Absolute Value and the Real Line - Revision history

Khanh at 19:18, 27 November 2021

2021-11-27T19:18:11Z

Khanh at 19:16, 27 November 2021

2021-11-27T19:16:22Z

Khanh at 05:03, 22 November 2021

2021-11-22T05:03:28Z

Khanh at 05:02, 22 November 2021

2021-11-22T05:02:23Z

Khanh: Created page with "== The Real Line == The real line In mathematics, the '''real line''', or '''real number line''' is the line whose points..."

2021-11-22T04:50:25Z

Created page with "== The Real Line == thumb|right|382x382px|The real line In mathematics, the '''real line''', or '''real number line''' is the line whose points..."

New page

== The Real Line ==
[[Image:Real number line.svg|thumb|right|382x382px|The real line]]

In mathematics, the '''real line''', or '''real number line''' is the line whose points are the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric 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.

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.

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, 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.

===As a linear continuum===
[[File:Number line with x smaller than y.svg|thumb|300px|The order on the number line]]
[[File:Illustration of supremum.svg|thumb|300px|Each set on the real number line has a supremum.]]
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]].

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.

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]].

===As a metric space===
[[File:Absolute difference.svg|thumb|300px|right|The [[metric space|metric]] on the real line is [[absolute difference]].]]
[[File:Epsilon Umgebung.svg|thumb|300px|An {{math|''ε''}}-[[Ball (mathematics)|ball]] around a number {{math|''a''}}]]
The real line forms a [[metric space]], with the [[distance function]] given by absolute difference:
: <math>d(x, y) = |x - y|.</math>
The [[metric tensor]] is clearly the 1-dimensional [[Euclidean metric]]. Since the {{mvar|n}}-dimensional Euclidean metric can be represented in matrix form as the {{mvar|n}}-by-{{mvar|n}} identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.

If {{math|''p'' ∈ '''R'''}} and {{math|''ε'' > 0}}, then the {{mvar|ε}}-[[Ball (mathematics)|ball]] in {{math|'''R'''}} centered at {{mvar|p}} is simply the open [[Interval (mathematics)|interval]] {{math|(''p'' − ''ε'', ''p'' + ''ε'')}}.

This real line has several important properties as a metric space:
* The real line is a [[complete metric space]], in the sense that any [[Cauchy sequence]] of points converges.
* The real line is [[path-connected]] and is one of the simplest examples of a [[geodesic metric space]].
* The [[Hausdorff dimension]] of the real line is equal to one.

===As a topological space===
[[Image:Real projective line.svg|right|thumb|150px|The real line can be [[Compactification (mathematics)|compactified]] by adding a [[point at infinity]].]]
The real line carries a standard [[topological space|topology]], which can be introduced in two different, equivalent ways.
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)}}.

The real line is trivially a [[topological manifold]] of [[dimension]] {{Num|1}}. Up to homeomorphism, it is one of only two different connected 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.)

The real line is a [[locally compact space]] and a [[paracompact space]], 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.

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.

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]].

===As a vector space===
[[File:Bijection between vectors and points on number line.svg|thumb|300px|The bijection between points on the real line and vectors]]
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 the usual multiplication as an [[inner product]], making it a [[Euclidean vector space]]. The [[Norm (mathematics)|norm]] defined by this inner product is simply the [[absolute value]].

==As a measure space==
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.

Lebesgue measure on the real line is one of the simplest examples of a [[Haar measure]] on a [[locally compact group]].

===In real algebras===
The real line is a one-dimensional [[subspace (linear algebra)|subspace]] of a [[algebra over a field|real algebra]] ''A'' where '''R''' ⊂ ''A''.{{clarify|reason= What does this mean? It seems to be the tautology "if A contains R, then A contains the real line?|date=May 2020}} 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
:''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k
has a real line in the subspace {''q'' : ''x'' = ''y'' = ''z'' = 0 }.

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.

