|
|
| Line 1: |
Line 1: |
| :''This article is about the generalization of the basic concept. For the basic concept, see [[Absolute value]]. For other uses, see [[Absolute value (disambiguation)]].'' | | fee to watch your webinar at their convenience. Once it is in place, [http://www.comoganhardinheiro101.com/?p=12 ganhe dinheiro] promotion and possibly answering questions will be your only tasks. <br><br><br>Getting paid money to work online isn't the easiest thing to do in the world, but it is possible. If this is something you wish to work with, then the tips presented above should have helped you. Take some time, do things the right ganhar dinheiro way and then you can succeed. |
|
| |
|
| In [[mathematics]], an '''absolute value''' is a [[function (mathematics)|function]] which measures the "size" of elements in a [[Field (mathematics)|field]] or [[integral domain]]. More precisely, if ''D'' is an integral domain, then an '''absolute value''' is any mapping | ''x'' | from ''D'' to the [[real numbers]] '''R''' satisfying:
| | Start your online [http://www.comoganhardinheiro101.com/2013/11/ como conseguir dinheiro] earning today by following the great advice discussed in this article. Earning money is not as hard as it may seem, you just need to know how to get started. By choosing to put your right foot forward, you are heading off to a great start earning money to make ends meet. Here's more on [http://ganhardinheironainternet.comoganhardinheiro101.com como ganhar dinheiro] have a look at http://ganhardinheironainternet.comoganhardinheiro101.com <br><br>Make money online by selling your [http://comoficarrico.comoganhardinheiro101.com talents]. Good music is always in demand and with today's technological advances, anyone with musical talent can make music and offer it for sale [http://www.comoganhardinheiro101.com/?p=90 ganhar dinheiro na internet] to a broad audience. By setting up your own website and using [http://comoficarrico.Comoganhardinheiro101.com/ social media] for promotion, you can share your music with others and sell downloads with a free PayPal account.<br><br>It's easy to make money online. There is truth to the fact that you can start making money on the Internet as soon as you're done with this article. After [http://comoficarrico.comoganhardinheiro101.com como ganhar dinheiro na internet] all, so many others are making money online, why not you? Keep your mind open and you can make a lot of money. As you [http://www.comoganhardinheiro101.com/ ganhar dinheiro pela internet] can [http://ganhedinheiro.comoganhardinheiro101.com/ ganhando dinheiro na internet] see, there are many ways to approach the world of online income. |
|
| |
|
| * | ''x'' | ≥ 0,
| | With various streams of income available, you are sure to find one, or two, that can help you with your income needs. Take this information to heart, put it to use and build your own online success story. |
| * | ''x'' | = 0 if and only if ''x'' = 0,
| |
| * | ''xy'' | = | ''x'' || ''y'' |,
| |
| * | ''x'' + ''y'' | ≤ | ''x'' | + | ''y'' |.
| |
| | |
| It follows from these axioms that | 1 | = 1 and | −1 | = 1. Furthermore, for any positive integer ''n'',
| |
| | |
| :| ''n'' | = | 1+1+...(''n'' times) | = | −1−1...(''n'' times) | ≤ ''n''.
| |
| | |
| Note that some authors use the terms '''valuation''', '''norm''',<ref>{{cite book|last=Koblitz|first=Neal|title=P-adic numbers, p-adic analysis, and zeta-functions|year=1984|publisher=Springer-Verlag|location=New York|isbn=978-0-387-96017-3|url=http://www.springer.com/mathematics/numbers/book/978-0-387-96017-3|edition=2nd ed.|accessdate=24 August 2012|p=1 |quote=The metrics we'll be dealing with will come from ''norms'' on the field ''F''...}}</ref> or '''magnitude''' instead of "absolute value". However, the word "[[norm (mathematics)|norm]]" usually refers to a specific kind of absolute value on a field (and which is also applied to other vector spaces).
| |
| | |
| The classical "[[absolute value]]" is one in which, for example, |2|=2. But many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value (but not the square thereof).
| |
| | |
| == Types of absolute value ==
| |
| The '''trivial''' absolute value is the absolute value with | ''x'' | = 0 when ''x'' = 0 and | ''x'' | = 1 otherwise.<ref>{{cite book|last=Koblitz|first=Neal|title=P-adic numbers, p-adic analysis, and zeta-functions|year=1984|publisher=Springer-Verlag|location=New York|isbn=978-0-387-96017-3|url=http://www.springer.com/mathematics/numbers/book/978-0-387-96017-3|edition=2nd ed.|accessdate=24 August 2012|page=3|quote=By the 'trivial' norm we mean the norm ‖ ‖ such that ‖0‖=0 and ‖x‖=1 for x≠0.}}</ref> Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a finite field because any element can be raised to some power to yield 1.
