratio]] <math>\phi</math> is algebraic since it is a root of the polynomial <math>x^2 - x - 1</math>.
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| **The numbers [[Pi|<math>\pi</math>]] and [[e (mathematical constant)|<math>e</math>]] are not algebraic numbers (see the [[Lindemann–Weierstrass theorem]]);<ref>Also [[Liouville number|Liouville's theorem]] can be used to "produce as many examples of transcendentals numbers as we please," cf Hardy and Wright p. 161ff</ref> hence they are transcendental.
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| == {{anchor|Degree of an algebraic number}} Properties ==
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| <!-- This Anchor tag serves to provide a permanent target for incoming section links. Please do not move it out of the section heading, even though it disrupts edit summary generation (you can manually fix the edit summary before saving your changes). Please do not modify it, even if you modify the section title. See [[Template:Anchor]] for details. (This text: [[Template:Anchor comment]]) -->
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| [[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree. (red=1, green=2, blue=3, yellow=4)]]
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| *The set of algebraic numbers is [[countable set|countable]] (enumerable).<ref>Hardy and Wright 1972:160</ref>
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| *Hence, the set of algebraic numbers has [[Lebesgue measure]] zero (as a subset of the complex numbers), i.e. "[[Almost everywhere|almost all]]" complex numbers are not algebraic.
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| *Given an algebraic number, there is a unique [[monic polynomial]] (with rational coefficients) of least [[degree of a polynomial|degree]] that has the number as a root. This polynomial is called its [[minimal polynomial (field theory)|minimal polynomial]]. If its minimal polynomial has degree <math>n</math>, then the algebraic number is said to be of ''degree <math>n</math>''. An algebraic number of degree 1 is a [[rational number]]. A real algebraic number of degree 2 is a [[quadratic irrational]].
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| *All algebraic numbers are [[computable number|computable]] and therefore [[definable number|definable]] and [[arithmetical numbers|arithmetical]].
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| *The set of real algebraic numbers is [[linearly ordered]], countable, [[densely ordered]], and without first or last element, so is [[order-isomorphic]] to the set of rational numbers.
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| ==The field of algebraic numbers==
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| [[File:Algebraic number in the complex plane.png|thumb|Algebraic numbers colored by degree (blue=4, cyan=3, red=2, green=1). The unit circle in black.]]
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| The sum, difference, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstrated using the [[resultant]]), and the algebraic numbers therefore form a [[field (mathematics)|field]], sometimes denoted by '''A''' (which may also denote the [[adele ring]]) or <span style="text-decoration: overline;">'''Q'''</span>. Every root of a polynomial equation whose coefficients are ''algebraic numbers'' is again algebraic. This can be rephrased by saying that the field of algebraic numbers is [[algebraically closed field|algebraically closed]]. In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the [[algebraic closure]] of the rationals.
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| ==Related fields==
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| ===Numbers defined by radicals===
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| All numbers that can be obtained from the integers using a [[finite set|finite]] number of integer [[addition]]s, [[subtraction]]s, [[multiplication]]s, [[division (mathematics)|division]]s, and taking ''n''th roots (where ''n'' is a positive integer) are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. All of these numbers are solutions to polynomials of degree ≥5. This is a result of [[Galois theory]] (see [[Quintic equation]]s and the [[Abel–Ruffini theorem]]). An example of such a number is the unique real root of the polynomial {{nowrap|''x''<sup>5</sup> − ''x'' − 1}} (which is approximately 1.167304).
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| ===Closed-form number===
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| {{Main|Closed-form number}}
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| Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "[[closed-form number]]s", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers ''explicitly'' defined in terms of polynomials, exponentials, and logarithms – this does not include algebraic numbers, but does include some simple transcendental numbers such as ''e'' or log(2).
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| ==Algebraic integers==
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| {{Main|Algebraic integer}}
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| [[Image:Leadingcoeff.png|thumb|Algebraic numbers colored by leading coefficient (red signifies 1 for an algebraic integer).]]
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| An '''[[algebraic integer]]''' is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial). Examples of algebraic integers are {{nowrap|5 + 13√{{overline|2}}}}, {{nowrap|2 − 6''i''}}, and {{nowrap|{{frac|1|2}}(1 + ''i''√{{overline|3}}).}} (Note, therefore, that the algebraic integers constitute a proper [[superset]] of the [[integer]]s, as the latter are the roots of monic polynomials {{nowrap|''x'' − ''k''}} for all {{nowrap|''k'' ∈ '''Z'''.)}}
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| The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a [[ring (algebra)|ring]]. The name ''algebraic integer'' comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any [[algebraic number field|number field]] are in many ways analogous to the integers. If ''K'' is a number field, its [[ring of integers]] is the subring of algebraic integers in ''K'', and is frequently denoted as ''O<sub>K</sub>''. These are the prototypical examples of [[Dedekind domain]]s. | |
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| ==Special classes of algebraic number==
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| *[[Gaussian integer]]
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| *[[Eisenstein integer]]
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| *[[Quadratic irrational]]
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| *[[Fundamental unit (number theory)|Fundamental unit]]
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| *[[Root of unity]]
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| *[[Gaussian period]]
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| *[[Pisot–Vijayaraghavan number]]
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| *[[Salem number]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Citation |last=Artin |first=Michael |author-link=Michael Artin |title=Algebra |publisher=[[Prentice Hall]] |isbn=0-13-004763-5 |mr=1129886 |year=1991}}
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| *{{Citation |last1=Ireland |first1=Kenneth |last2=Rosen |first2=Michael |title=A Classical Introduction to Modern Number Theory |edition=Second |publisher=[[Springer-Verlag]] |location=Berlin, New York |series=Graduate Texts in Mathematics |isbn=0-387-97329-X |mr=1070716 |year=1990 |volume=84}}
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| *[[G.H. Hardy]] and [[E.M. Wright]] 1978, 2000 (with general index) ''An Introduction to the Theory of Numbers: 5th Edition'', Clarendon Press, Oxford UK, ISBN 0-19-853171-0
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| *{{Lang Algebra}}
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| *[[Øystein Ore]] 1948, 1988, ''Number Theory and Its History'', Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)
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| {{Number Systems}}
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| {{DEFAULTSORT:Algebraic Number}}
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| [[Category:Algebraic numbers| ]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
If you have any kind of inquiries pertaining to where and the best ways to make use of Washington DC Dentist, you can call us at the site.