5.2 Mathematical text - Revision history

Tek: Phil Ramsden changes

Tek — Mon, 16 Mar 2009 13:06:15 GMT

Phil Ramsden changes

←Older revision		Revision as of 13:06, 16 March 2009
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	<center>Explain your solution.</center>		<center>Explain your solution.</center>

-	The solution must not only be a statement of the ~~formulas that~~ you used, but also a description of how you reasoned. Use words to do this! ~~To achieve the appropriate level of the solution:~~ Imagine you are explaining the solution for a ~~class mate~~ who has difficulty in keeping up with all the steps. You need not explain every little calculation ~~but~~ nonetheless do not skip ~~over~~ important steps.	+	The solution must contain not only a statement of the formulae you used, but also a description of how you reasoned. Use words to do this! Imagine you are explaining the solution for a classmate who has difficulty in keeping up with all the steps. You need not explain every little calculation; nonetheless, do not skip important steps. If you simply follow the above advice, you will have done 80% of what is required to provide an adequate solution.
-	If you ~~just~~ follow the above advice, you will have done 80% of what is required to provide an adequate solution.	+

	====Write good English====		====Write good English====
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	====Make a clean copy of the solution====		====Make a clean copy of the solution====
-	After you have solved the problem, you should rewrite the solution. Then you can ~~rather~~ concentrate on ~~the~~ presentation ~~of the solution and~~ this even may lead to ~~improving your original~~ solution. A tip is to ask someone else to read your solution to detect ambiguities. It is better to postpone the presentation phase for a later date so that when you solve the problem the first time you are able to work freely and not commit yourself too early to a specific ~~way to solve the problem~~.	+	After you have solved the problem, you should rewrite the solution. Then you can concentrate on its presentation; this even may lead to improvements in the solution itself. A tip is to ask someone else to read your solution to detect ambiguities. It is better to postpone the presentation phase for a later date, so that when you solve the problem the first time you are able to work freely and not commit yourself too early to a specific solution method.
-		+
-	When you enter the solution ~~create a text document and not~~ as screen ~~dumps~~ from ~~your~~ word processor. ~~While it~~ may be easier to write the solution on your own computer using your ~~favorite~~ program, you should remember that in the next phase your solution is to be included as part of a group project and thus it is important that the solution ~~is in editable state~~.	+	When you enter the solution, be sure to enter it as text, rather than (say) a screen capture from a word processor. It may be easier to write the solution on your own computer using your favourite program, but you should remember that in the next phase your solution is to be included as part of a group project and thus it is important that the solution can be edited, which a screen capture cannot.

	====A clear answer====		====A clear answer====
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	</div>		</div>

-	An important part of the solution is to think ~~out~~ reasonable methods to check the answer. For example, ~~inserting~~ the solution of ~~the~~ equation and make sure that it really is a solution because one may well have calculated wrongly (do not confuse this with the investigation of spurious roots). This may also be done for the sub-steps in a solution.	+	An important part of the solution process is to think of reasonable methods to check the answer. For example, one might substitute the solution of an equation back into that equation, and make sure that it really is a solution, because one may well have calculated wrongly (do not confuse this with the investigation of spurious roots). This may also be done for the sub-steps in a solution.

	Another point is to assess if the answer is reasonable. Insert values for some of the parameters and ensure that you get the right answer. Eg what happens if <math>a = 0</math>, <math>a = 1</math> or a goes to infinity?		Another point is to assess if the answer is reasonable. Insert values for some of the parameters and ensure that you get the right answer. Eg what happens if <math>a = 0</math>, <math>a = 1</math> or a goes to infinity?
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	Simple arrow (<math>\rightarrow</math>) is used in mathematics often to handle different kinds of limits: <math>a \to \infty</math> means that a increases without limit (goes towards infinity). You will probably not need to use a simple arrow in this course.		Simple arrow (<math>\rightarrow</math>) is used in mathematics often to handle different kinds of limits: <math>a \to \infty</math> means that a increases without limit (goes towards infinity). You will probably not need to use a simple arrow in this course.

