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| {{hatnote|This article covers advanced topics. For basic topics, see [[Number line]].}}
| | However, a large percent of the population is not happy with their height. I didn't believe my playing partner when he said I couldn't clear it now because I didn't have a club in my bag that I could make it with. ' Only one subspecies, the Uganda giraffe, is considered endangered. Coming up with your own poems for toddlers are a great way to add your touch to any holiday or classroom themed lesson plan. The diet ought to comprise adequate amounts of proteins, for the reason that the function of proteins in the repairing of tissues that are worn out and the increase in height is extremely important. <br><br>To accommodate the multi-passenger crew of the Voskhod, potentially disastrous steps were made. However there are thousands of people the world who are struggling to extend height. Therefore, it cannot rest on it's own it need a firm place of gravitational pull while resting unlike a bed. The high density and heavier cold air will be always pushing the low density and lighter hot gases up. Protein: This is a very essential nutrient to aid growth of the body. <br><br>Nowadays I have found the funny side and I can even laugh at myself for being that clumsy bloke, this is fine now but through my younger years it was not so easy. It demonstrates physical attributes and contributes to self confidence and self esteem. Simply height increase can show a significant change in anyone's life. That means each tree will get the moisture from a 5 foot diameter ground area without competition from the tree beside it. Recall that comparative adjectives are used in Latin to compare two persons or objects on some quality. <br><br>If you are asking "how can I get taller", then you must consider following a healthy diet. The ground being harder makes you back and neck firm which promotes health. One would think that tall mothers and fathers would produce tall children, or that a short mother would have short daughters even though dad and son are both tall. Stretching is very good to feel better in your body and it will allow you to not slouch and feel better which will make you grow taller. Ask him if he would have preferred to be slightly taller than what he is and note his answer. <br><br>As per the studies, if any child receives HGH hormone injections at least three times weekly then he can easily gain an average increase of more than one inch and when he enhances the dosage to six times a week then better results are produced and the average height gets increases by three inches. No-mow brings to mind something plastic like the artificial turf on a football field or the Brady's lawn from The Brady Bunch, but no-mow is a live grass. Some girls claim platform shoes are rather comfortable, and they most likely are because the entire length of the foot is on a flat plane instead of the heel being higher. Big Tex's clothing, a set of denim jeans and red-white-and blue cowboy shirt, contributed to the fast-burning blaze. The pictures of female bodies are seen being displayed all over the place.<br><br>Here is more info about how to grow taller exercises ([http://www.estampas.info/sitemap/ http://www.estampas.info]) visit the webpage. |
| {{no footnotes|date=January 2013}}
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| [[Image:Real number line.svg|thumb|right|350px|The real line]]
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| In [[mathematics]], the '''real line''', or '''real number line''' is the [[line (geometry)|line]] whose [[Point (geometry)|points]] are the [[real number]]s. That is, the real line is the [[set (mathematics)|set]] {{math|'''R'''}} of all real numbers, viewed as a [[geometry|geometric]] [[space (mathematics)|space]], namely the [[Euclidean space]] of [[dimension]] one. It can be thought of as a [[vector space]] (or [[affine space]]), a [[metric space]], a [[topological space]], a [[measure space]], or a [[linear continuum]].
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| Just like the set of real numbers, the real line is usually denoted by the symbol {{math|'''R'''}} (or alternatively, <math> \mathbb{R} </math>, the letter “[[R]]” in [[blackboard bold]]). However, it is sometimes denoted {{math|'''R'''<sup>1</sup>}} in order to emphasize its role as the first Euclidean space.
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| This article focuses on the aspects of {{math|'''R'''}} as a geometric space in [[topology]], geometry, and [[real analysis]]. The real numbers also play an important role in [[algebra]] as a [[field (mathematics)|field]], but in this context {{math|'''R'''}} is rarely referred to as a line. For more information on {{math|'''R'''}} in all of its guises, see [[real number]].
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| ==As a linear continuum==
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| The real line is a [[linear continuum]] under the standard {{math|<}} ordering. Specifically, the real line is [[linearly ordered set|linearly ordered]] by {{math|<}}, and this ordering is [[dense order|dense]] and has the [[least-upper-bound property]].
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| In addition to the above properties, the real line has no [[Greatest element|maximum]] or [[least element|minimum element]]. It also has a [[countable set|countable]] [[dense set|dense]] [[subset]], namely the set of [[rational number]]s. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is [[Order isomorphism|order-isomorphic]] to the real line.