| |
| | |
| If | ''x'' + ''y'' | satisfies the stronger property | ''x'' + ''y'' | ≤ max(|''x''|, |''y''|), then | ''x'' | is called an [[ultrametric]] or '''non-Archimedean absolute value''', and otherwise an '''Archimedean absolute value'''.
| |
| | |
| == Places ==
| |
| If | ''x'' |<sub>1</sub> and | ''x'' |<sub>2</sub> are two absolute values on the same integral domain ''D'', then the two absolute values are ''equivalent'' if {{nowrap begin}}| ''x'' |<sub>1</sub> < 1{{nowrap end}} if and only if {{nowrap begin}} | ''x'' |<sub>2</sub> < 1.{{nowrap end}} If two nontrivial absolute values are equivalent, then for some exponent ''e'', we have | ''x'' |<sub>1</sub><sup>''e''</sup> = | ''x'' |<sub>2</sub>. Raising an absolute value to a power less than 1 results in another absolute value, but raising to a power greater than 1 does not necessarily result in an absolute value. (For instance, squaring the usual absolute value on the real numbers yields a function which is not an absolute value because it would violate the rule |x+y|≤|x|+|y|.) Absolute values up to equivalence, or in other words, an equivalence class of absolute values, is called a '''[[prime place|place]]'''.
| |
| | |
| [[Ostrowski's theorem]] states that the nontrivial places of the [[rational numbers]] '''Q''' are the ordinary [[absolute value]] and the [[p-adic number|''p''-adic absolute value]] for each prime ''p''.<ref>Cassels (1986) p.16</ref> For a given prime ''p'', any rational number ''q'' can be written as ''p''<sup>''n''</sup>(''a''/''b''), where ''a'' and ''b'' are integers not divisible by ''p'' and ''n'' is an integer. The ''p''-adic absolute value of ''q'' is
| |
| :<math>\left|p^n \frac{a}{b}\right|_p = p^{-n}.</math>
| |
| Since the ordinary absolute value and the ''p''-adic absolute values are absolute values according to the definition above, these define places.
| |
| | |
| == Valuations ==
| |
| {{main|Valuation (algebra)}}
| |
| If for some ultrametric absolute value and any base ''b''>1, we define ν(''x'') = -log<sub>''b''</sub> |''x''| for x ≠ 0 and ν(0) = ∞, where ∞ is ordered to be greater than all real numbers, then we obtain a function from ''D'' to '''R''' ∪ {∞}, with the following properties:
| |
| | |
| * ν(''x'') = ∞ ⇒ ''x'' = 0,
| |
| * ν(''xy'') = ν(''x'') + ν(''y''),
| |
| * ν(''x'' + ''y'') ≥ min(ν(''x''), ν(''y'')).
| |
| | |
| Such a function is known as a ''[[valuation (algebra)|valuation]]'' in the terminology of [[Bourbaki]], but other authors use the term ''valuation'' for ''absolute value'' and then say ''exponential valuation'' instead of ''valuation''.
| |
| | |
| == Completions ==
| |
| Given an integral domain ''D'' with an absolute value, we can define the [[Cauchy sequence]]s of elements of ''D'' with respect to the absolute value by requiring that for every ''r'' > 0 there is a positive integer ''N'' such that for all integers ''m'', ''n'' > ''N'' one has | ''x''<sub>''m'' </sub> − ''x''<sub>''n''</sub> | < ''r''. It is not hard to show that Cauchy sequences under pointwise addition and multiplication form a [[ring (mathematics)|ring]]. One can also define null sequences as sequences of elements of ''D'' such that | ''a''<sub>''n''</sub> | converges to zero. Null sequences are a [[prime ideal]] in the ring of Cauchy sequences, and the [[quotient ring]] is therefore an integral domain. The domain ''D'' is [[embedding|embedded]] in this quotient ring, called the [[complete metric space|completion]] of ''D'' with respect to the absolute value | ''x'' |.