-	====Do not be careless with ~~parenthesis~~====	+	====Do not be careless with brackets====
	Since multiplication and division have higher priority than addition and subtraction, one must use brackets when addition and/or subtraction is to be carried out first.		Since multiplication and division have higher priority than addition and subtraction, one must use brackets when addition and/or subtraction is to be carried out first.

Tek at 08:12, 13 March 2009

Tek — Fri, 13 Mar 2009 08:12:14 GMT

←Older revision		Revision as of 08:12, 13 March 2009
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	====Explain your solution====		====Explain your solution====
-	The main advice is:	+	The main advice is:
-	Explain your solution.	+
		+	<center>Explain your solution.</center>

	The solution must not only be a statement of the formulas that you used, but also a description of how you reasoned. Use words to do this! To achieve the appropriate level of the solution: Imagine you are explaining the solution for a class mate who has difficulty in keeping up with all the steps. You need not explain every little calculation but nonetheless do not skip over important steps.		The solution must not only be a statement of the formulas that you used, but also a description of how you reasoned. Use words to do this! To achieve the appropriate level of the solution: Imagine you are explaining the solution for a class mate who has difficulty in keeping up with all the steps. You need not explain every little calculation but nonetheless do not skip over important steps.
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	====Write good English====		====Write good English====
	Although this is not an assignment in English and, of course, it is the		Although this is not an assignment in English and, of course, it is the
-	mathematical content that is the most important, you should nonetheless think about things like typos, grammatical errors, etc. If your solution has too many language errors it can give a very negative impression and affect the credibility of the solution. Impression is important!	+	mathematical content that is the most important, you should nonetheless think about things like typos, grammatical errors, etc. If your solution has too many language errors it can give a very negative impression and affect the credibility of the solution. Impression is important!

	====Make a clean copy of the solution====		====Make a clean copy of the solution====
-	After you have solved the problem, you should rewrite the solution. Then you can rather concentrate on the presentation of the solution and this even may lead to improving your original solution. A tip is to ask someone else to read your solution to detect ambiguities. It is better to postpone the presentation phase for a later date so that when you solve the problem the first time you are able to work freely and not commit yourself too early to a specific way to solve the problem.	+	After you have solved the problem, you should rewrite the solution. Then you can rather concentrate on the presentation of the solution and this even may lead to improving your original solution. A tip is to ask someone else to read your solution to detect ambiguities. It is better to postpone the presentation phase for a later date so that when you solve the problem the first time you are able to work freely and not commit yourself too early to a specific way to solve the problem.
		+
	When you enter the solution create a text document and not as screen dumps from your word processor. While it may be easier to write the solution on your own computer using your favorite program, you should remember that in the next phase your solution is to be included as part of a group project and thus it is important that the solution is in editable state.		When you enter the solution create a text document and not as screen dumps from your word processor. While it may be easier to write the solution on your own computer using your favorite program, you should remember that in the next phase your solution is to be included as part of a group project and thus it is important that the solution is in editable state.

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	<li>Do not return the answer <math>\sqrt8</math>, but give <math>2\sqrt2</math> as the answer. </li>		<li>Do not return the answer <math>\sqrt8</math>, but give <math>2\sqrt2</math> as the answer. </li>
	<li>Do not give the answer as <math>\sin^2 x + \cos^2x + 2\sin 2x</math> but as <math>1 + 2\sin 2x</math>.</li>		<li>Do not give the answer as <math>\sin^2 x + \cos^2x + 2\sin 2x</math> but as <math>1 + 2\sin 2x</math>.</li>
-	<li> Do not give the answer as <math>x = \left\{\begin{align}&\pi/4+ n\pi\\ &3\pi / 4 + n\pi\end{align}\right.</math> but as <math>x = \pi / 4 + n\pi / 2</math>.</li>	+	<li> Do not give the answer as <math>x = \left\{\begin{align}&\pi/4+ n\pi\\ &3\pi / 4 + n\pi\end{align}\right.\ \ (n\ \text{integer})\ </math> but as <math>\ x = \pi / 4 + n\pi / 2\ \ (n\ \text{integer})</math>.</li>
	</ol>		</ol>

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	It can happen that when you solve some equations so called spurious roots turn up as a consequence of the method of solution that is being used. In these cases, explain why spurious roots may have appeared and test the solutions to see which are real solutions and which are spurious roots.		It can happen that when you solve some equations so called spurious roots turn up as a consequence of the method of solution that is being used. In these cases, explain why spurious roots may have appeared and test the solutions to see which are real solutions and which are spurious roots.