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| The real line also satisfies the [[countable chain condition]]: every collection of mutually [[disjoint sets|disjoint]], [[nonempty]] open [[interval (mathematics)|interval]]s in {{math|'''R'''}} is countable. In [[order theory]], the famous [[Suslin's problem|Suslin problem]] asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to {{math|'''R'''}}. This statement has been shown to be [[independence (mathematical logic)|independent]] of the standard axiomatic system of [[set theory]] known as [[ZFC]].
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| ==As a metric space==
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| [[File:Absolute difference.svg|thumb|300px|right|The [[metric space|metric]] on the real line is [[absolute difference]].]]
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| The real line forms a [[metric space]], with the [[distance function]] given by absolute difference:
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| :{{math|''d''(''x'', ''y'') {{=}} {{!}} ''x'' − ''y'' {{!}} }}.
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| The [[metric tensor]] is clearly the 1-dimensional [[Euclidean metric]]. Since the'' {{math|n}}''-dimensional Euclidean metric can be represented in matrix form as the ''{{math|n}} ''by '' {{math|n}}'' identity matrix, the metric on the real line is simply the 1 by 1 identity matrix, i.e. 1.
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| If {{math|''p'' ∈ '''R'''}} and {{math|''ε'' > 0}}, then the {{math|''ε''}}-[[Ball (mathematics)|ball]] in {{math|'''R'''}} centered at {{math|''p''}} is simply the open [[Interval (mathematics)|interval]] {{math|(''p'' − ''ε'', ''p'' + ''ε'')}}.
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| This real line has several important properties as a metric space:
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| * The real line is a [[complete metric space]], in the sense that any [[Cauchy sequence]] of points converges.
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| * The real line is [[path-connected]], and is one of the simplest examples of a [[geodesic metric space]]
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| * The [[Hausdorff dimension]] of the real line is equal to one.
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| ==As a topological space==
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| [[Image:Real projective line.svg|right|thumb|150px|The real line can be [[Compactification (mathematics)|compactified]] by adding a [[point at infinity]].]]
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| The real line carries a standard [[topological space|topology]] which can be introduced in two different, equivalent ways.
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| First, since the real numbers are [[total order|totally ordered]], they carry an [[order topology]]. Second, the real numbers inherit a [[metric topology]] from the metric defined above. The order topology and metric topology on {{math|'''R'''}} are the same. As a topological space, the real line is [[homeomorphism|homeomorphic]] to the open interval {{math|(0, 1)}}.
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| The real line is trivially a [[topological manifold]] of [[dimension]] {{Num|1}}. Up to homeomorphism, it is one of only two different 1-manifolds without [[manifold with boundary|boundary]], the other being the [[circle]]. It also has a standard differentiable structure on it, making it a [[differentiable manifold]]. (Up to [[diffeomorphism]], there is only one differentiable structure that the topological space supports.) | |
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| The real line is [[locally compact]] and [[paracompact]], as well as [[second-countable space|second-countable]] and [[normal space|normal]]. It is also [[path-connected]], and is therefore [[connected space|connected]] as well, though it can be disconnected by removing any one point. The real line is also [[contractible]], and as such all of its [[homotopy group]]s and [[reduced homology]] groups are zero.
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| As a [[locally compact space]], the real line can be compactified in several different ways. The [[one-point compactification]] of {{math|'''R'''}} is a circle (namely the [[real projective line]]), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two [[End (topology)|ends]], and the resulting end compactification is the [[Extended real number line|extended real line]] {{math|[−∞, +∞]}}. There is also the [[Stone–Čech compactification]] of the real line, which involves adding an infinite number of additional points. | |
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| In some contexts, it is helpful to place other topologies on the set of real numbers, such as the [[lower limit topology]] or the [[Zariski topology]]. For the real numbers, the latter is the same as the [[finite complement topology]].
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| ==As a vector space==
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| The real line is a [[vector space]] over the [[field (mathematics)|field]] {{math|'''R'''}} of real numbers (that is, over itself) of [[dimension]] {{Num|1}}. It has a standard [[inner product]], making it a [[Euclidean space]]. (The inner product is simply ordinary [[multiplication]] of real numbers.) The standard [[Norm (mathematics)|norm]] on {{math|'''R'''}} is simply the [[absolute value]] function. | |
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| ==As a measure space==
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| The real line carries a canonical [[Measure (mathematics)|measure]], namely the [[Lebesgue measure]]. This measure can be defined as the [[Complete measure|completion]] of a [[Borel measure]] defined on {{math|'''R'''}}, where the measure of any interval is the length of the interval.