| |
| | |
| Since fields are integral domains, this is also a construction for the completion of a field with respect to an absolute value. To show that the result is a field, and not just an integral domain, we can either show that null sequences form a [[maximal ideal]], or else construct the inverse directly. The latter can be easily done by taking, for all nonzero elements of the quotient ring, a sequence starting from a point beyond the last zero element of the sequence. Any nonzero element of the quotient ring will differ by a null sequence from such a sequence, and by taking pointwise inversion we can find a representative inverse element.
| |
| | |
| Another theorem of [[Alexander Ostrowski]] has it that any field complete with respect to an [[Archimedes|Archimedean]] absolute value is isomorphic to either the real or the complex numbers and the valuation is equivalent to the usual one.<ref>Cassels (1986) p.33</ref> The '''Gelfand-Tornheim theorem''' states that any field with an Archimedean valuation is isomorphic to a subfield of '''C''', the valuation being equivalent to the usual absolute value on '''C'''.<ref>http://modular.fas.harvard.edu/papers/ant/html/node60.html</ref>
| |
| | |
| == Fields and integral domains ==
| |
| If ''D'' is an integral domain with absolute value | ''x'' |, then we may extend the definition of the absolute value to the [[field of fractions]] of ''D'' by setting
| |
| | |
| :<math>|x/y| = |x|/|y|.\,</math>
| |
| | |
| On the other hand, if ''F'' is a field with ultrametric absolute value | ''x'' |, then the set of elements of ''F'' such that | ''x'' | ≤ 1 defines a [[valuation ring]], which is a subring ''D'' of ''F'' such that for every nonzero element ''x'' of ''F'', at least one of ''x'' or ''x''<sup>−1</sup> belongs to ''D''. Since ''F'' is a field, ''D'' has no zero divisors and is an integral domain. It has a unique [[maximal ideal]] consisting of all ''x'' such that | ''x'' | < 1, and is therefore a [[local ring]].
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| {{refbegin}}
| |
| *{{cite book | last=Bourbaki | first=Nicolas
| |
| | authorlink = Nicolas Bourbaki
| |
| | title = Commutative Algebra
| |
| | publisher = Addison-Wesley
| |
| | year = 1972}}
| |
| * {{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Local Fields | series=London Mathematical Society Student Texts | volume=3 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-31525-5 | zbl=0595.12006 }}
| |
| *{{cite book | last=Jacobson | first=Nathan
| |
| | authorlink = Nathan Jacobson
| |
| | title = Basic algebra II
| |
| | publisher = W H Freeman
| |
| | year = 1989
| |
| | isbn = 0-7167-1933-9
| |
| | edition = 2nd}} Chapter 9, paragraph 1 "''Absolute values''".
| |
| *{{cite book | last=Janusz | first = Gerald J.
| |
| | title = Algebraic Number Fields
| |
| | publisher = American Mathematical Society
| |
| | year = 1996, 1997
| |
| | isbn = 0-8218-0429-4
| |
| | edition = 2nd }}
| |
| {{refend}}
| |
| | |
| [[Category:Abstract algebra]]
| |
fee to watch your webinar at their convenience. Once it is in place, ganhe dinheiro promotion and possibly answering questions will be your only tasks.
Getting paid money to work online isn't the easiest thing to do in the world, but it is possible. If this is something you wish to work with, then the tips presented above should have helped you. Take some time, do things the right ganhar dinheiro way and then you can succeed.
Start your online como conseguir dinheiro earning today by following the great advice discussed in this article. Earning money is not as hard as it may seem, you just need to know how to get started. By choosing to put your right foot forward, you are heading off to a great start earning money to make ends meet. Here's more on como ganhar dinheiro have a look at http://ganhardinheironainternet.comoganhardinheiro101.com
Make money online by selling your talents. Good music is always in demand and with today's technological advances, anyone with musical talent can make music and offer it for sale ganhar dinheiro na internet to a broad audience. By setting up your own website and using social media for promotion, you can share your music with others and sell downloads with a free PayPal account.
It's easy to make money online. There is truth to the fact that you can start making money on the Internet as soon as you're done with this article. After como ganhar dinheiro na internet all, so many others are making money online, why not you? Keep your mind open and you can make a lot of money. As you ganhar dinheiro pela internet can ganhando dinheiro na internet see, there are many ways to approach the world of online income.
With various streams of income available, you are sure to find one, or two, that can help you with your income needs. Take this information to heart, put it to use and build your own online success story.