		+	<!--
	<div class="regel">		<div class="regel">
	'''Spurious roots '''		'''Spurious roots '''
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	*Roots outside the functions definition volume are false		*Roots outside the functions definition volume are false
	</div>		</div>
		+	-->

	Lost solutions. Eg a factor on both sides of the equation is cancelled out and one does not realize that the equation obtained by setting this factor to zero provides additional solutions.		Lost solutions. Eg a factor on both sides of the equation is cancelled out and one does not realize that the equation obtained by setting this factor to zero provides additional solutions.
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	====Draw clear figures====		====Draw clear figures====
	A figure may explain introduced symbols or reasoning many times		A figure may explain introduced symbols or reasoning many times
-	better than text, so please use figures. Bear in mind to draw them clearly and do not overload a figure with too many ~~things~~. It may be better to have several nearly identical figures where each illustrates one point rather than have a great combination-figure which illustrates everything.	+	better than text, so please use figures. Bear in mind to draw them clearly and do not overload a figure with too many details. It may be better to have several nearly identical figures where each illustrates one point rather than have a great combination-figure which illustrates everything.


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	''' Don't '''		''' Don't '''

-	formula (text text text text text text ...)	+	''formula'' (text text text text text text ...)

-	formula (text text text text text text ...)	+	''formula'' (text text text text text text ...)

	</div>		</div>
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	Text text text text		Text text text text

-	:formula.	+	::''formula''.

	Text text text text		Text text text text

-	:formula.	+	::''formula''.

	</div>		</div>
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	''' Do '''		''' Do '''

-	...text text text text text text text text	+	...text text text ''<span style="color:green">formula</span>'' text text text text.
		+
		+	Text text text

-	:formula	+	::''<span style="color:green">formula</span>''

	text text text text text text text text...		text text text text text text text text...
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	...which provides that'''<span style="color:red">:</span>'''		...which provides that'''<span style="color:red">:</span>'''

-	:formula	+	::''formula''

	We start...		We start...
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	... and it is		... and it is

-	:formula'''<span style="color:green;">.</span>'''	+	::''formula'''''<span style="color:green;">.</span>'''

	The next step is ...		The next step is ...
		+
		+	(Note the full stop after the formula above.)

	</div>		</div>

-	A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You seldom need to refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.	+	A bad habit is excessive numbering. For example, to put a number in front of each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You seldom need to refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.

	<div class="exempel">		<div class="exempel">
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	'''<span style="color:red;">3.</span>''' text text text text text text text text ...		'''<span style="color:red;">3.</span>''' text text text text text text text text ...

-	:formula	+	::''formula''

	'''<span style="color:red;">4.</span>''' text text text text text text text text ...		'''<span style="color:red;">4.</span>''' text text text text text text text text ...
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	...text text text text text text text text		...text text text text text text text text

-	:{\|width="100%"	+	::{\|width="100%"
-	\|align="left" width="100%"\|formula.	+	\|align="left" width="100%"\|''formula''.
	\|\|<span style="color:green;">(1)</span>		\|\|<span style="color:green;">(1)</span>
	\|}		\|}
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	Text text <span style="color: green;">(1)</span> text text text text text text		Text text <span style="color: green;">(1)</span> text text text text text text

-	:formula.	+	::''formula''.

	Text text text text text text text text...		Text text text text text text text text...
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	There is a difference between <math>\Rightarrow</math> (implication arrow), <math>\Leftrightarrow</math> (equivalence arrow) and <math>=</math> (equals sign). For two equations that are known a priori to have the same solutions one uses the equivalence arrow <math>\Leftrightarrow</math> to represent this.		There is a difference between <math>\Rightarrow</math> (implication arrow), <math>\Leftrightarrow</math> (equivalence arrow) and <math>=</math> (equals sign). For two equations that are known a priori to have the same solutions one uses the equivalence arrow <math>\Leftrightarrow</math> to represent this.