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| Lebesgue measure on the real line is one of the simplest examples of a [[Haar measure]] on a [[locally compact group]].
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| ==In real algebras==
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| The real line is a one-dimensional [[subspace (linear algebra)|subspace]] of a [[algebra over a field|real algebra]] ''A'' where '''R''' ⊂ ''A''. For example, in the [[complex plane]] ''z'' = ''x'' + i''y'', the subspace {''z'' : ''y'' = 0} is a real line. Similarly, the algebra of [[quaternion]]s
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| :''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k
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| has a real line in the subspace {''q'' : ''x'' = ''y'' = ''z'' = 0 }.
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| When the real algebra is a [[direct sum of modules|direct sum]] <math>A = R \oplus V,</math> then a '''conjugation''' on ''A'' is introduced by the mapping <math>v \mapsto -v</math> of subspace ''V''. In this way the real line consists of the [[fixed point (mathematics)|fixed point]]s of the conjugation.
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| ==See also==
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| * [[Line (geometry)]]
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| * [[Imaginary line (mathematics)]]
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| * [[Real projective line]]
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| ==References==
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| * {{cite book
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| | first = James
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| | last = Munkres
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| | author-link = James Munkres
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| | year = 1999
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| | title = Topology
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| | edition = 2nd
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| | publisher = [[Prentice Hall]]
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| | isbn = 0-13-181629-2
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| }}
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| * Walter Rudin, ''Real and Complex Analysis'', McGraw-Hill, 1966, ISBN 0-07-100276-6.
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| {{DEFAULTSORT:Real Line}}
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| [[Category:Real numbers]]
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| [[Category:Topological spaces]]
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| [[es:Recta real]]
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| [[eu:Zuzen erreal]]
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However, a large percent of the population is not happy with their height. I didn't believe my playing partner when he said I couldn't clear it now because I didn't have a club in my bag that I could make it with. ' Only one subspecies, the Uganda giraffe, is considered endangered. Coming up with your own poems for toddlers are a great way to add your touch to any holiday or classroom themed lesson plan. The diet ought to comprise adequate amounts of proteins, for the reason that the function of proteins in the repairing of tissues that are worn out and the increase in height is extremely important.
To accommodate the multi-passenger crew of the Voskhod, potentially disastrous steps were made. However there are thousands of people the world who are struggling to extend height. Therefore, it cannot rest on it's own it need a firm place of gravitational pull while resting unlike a bed. The high density and heavier cold air will be always pushing the low density and lighter hot gases up. Protein: This is a very essential nutrient to aid growth of the body.
Nowadays I have found the funny side and I can even laugh at myself for being that clumsy bloke, this is fine now but through my younger years it was not so easy. It demonstrates physical attributes and contributes to self confidence and self esteem. Simply height increase can show a significant change in anyone's life. That means each tree will get the moisture from a 5 foot diameter ground area without competition from the tree beside it. Recall that comparative adjectives are used in Latin to compare two persons or objects on some quality.
If you are asking "how can I get taller", then you must consider following a healthy diet. The ground being harder makes you back and neck firm which promotes health. One would think that tall mothers and fathers would produce tall children, or that a short mother would have short daughters even though dad and son are both tall. Stretching is very good to feel better in your body and it will allow you to not slouch and feel better which will make you grow taller. Ask him if he would have preferred to be slightly taller than what he is and note his answer.
As per the studies, if any child receives HGH hormone injections at least three times weekly then he can easily gain an average increase of more than one inch and when he enhances the dosage to six times a week then better results are produced and the average height gets increases by three inches. No-mow brings to mind something plastic like the artificial turf on a football field or the Brady's lawn from The Brady Bunch, but no-mow is a live grass. Some girls claim platform shoes are rather comfortable, and they most likely are because the entire length of the foot is on a flat plane instead of the heel being higher. Big Tex's clothing, a set of denim jeans and red-white-and blue cowboy shirt, contributed to the fast-burning blaze. The pictures of female bodies are seen being displayed all over the place.
Here is more info about how to grow taller exercises (http://www.estampas.info) visit the webpage.