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__NOTOC__ {| border="0" cellspacing="0" cellpadding="0" height="30" width="100%" | style="border-bottom:1px solid #797979" width="5px" | &nbsp; {{Selected tab|[[5.2 Mathematical text|Theory]]}} {{Not selected tab|[[5.2 Exercises|Exercises]]}} | style="border-bottom:1px solid #797979" width="100%"| &nbsp; |} {{Info| '''Contents:''' * General advice * Mixing formulas and text * Common errors }} {{Info| '''Learning outcomes:''' After this section you will have learned how to: * Express mathematics * Explain mathematics }} ==Advice== ====Explain your solution==== The main advice is: Explain your solution. 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. If you just follow the above advice, you will have done 80% of what is required to provide an adequate solution. ====Write good English==== 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! ====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. 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. ====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 then of course no separate answer at the end is required. Simplify the answer as far as possible. <div class="exempel"> ''' Example 1''' <ol type="a"> <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>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> </ol> </div> ====Try and check sub-steps and answers==== 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"> '''Spurious roots ''' *Can occur by squaring *Can occur by multiplying equations with factors containing <math>x</math> *Roots outside the functions definition volume are false </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. <!-- Example --> <div class="exempel"> ''' Example 2''' 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>}} and then cancel <math>x</math> on both sides, {{Displayed math||<math>2x=5\,,</math>}} you will lose the solution <math>x=0</math>. If you instead factorize the left-hand side, {{Displayed math||<math>x(2x-5) = 0\,\textrm{,}</math>}} 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> 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==== 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. ==Treat formulas as part of the text== It is important that you write your solution in a way that makes it easy for others to follow. To help you we will present some don'ts and dos below to illustrate some tips and common errors that can occur when you mix formulas and text. <div class="regel"> '''Advice about mixing formulas and text:''' # Write the explanatory text on previos line # Think about the punctuation # Write displayed equations with indentation (or centered) </div> Formulas should not be seen as something that is extraneous to the text (or vice versa), but both text and formulas are to be integrated together in a linear flow. Therefore don´t write the text inside brackets behind the formulas. Instead, write the explanatory text on the previous line. <div class="exempel"> ''' Don't ''' formula (text text text text text text ...) formula (text text text text text text ...) </div> <div class="exempel"> ''' Do ''' Text text text text :formula. Text text text text :formula. </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. <div class="exempel"> ''' Do ''' ...text text text text text text text text :formula text text text text text text text text... (Note how the indentation highlights both the explanatory text and the formula. ) </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> 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"> ''' Do ''' ... and it is :formula'''<span style="color:green;">.</span>''' The next step is ... </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. <div class="exempel"> ''' Don't''' '''<span style="color:red;">3.</span>''' text text text text text text text text ... :formula '''<span style="color:red;">4.</span>''' text text text text text text text text ... </div> Sometimes one wants to refer back to a separate formula or equation, and in this case it can be given a number (or star) in brackets in the right or left margin. <div class="exempel"> ''' Do''' ...text text text text text text text text :{|width="100%" |align="left" width="100%"|formula. ||<span style="color:green;">(1)</span> |} Text text <span style="color: green;">(1)</span> text text text text text text :formula. Text text text text text text text text... </div> ==Common errors== ====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. 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"> ''' Example 3''' <ol type="a"> <li><math>x + 5 = 3\quad \Leftrightarrow\quad x = -2</math></li> <li><math>x^2-4x-1=0\quad\Leftrightarrow\quad (x-2)^2-5=0</math></li> <li><math>\sqrt x = x - 2\quad\Rightarrow\quad x = (x - 2)^2</math></li> </ol> </div> One often does not bother to write the symbol <math>\Leftrightarrow</math> between the different steps in a solution when they are on different lines (and thus the equivalence is implied). It is also often better to use explanatory text instead of arrows between the different steps in the solution. Do not use the implication arrow as a general continuation symbol (in the sense "The next step is"). The equal sign (<math>=</math>) is commonly used in two senses, firstly between things that are identical, eg <math>(x - 2)^2 = x^2-4x + 4</math> which is true for all <math>x</math>, and secondly in equations in which both sides are equal for some <math>x</math>, such as <math>(x - 2) ^2 = 4</math>, which only is satisfied if <math>x = 0</math> or <math>x = 4</math>. You should not mix these two different uses of the same symbol. <div class="exempel"> ''' Example 4''' Don't write :<math>x^2 - 2x + 1 = (x - 1)^2 = 4</math> when solving the equation <math>x^2 - 2x + 1 = 4</math>, since it can lead to misinterpretations. Write rather :<math>x^2 - 2x + 1 = 4\quad \Leftrightarrow\quad (x - 1) ^2 = 4</math>. </div> (There is also a third use of the equals sign, which occurs when defining an expression or for example an operation.) 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==== 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. <div class="exempel"> ''' Example 5''' <ol type="a"> <li>Do not write <math>1 + x / \cos x</math> when you really mean <math>(1 + x) / \cos x</math>.</li> <li>Do not write <math>1 + (1/\sin x)</math> when <math>1 + 1/\sin x</math> will do (even if the first expression is, formally, not wrong).</li> </ol> </div> When dealing with algebraic expressions one usually omits the multiplication sign. For example, one almost never would write <math>4\times x\times y\times z</math> but rather <math>4xyz</math>. This omission of the multiplication gives precedence over other multiplication and division (but not exponentiation). When one therefore writes <math>1/2R</math> it means <math>1 / (2R)</math> and not <math>(1 / 2) R</math>. Since this can be a source of misunderstanding, it is not entirely unusual to print the brackets in both situations. Arguments to the basic elementary functions are written without parentheses. Therefore, you should not write <center><math>\cos (x)</math>, <math>\sin (x)</math>, <math>\tan (x)</math>, <math>\cot (x)</math>, <math>\lg (x)</math> and <math>\ln (x)</math></center> but <center><math>\cos x</math>, <math>\sin x</math>, <math>\tan x</math>, <math>\cot x</math>, <math>\lg x</math> and <math>\ln x</math>.</center> In fact you should write <math>\cos 2x</math> and not <math>\cos (2x)</math> (since the argument <math>2x</math> is tightly linked together via a juxtaposition), but brackets are necessary when you write <math>\sin (x + y)</math>; <math>\sin(x / 2)</math> or <math>(\sin x)^2</math> (which you, alternatively, can write as <math>\sin ^2\!x</math>). <div class="inforuta" style="width:580px;"> '''Study advice''' '''Useful web sites''' * [http://scpd.stanford.edu/knuth/index.jsp A video course] in mathematical writing by Donald Knuth (A compendium accompaning the course is avalable in [http://www-cs-faculty.stanford.edu/~knuth/papers/mathwriting.tex.gz source form] or in excerpts from [http://books.google.com/books?id=dDOehHMbUMcC&printsec=frontcover&dq=inauthor:Donald+inauthor:Ervin+inauthor:Knuth&lr=&ei=JbN1SZfvFZysMqPPhM8M&hl=sv#PPP9,M1 Google books]). </div>

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