-	However, if we write Equation 1 <math>\Rightarrow</math> Equation 2, it means that all solutions that Equation 1 has, Equation 2 also has, (but Equation 2 may have more solutions).	+	However, if we write "Equation 1 <math>\Rightarrow</math> Equation 2", it means that all solutions that Equation 1 has, Equation 2 also has, (but Equation 2 may have more solutions).

	<div class="exempel">		<div class="exempel">

Lina at 12:59, 3 February 2009

Lina — Tue, 03 Feb 2009 12:59:51 GMT

←Older revision		Revision as of 12:59, 3 February 2009
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	</div>		</div>

-	A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You ~~need~~ seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.	+	A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You seldom need to refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.

	<div class="exempel">		<div class="exempel">

Lina at 12:56, 3 February 2009

Lina — Tue, 03 Feb 2009 12:56:02 GMT

←Older revision		Revision as of 12:56, 3 February 2009
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	<div class="regel">		<div class="regel">
	'''Spurious roots '''		'''Spurious roots '''
-	#Can occur by squaring	+	*Can occur by squaring
-	#Can occur by multiplying equations with factors containing <math>x</math>	+	*Can occur by multiplying equations with factors containing <math>x</math>
-	#Roots outside the functions definition volume are false	+	*Roots outside the functions definition volume are false
	</div>		</div>

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	</div>		</div>

-	An important part of the solution is to think out reasonable methods to check the answer. For example, inserting the solution of the equation and make sure that it really is a solution because one may well have calculated wrongly (do not confuse this with the investigation of spurious roots). This may also be done for the sub-steps in a solution. Another point is to assess if the answer is reasonable. Insert values for some of the parameters and ensure that you get the right answer (what happens if <math>a = 0</math>, <math>a = 1</math> or a goes to infinity?)	+	An important part of the solution is to think out reasonable methods to check the answer. For example, inserting the solution of the equation and make sure that it really is a solution because one may well have calculated wrongly (do not confuse this with the investigation of spurious roots). This may also be done for the sub-steps in a solution.
		+
		+	Another point is to assess if the answer is reasonable. Insert values for some of the parameters and ensure that you get the right answer. Eg what happens if <math>a = 0</math>, <math>a = 1</math> or a goes to infinity?

	====Draw clear figures====		====Draw clear figures====
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	# Write the explanatory text on previos line		# Write the explanatory text on previos line
	# Think about the punctuation		# Think about the punctuation
-	# Write ~~displyed~~ equations with indentation (or centered)	+	# Write displayed equations with indentation (or centered)
	</div>		</div>

Lina at 12:43, 3 February 2009

Lina — Tue, 03 Feb 2009 12:43:24 GMT

←Older revision		Revision as of 12:43, 3 February 2009
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	====A clear answer====		====A clear answer====
-	Write a clear answer at the end. This is especially important if the solution is long and the answer is scattered in various parts of the text. However, there are problems where the actual solution constitutes the answer (e.g. "Show that ...") and of course no separate answer at the end is required.	+	Write a clear answer at the end. This is especially important if the solution is long and the answer is scattered in various parts of the text. However, there are problems where the actual solution constitutes the answer (e.g. "Show that ...") and then of course no separate answer at the end is required.

	Simplify the answer as far as possible.		Simplify the answer as far as possible.

Tek at 07:57, 29 January 2009

Tek — Thu, 29 Jan 2009 07:57:44 GMT

←Older revision		Revision as of 07:57, 29 January 2009
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	<div class="exempel">		<div class="exempel">
-	''' Example '''	+	''' Example 1'''

	<ol type="a">		<ol type="a">
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	<!-- Example -->		<!-- Example -->
	<div class="exempel">		<div class="exempel">
-	''' Example '''	+	''' Example 2'''

-	If you solve the ~~expression~~ <math>2x^2-5x=0</math> by first moving <math>5x</math> to the ~~left~~ side	+	If you solve the equation <math>2x^2-5x=0</math> by first moving <math>5x</math> to the right-hand side,

-	{{Displayed math\|\|<math>2x^2=5x</math>.}}	+	{{Displayed math\|\|<math>2x^2=5x\,,</math>}}

-	~~Then divide both sides with~~ <math>x</math>	+	and then cancel <math>x</math> on both sides,

-	{{Displayed math\|\|<math>2x=5</math>.}}	+	{{Displayed math\|\|<math>2x=5\,,</math>}}

-	~~You~~ will lose the solution <math>x=0</math>.	+	you will lose the solution <math>x=0</math>.

