A Guide to Writing Mathematics

1 A Guide to Writing MathematicsDr. Kevin P. LeeIntroductionThis is a math class! Why are we Writing ?There is a good chance that you have never written a paper in a math class before. So youmight be wondering why Writing is required in your math class Greek wordmathemas, from which we derive the word Mathematics , embodies thenotions of knowledge, cognition, understanding, and perception. In the end, Mathematics isaboutideas. In math classes at the university level, the ideas and concepts encountered aremore complex and sophisticated. The Mathematics learned in college will include conceptswhich cannot be expressed using just equations and formulas. Puttingmathemason paperwill require Writing sentences and paragraphs in addition to the equations and actually spend a great deal of time Writing .

2 If a mathematician wantsto contribute to the greater body of mathematical knowledge, she must be ablecommunicate her ideas in a way which is comprehensible to others. Thus, being able towrite clearly is as important a mathematical skill as being able to solve the ability to write clear mathematical explanations is important fornon-mathematicians as well. As you continue taking math courses in college, you will cometo know more Mathematics than most other people. When you use your mathematicalknowledge in the future, you may be required to explain your thinking process to anotherperson (like your boss, a co-worker, or an elected o cial), and it will be quite likely thatthis other person will know less math than you do. Learning how to communicatemathematical ideas clearly can help you advance in your will nd that Writing good mathematical explanations will improve your knowledgeand understanding of the mathematical ideas you encounter.

3 Putting an idea on paperrequires careful thought and attention. Hence, Mathematics which is written clearly andcarefully is more likely to be correct. The process of Writing will help you learn and retainthe concepts which you will be exploring in your math does good mathematical Writing look like?As you learn more math , being able to express mathematical ideas will become moreimportant. It will no longer be su cient just to be able to write down some nal \answer".There is a good reason why Herman Melville wroteMoby Dickas a novel and not as thesingle this same reason, just Writing down your nal conclusions in an assignment will not besu cient for a university math should not confuse Writing Mathematics with \showing your work".

4 You will notbe Writing math papers to demonstrate that you have done the homework. Rather, youwill be Writing to demonstrate how well you understand mathematical ideas and list of calculations without any context or explanation demonstrates that you've spentsome time doing computations; however, a list of calculations without any explanationsomits ideas. The ideas are the Mathematics . So a page of computations without anywriting or explanationcontains no you write a paper in a math class, your goal will be to communicatemathematical reasoning and ideas clearly to another person. The Writing done in a mathclass is very similar to the Writing done for other classes. Your are probably already usedto Writing papers in other subjects like psychology, history, and literature.

5 You can followmany of the same guidelines in a Mathematics paper as you would in a paper written aboutthese other : Combining Words and EquationsFollowing the rules of Writing observes the rules of grammar. This applies to Writing in mathematicspapers as well! When you write in a math class, you are expected to use correct grammarand spelling. Your Writing should be clear and professional. Do not use any irregularabbreviations or shorthand forms which do not conform to standard Writing is written with sentences in paragraphs. (And yes, paragraphs are is not amusing to read a three-page paper consisting of just one paragraph.)There is however one element in mathematical Writing which is not found in other typesof Writing : formulas.

6 However, it may surprise you to know that in a math paper, formulasand equations follow the standard grammatical rules that apply to words. Mathematical2symbols can correspond to di erent parts of speech. For instance, below is a perfectly goodcomplete +1=2:The symbol \=" acts like a verb. Below are a couple more examples of complete < 2:5z2R:9 s6=t:Can you identify the verbs? On the other hand, an expression like5x2z 10yis not a complete sentence. There is no verb. Such an expression should be treated as anoun. Can you identify the nouns in the previous examples?Formulas and equations need to be contained in complete sentences with properpunctuation. Here is an example:The total revenue,R, made from selling widgets is givenby the equationR=pq;wherepis the price at which each widget is sold andqisthe number of widgets sold.

7 Based on past experience,we know that when widgets are priced at $15 each, 2000widgets will be sold. We also know that for every dollarincrease in price, 150 fewer widgets are sold. Hence, ifthe price is increased byxdollars, then the revenue isR=(15+x)(2000 150x)= 150x2 250x+30;000:Notice how punctuation follows each of equations. A computation which ends a sentenceneeds to end with a period. Computations which do not end sentences are followed good way to improve your mathematical Writing is by reading your Writing ,includingall of the equations, out loud. Your ears can often pick out sentence fragments andgrammatical errors better than your eyes. If you nd yourself saying a series of fragmentedsentences and equations, you should do some are a couple of other important things to observe in the above example.

8 Noticehow \we" is used. The use of rst person is common in Mathematics , especially the plural\we", so don't be afraid to use the word \we" in the papers you write in your math thing to notice is that important or long formulas are written on separatelines. You can make your mathematical Writing easier to read if you place each importantformula on a line of its own. It's hard to pick out the important formulas below:Ifdis Bob's distance above the ground in feet, thend=100 16t2;wheretis the number of seconds after Bob'sFlugelputz-Levitator is activated. Solving fortin theequation 100 16t2= 0, we nd thatt=2:5. Bob hitsthe ground after 2:5 is clearer:Ifdis Bob's distance above the ground in feet, thend= 100 16t2;wheretis the number of seconds after Bob'sFlugelputz-Levitator is activated.

9 Solving fortin theequation100 16t2=0;we nd thatt=2:5. Bob hits the ground after 2:5 and is important to use words and symbols appropriately. Part of being able to writemathematics well is knowing when to use symbols and knowing when to use 't use mathematical symbols when you really mean something else. A commonmistake is to misuse the \=" symbol. For instance:32x 2(3x)= 1=(3x)2 2(3x)+1=0 =(3x 1)2=0 = 3x=1 =x=0:!!Do not use the equal sign when you really mean \the next step is" or \implies". The aboveexample is really saying that 1 = 0 = 1! Using arrows instead of equal signs is a slightimprovement, but still not desirable:432x 2(3x)= 1!(3x)2 2(3x)+1=0!(3x 1)2=0!3x=1!x=0:!With a sequence of calculations, sometimes it is best to just place each equation on aseparate 2(3x)= 1(3x)2 2(3x)+1=0(3x 1)2=03x=1x=0:For a di cult computation where the reader might not readily follow each step, you caninclude words to describe the steps you wa nt t o s o l ve f o rxin the equation32x 2(3x)= 1:We can rewrite this equation in terms of 3x:(3x)2 2(3x)+1=0:After factoring, this becomes(3x 1)2=1and it follows that 3x=1,orx= , make sure that your paper has a single ow.

10 Don't explain a calculation usingthe \two-column method".32x 2(3x)= 1 Solve this equation.(3x)2 2(3x) + 1 = 0 Collect the terms on one side.(3x 1)2= 1 Use the Zero Factor is hard to read through. It's also bad things are best expressed with words. But other things are best expressed withmathematical notation. For instance, it hard to read:It follows thatxplus two is larger than , mathematical notation is more follows thatx+2> are a couple of other pointers to help you get started with your mathematical Writing . Don't start a sentence with a formula. While it may be grammatically correct, itlooks 2000, so we can conclude that the newfactory will be completely overrun with cockroaches in globber uxible atx= just a word or two can x these 2000, we can conclude that thenew factory will be completely overrun with cockroachesin 5 functionfis globber uxible atx= Don't turn in pages of unreadable scribbles to your professor.