== Licensing ==
Content obtained and/or adapted from:
* [http://mathonline.wikidot.com/the-absolute-value-of-a-real-number The Absolute Value of a Real Number, mathonline.wikidot.com] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Real_line Real line, Wikipedia] under a CC BY-SA license

@@ Line 12: / Line 12: @@
 :*'''Proof:''' We will split this proof up into three cases.
-:*'''Case 1:</strong> Suppose that <math>a > 0</math>. Then <math>-a < 0</math>. Therefore by the definition of the absolute value of a number, <math>\mid a \mid = a</math>, and <math>\mid -a \mid = -(-a) = a</math>, and so <math>\mid a \mid = \mid -a \mid</math>.
+:*'''Case 1:''' Suppose that <math>a > 0</math>. Then <math>-a < 0</math>. Therefore by the definition of the absolute value of a number, <math>\mid a \mid = a</math>, and <math>\mid -a \mid = -(-a) = a</math>, and so <math>\mid a \mid = \mid -a \mid</math>.
 :*'''Case 2:''' Now suppose that <math>a = 0</math>. Therefore <math>-a = 0</math> and clearly <math>\mid a \mid = 0</math> and <math>\mid -a \mid = 0</math>, and so <math>\mid a \mid = \mid -a \mid</math>.

← Older revision		Revision as of 19:16, 27 November 2021
Line 1:		Line 1:
		+	== The Absolute Value of a Real Number ==
		+
		+	<blockquote style="background: white; border: 1px solid black; padding: 1em;">
		+	:'''Definition:''' If <math>a</math> is a real number, then we define the '''absolute value of the number <math>a</math>''' denoted <math>\mid a \mid</math> or <math>\mathrm{abs}(a)</math> as: <math>\mid a \mid = \left\{\begin{matrix} a & \mathrm{if\,a>0,}\\ 0 & \mathrm{if\,a = 0,}\\ -a & \mathrm{if\,a <0.} \end{matrix}\right.</math>.
		+	</blockquote>
		+
		+	For example, suppose we want to find the absolute value of <math>5</math>. Well since <math>5 > 0</math>, we note that <math>\mid 5 \mid = 5</math>. If we wanted to find the absolute value of <math>-5</math> then since <math>-5 < 0</math> we note that <math>\mid -5 \mid = -(-5) = 5</math>. We will now look at some important properties of the absolute values of real numbers utilizing The Order Properties of Real Numbers.
		+
		+
		+	*'''Theorem 1:''' If <math>a</math> is a real number then <math>\mid a \mid = \mid -a \mid</math>.
		+
		+	:*'''Proof:''' We will split this proof up into three cases.
		+
		+	:*'''Case 1:</strong> Suppose that <math>a > 0</math>. Then <math>-a < 0</math>. Therefore by the definition of the absolute value of a number, <math>\mid a \mid = a</math>, and <math>\mid -a \mid = -(-a) = a</math>, and so <math>\mid a \mid = \mid -a \mid</math>.
		+
		+	:*'''Case 2:''' Now suppose that <math>a = 0</math>. Therefore <math>-a = 0</math> and clearly <math>\mid a \mid = 0</math> and <math>\mid -a \mid = 0</math>, and so <math>\mid a \mid = \mid -a \mid</math>.
		+
		+	:*'''Case 3:''' Lastly suppose that <math>a < 0</math>. Then <math>-a > 0</math>. We obtain that <math>\mid a \mid = -a</math> and <math>\mid -a \mid = -a</math>, and so <math>\mid a \mid = \mid -a \mid</math>.