-	If we instead factorize the ~~expression we will get two solutions to the expression~~	+	If you instead factorize the left-hand side,

-	{{Displayed math\|\|<math>x(2x-5) = 0\,\textrm{.}</math>}}	+	{{Displayed math\|\|<math>x(2x-5) = 0\,\textrm{,}</math>}}

-	~~To practice factorise expressions go~~ to ~~[[2~~.~~1 Exercises#Exercise 2~~.~~1:3 \|Exercise~~ 2.~~1:3]]~~	+	you will be able to read off both solutions: <math>x=0</math> and <math>2x-5=0</math> (i.e. <math>x=\tfrac{5}{2}</math>).

		+	Go to [[2.1 Exercises#Exercise 2.1:3 \|Exercise 2.1:3]] to practice factorization.
	</div>		</div>

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	A figure may explain introduced symbols or reasoning many times		A figure may explain introduced symbols or reasoning many times
	better than text, so please use figures. Bear in mind to draw them clearly and do not overload a figure with too many things. It may be better to have several nearly identical figures where each illustrates one point rather than have a great combination-figure which illustrates everything.		better than text, so please use figures. Bear in mind to draw them clearly and do not overload a figure with too many things. It may be better to have several nearly identical figures where each illustrates one point rather than have a great combination-figure which illustrates everything.
		+

	==Treat formulas as part of the text==		==Treat formulas as part of the text==
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	</div>		</div>

-	As a formula is to be part of the text, it must be treated as part of the sentence. Think therefore about the punctuation. For example, do not forget the full stop after ~~the~~ formula.	+	As a formula is to be part of the text, it must be treated as part of the sentence. Think therefore about the punctuation. For example, do not forget the full stop after a formula if it ends a sentence.

	<div class="exempel">		<div class="exempel">
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	</div>		</div>
-

	A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You need seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.		A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You need seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.
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-	==~~5.3~~ Common errors==	+	==Common errors==
	====Be careful with arrows and similarities====		====Be careful with arrows and similarities====
	There is a difference between <math>\Rightarrow</math> (implication arrow), <math>\Leftrightarrow</math> (equivalence arrow) and <math>=</math> (equals sign). For two equations that are known a priori to have the same solutions one uses the equivalence arrow <math>\Leftrightarrow</math> to represent this.		There is a difference between <math>\Rightarrow</math> (implication arrow), <math>\Leftrightarrow</math> (equivalence arrow) and <math>=</math> (equals sign). For two equations that are known a priori to have the same solutions one uses the equivalence arrow <math>\Leftrightarrow</math> to represent this.
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	<div class="exempel">		<div class="exempel">
-	''' Example '''	+	''' Example 3'''

	<ol type="a">		<ol type="a">
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	<div class="exempel">		<div class="exempel">
-	''' Example '''	+	''' Example 4'''

	Don't write		Don't write
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	<div class="exempel">		<div class="exempel">
-	''' Example '''	+	''' Example 5'''

	<ol type="a">		<ol type="a">

Lina at 15:12, 27 January 2009

Lina — Tue, 27 Jan 2009 15:12:45 GMT

←Older revision		Revision as of 15:12, 27 January 2009
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	:formula.		:formula.

-	</div>	+	</div>

-	~~A common mistake is to use a colon in front~~ of ~~all displayed~~ formulas.	+	Formulas can be either written as part of the text or as separate formulas. When formulas are separated from the text they appear on their own line and are either centered or slightly indented.

	<div class="exempel">		<div class="exempel">
-	''' ~~Don't~~'''	+	''' Do '''

-	...~~which provides that'''<span style="color:red">:</span>'''~~	+	...text text text text text text text text

	:formula		:formula

-	~~We start~~...	+	text text text text text text text text...

-	(Note ~~also there should be a full stop after~~ the formula ~~above~~.)	+	(Note how the indentation highlights both the explanatory text and the formula. )

-	</div>	+	</div>

-	~~Formulas can be either written as part~~ of ~~the text or as separate~~ formulas~~. When formulas are separated from the text they appear on their own line and are either centered or slightly indented~~.	+	A common mistake is to use a colon in front of all displayed formulas.

	<div class="exempel">		<div class="exempel">
-	''' Do '''	+	''' Don't'''

-	...~~text text text text text text text text~~	+	...which provides that'''<span style="color:red">:</span>'''

	:formula		:formula

-	~~text text text text text text text text~~...	+	We start...

-	(Note ~~how the indentation highlights both the explanatory text and~~ the formula. )	+	(Note also there should be a full stop after the formula above.)

	</div>		</div>

Lina at 15:11, 27 January 2009

Lina — Tue, 27 Jan 2009 15:11:43 GMT

←Older revision		Revision as of 15:11, 27 January 2009
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	</div>		</div>
		+
		+	A common mistake is to use a colon in front of all displayed formulas.
		+
		+	<div class="exempel">
		+	''' Don't'''
		+
		+	...which provides that'''<span style="color:red">:</span>'''
		+
		+	:formula
		+
		+	We start...
		+
		+	(Note also there should be a full stop after the formula above.)
		+
		+	</div>

	Formulas can be either written as part of the text or as separate formulas. When formulas are separated from the text they appear on their own line and are either centered or slightly indented.		Formulas can be either written as part of the text or as separate formulas. When formulas are separated from the text they appear on their own line and are either centered or slightly indented.
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	</div>		</div>

-	A common mistake is to use a colon in front of all displayed formulas.
-
-	<div class="exempel">
-	''' Don't'''
-
-	...which provides that'''<span style="color:red">:</span>'''
-
-	:formula
-
-	We start...
-
-	(Note also there should be a full stop after the formula above.)
-
-	</div>

	A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You need seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.		A bad habit is numbering each step in a solution (numbering should be used for enumeration). The extra digits do not add anything but rather distract. You need seldom refer back to the individual steps, and when you need to, you can often write something of the sort "when we squared the equation" etc.

Lina at 14:16, 27 January 2009

Lina — Tue, 27 Jan 2009 14:16:08 GMT

←Older revision		Revision as of 14:16, 27 January 2009
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	{{Displayed math\|\|<math>x(2x-5) = 0\,\textrm{.}</math>}}		{{Displayed math\|\|<math>x(2x-5) = 0\,\textrm{.}</math>}}
-
-	From this factorized expression, we read off that the solutions are <math>x=0</math> and <math>x=5/2\,</math>.

	To practice factorise expressions go to [[2.1 Exercises#Exercise 2.1:3 \|Exercise 2.1:3]]		To practice factorise expressions go to [[2.1 Exercises#Exercise 2.1:3 \|Exercise 2.1:3]]

Lina at 14:13, 27 January 2009

Lina — Tue, 27 Jan 2009 14:13:57 GMT

←Older revision		Revision as of 14:13, 27 January 2009
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	<div class="regel">		<div class="regel">
	'''Spurious roots '''		'''Spurious roots '''
-	#Can occur by ~~Squaring~~	+	#Can occur by squaring
	#Can occur by multiplying equations with factors containing <math>x</math>		#Can occur by multiplying equations with factors containing <math>x</math>
	#Roots outside the functions definition volume are false		#Roots outside the functions definition volume are false
Line 90:		Line 90:

	To practice factorise expressions go to [[2.1 Exercises#Exercise 2.1:3 \|Exercise 2.1:3]]		To practice factorise expressions go to [[2.1 Exercises#Exercise 2.1:3 \|Exercise 2.1:3]]
-
-	See also [[4.4 Exercises#Exercise 4.4:6 \|Exercise 4.4:6a]]

	</div>		</div>