		+
		+	:*In all three cases we get that <math>\mid a \mid = \mid a \mid</math>. <math>\blacksquare</math>
		+
		+
		+	*'''Theorem 2:''' If <math>a</math> and <math>b</math> are real numbers then <math>\mid ab \mid = \mid a \mid \mid b \mid</math>.
		+
		+	:*'''Proof:''' We will split this proof up into three cases.</li>
		+
		+	:*'''Case 1:''' Suppose that <math>a = 0</math> or <math>b = 0</math> or both <math>a, b = 0</math>. Then <math>a \cdot b = 0</math>, and so <math>\mid a b \mid = 0</math>. Similarly <math>\mid a \mid \mid b \mid</math> is either <math>0 \cdot \mid b \mid</math> or <math>\mid a \mid \cdot 0</math> or <math>0 \cdot 0</math>, all of which equal <math>0</math>, so <math>\mid ab \mid = \mid a \mid \mid b \mid</math>.
		+
		+	:*'''Case 2:''' Suppose that <math>a, b > 0</math>. Then <math>ab > 0</math> and so <math>\mid ab \mid = ab</math> and <math>\mid a \mid \mid b \mid = ab</math>, so <math>\mid ab \mid = \mid a \mid \mid b \mid</math>.
		+
		+	:*'''Case 3:''' Suppose that one of <math>a > 0</math> and <math>b < 0</math>. Then <math>ab < 0</math>. So <math>\mid ab \mid = -ab</math>, and <math>\mid a \mid \mid b \mid = a \cdot -b = -ab</math>, so <math>\mid ab \mid = \mid a \mid \mid b \mid</math>.
		+
		+	:*'''Case 4:''' Suppose that <math>a, b < 0</math>. Then <math>ab > 0</math> and so <math>\mid ab \mid = ab</math> and <math>\mid a \mid \mid b \mid = -a \cdot -b = (-1)(-1)ab = ab</math>. So <math>\mid ab \mid = \mid a \mid \mid b \mid</math>.
		+
		+	:*In all four cases we get that <math>\mid ab \mid = \mid a \mid \mid b \mid</math>. <math>\blacksquare</math>
		+
		+
		+	*'''Theorem 3:''' If <math>a</math> is a real number then <math>\mid a \mid ^2 = a^2</math>.
		+
		+	:*'''Proof:''' We know that <math>a^2 > 0</math> and there by applying Theorem 2 we get that <math>a^2 = \mid a^2 \mid = \mid a\cdot a \mid = \mid a \mid \mid a \mid = \mid a \mid ^2</math>. <math>\blacksquare</math>
		+
		+
		+	*'''Theorem 4:''' If <math>c \geq 0</math> then <math>\mid a \mid \leq c</math> if and only if <math>-c \leq a \leq c</math>.
		+
		+	:*'''Proof:''' <math>\Rightarrow</math> If <math>\mid a \mid \leq c</math> then we have that both <math>a \leq c</math> and <math>-a \leq c</math> or rather <math>a \geq -c</math> which is equivalent to saying that <math>-c \leq a \leq c</math>.
		+
		+	:*<math>\Leftarrow</math> Suppose that <math>-c \leq a \leq c</math>. Then <math>a \leq c</math> and <math>-c \leq a \Leftrightarrow c \geq -a</math> so then <math>\mid a \mid \leq c</math>. <math>\blacksquare</math>
		+
		+
		+	*'''Theorem 5:''' If <math>a</math> is a real number then <math>-\mid a \mid \leq a \leq \mid a \mid</math>.
		+
		+
	== The Real Line ==		== The Real Line ==
	[[Image:Real number line.svg\|thumb\|right\|382x382px\|The real line]]		[[Image:Real number line.svg\|thumb\|right\|382x382px\|The real line]]

@@ Line 32: / Line 32: @@
 ===As a topological space===
-[[Image:Real projective line.svg|right|thumb|150px|The real line can be [[Compactification (mathematics)|compactified]] by adding a  [[point at infinity]].]]
+[[Image:Real projective line.svg|right|thumb|150px|The real line can be compactified by adding a  point at infinity.]]
 The real line carries a standard topology, which can be introduced in two different, equivalent ways. First, since the real numbers are 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 homeomorphic to the open interval (0, 1).

@@ Line 11: / Line 11: @@
 [[File:Number line with x smaller than y.svg|thumb|300px|The order on the number line]]
 [[File:Illustration of supremum.svg|thumb|300px|Each set on the real number line has a supremum.]]
-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]].
+The real line is a linear continuum under the standard < ordering. Specifically, the real line is linearly ordered by <, and this ordering is dense and has the least-upper-bound property.
-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.
+In addition to the above properties, the real line has no maximum or minimum element. It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line.
-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]].
+The real line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in {{math|'''R'''}} is countable. In order theory, the famous 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 independent of the standard axiomatic system of set theory known as ZFC.
 ===As a metric space===
-[[File:Absolute difference.svg|thumb|300px|right|The [[metric space|metric]] on the real line is [[absolute difference]].]]
+[[File:Absolute difference.svg|thumb|300px|right|The metric on the real line is absolute difference.]]
-[[File:Epsilon Umgebung.svg|thumb|300px|An {{math|''ε''}}-[[Ball (mathematics)|ball]] around a number {{math|''a''}}]]
+[[File:Epsilon Umgebung.svg|thumb|300px|An {{math|''ε''}}-ball around a number {{math|''a''}}]]
-The real line forms a [[metric space]], with the [[distance function]] given by absolute difference:
+The real line forms a metric space, with the distance function given by absolute difference:
 : <math>d(x, y) = |x - y|.</math>
-The [[metric tensor]] is clearly the 1-dimensional [[Euclidean metric]]. Since the {{mvar|n}}-dimensional Euclidean metric can be represented in matrix form as the {{mvar|n}}-by-{{mvar|n}} identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.
+The metric tensor is clearly the 1-dimensional Euclidean metric. Since the {{mvar|n}}-dimensional Euclidean metric can be represented in matrix form as the {{mvar|n}}-by-{{mvar|n}} identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.
-If {{math|''p'' ∈ '''R'''}} and {{math|''ε'' > 0}}, then the {{mvar|ε}}-[[Ball (mathematics)|ball]] in {{math|'''R'''}} centered at {{mvar|p}} is simply the open [[Interval (mathematics)|interval]] {{math|(''p'' − ''ε'', ''p'' + ''ε'')}}.
+If {{math|''p'' ∈ '''R'''}} and {{math|''ε'' > 0}}, then the {{mvar|ε}}-ball in {{math|'''R'''}} centered at {{mvar|p}} is simply the open interval {{math|(''p'' − ''ε'', ''p'' + ''ε'')}}.
 This real line has several important properties as a metric space:
-* The real line is a [[complete metric space]], in the sense that any [[Cauchy sequence]] of points converges.
+* The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.
-* The real line is [[path-connected]] and is one of the simplest examples of a [[geodesic metric space]].
+* The real line is path-connected and is one of the simplest examples of a geodesic metric space.
-* The [[Hausdorff dimension]] of the real line is equal to one.
+* The Hausdorff dimension of the real line is equal to one.
 ===As a topological space===
 [[Image:Real projective line.svg|right|thumb|150px|The real line can be [[Compactification (mathematics)|compactified]] by adding a  [[point at infinity]].]]
-The real line is trivially a [[topological manifold]] of [[dimension]] {{Num|1}}.  Up to homeomorphism, it is one of only two different connected 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.)
+The real line carries a standard topology, which can be introduced in two different, equivalent ways. First, since the real numbers are 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 homeomorphic to the open interval (0, 1).
-The real line is a [[locally compact space]] and a [[paracompact space]], 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.
+The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without 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.)
-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.
+The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore 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 groups and reduced homology groups are zero.
-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]].
+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 ends, and the resulting end compactification is the extended real line {{math|[−∞, +∞]}}.  There is also the Stone–Čech compactification of the real line, which involves adding an infinite number of additional points.
 ===As a vector space===
 [[File:Bijection between vectors and points on number line.svg|thumb|300px|The bijection between points on the real line and vectors]]
-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 the usual multiplication as an [[inner product]], making it a [[Euclidean vector space]]. The [[Norm (mathematics)|norm]] defined by this inner product is simply the [[absolute value]].
+The real line is a vector space over the field {{math|'''R'''}} of real numbers (that is, over itself) of dimension 1. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.
 ==As a measure space==
-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.
+The real line carries a canonical measure, namely the Lebesgue measure. This measure can be defined as the completion of a Borel measure defined on {{math|'''R'''}}, where the measure of any interval is the length of the interval.
-Lebesgue measure on the real line is one of the simplest examples of a [[Haar measure]] on a [[locally compact group]].
+Lebesgue measure on the real line is one of the simplest examples of a Haar measure on a locally compact group.
 ===In real algebras===
-The real line is a one-dimensional [[subspace (linear algebra)|subspace]] of a [[algebra over a field|real algebra]] ''A'' where '''R''' ⊂ ''A''.{{clarify|reason= What does this mean? It seems to be the tautology "if A contains R, then A contains the real line?|date=May 2020}} 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
+The real line is a one-dimensional subspace of a 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 quaternions
 :''q'' = ''w'' + ''x'' i + ''y'' j + ''z'' k
 has a real line in the subspace {''q'' : ''x'' = ''y'' = ''z'' = 0 }.
-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.
+When the real algebra is a 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 points of the conjugation.
 == Licensing ==