# writingman.dvi

## Transcript

3 symbols can correspond to di erent parts of speech. For instance, below is a perfectly good complete sentence. : 1+1=2 The symbol \=" acts like a verb. Below are a couple more examples of complete sentences. : 3 2 xy < 2 R : 5 z s 6 = t: 9 Can you identify the verbs? On the other hand, an expression like 2 x 5 y z 10 is not a complete sentence. There is no verb. Such an expression should be treated as a noun. Can you identify the nouns in the previous examples? Formulas and equations need to be contained in complete sentences with proper punctuation. Here is an example: R , made from selling widgets is given The total revenue, by the equation R = pq; where p is the price at which each widget is sold and q is the number of widgets sold. Based on past experience, we know that when widgets are priced at \$15 each, 2000 widgets will be sold. We also know that for every dollar increase in price, 150 fewer widgets are sold. Hence, if the price is increased by x dollars, then the revenue is =(15+ x )(2000 150 x ) R 2 150 x = x +30 ; 000 : 250 Notice how punctuation follows each of equations. A computation which ends a sentence needs to end with a period. Computations which do not end sentences are followed by commas. A good way to improve your mathematical writing is by reading your writing, including all of the equations , out loud. Your ears can often pick out sentence fragments and grammatical errors better than your eyes. If you nd yourself saying a series of fragmented sentences and equations, you should do some rewriting. 3

4 There are a couple of other important things to observe in the above example. Notice how \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 class. Another thing to notice is that important or long formulas are written on separate lines. You can make your mathematical writing easier to read if you place each important formula on a line of its own. It's hard to pick out the important formulas below: is Bob's distance above the ground in feet, then = If d d 2 100 t 16 t is the number of seconds after Bob's ; where t in the Flugelputz-Levitator is activated. Solving for 2 equation 100 16 t t =2 : 5. Bob hits = 0, we nd that : the ground after 2 5 seconds. This is clearer: If d is Bob's distance above the ground in feet, then 2 t = 100 16 d ; t is the number of seconds after Bob's where t in the Flugelputz-Levitator is activated. Solving for equation 2 16 t 100 =0 ; t 5 sec- : 5. Bob hits the ground after 2 : we nd that =2 onds. Symbols and words. It is important to use words and symbols appropriately. Part of being able to write mathematics well is knowing when to use symbols and knowing when to use words. Don't use mathematical symbols when you really mean something else. A common mistake is to misuse the \=" symbol. For instance: x 2 x 2 x x )= 2(3 ) 1=(3 2(3 3 )+1=0 = !! x x 2 (3 1) =1 = x =0 : =0 = 3 Do not use the equal sign when you really mean \the next step is" or \implies". The above example is really saying that 1 = 0 = 1! Using arrows instead of equal signs is a slight improvement, but still not desirable: 4

5 x x x 2 x 2 1 ! (3 2(3 ) 3 2(3 )= )+1=0 ! ! x 2 x (3 3 =1 1) x =0 : =0 ! ! With a sequence of calculations, sometimes it is best to just place each equation on a separate line. 2 x x 2(3 1 3 )= x x 2 (3 ) 2(3 )+1=0 x 2 (3 1) =0 x 3 =1 x =0 : For a dicult computation where the reader might not readily follow each step, you can include words to describe the steps you take. We wa nt t o s o l ve f o r x in the equation 2 x x 3 2(3 1 : )= x We can rewrite this equation in terms of 3 : x 2 x (3 2(3 ) )+1=0 : After factoring, this becomes x 2 (3 1) =1 x and it follows that 3 =1,or x =0. However, make sure that your paper has a single ow. Don't explain a calculation using the \two-column method". x 2 x 2(3 )= 1 Solve this equation. 3 x 2 x (3 2(3 ) + 1 = 0 Collect the terms on one side. ) 2 x (3 =0 Factor. 1) x 3 = 1 Use the Zero Factor Property. x =0 Solvefor x . 5

6 This is hard to read through. It's also bad style. Some things are best expressed with words. But other things are best expressed with mathematical notation. For instance, it hard to read: It follows that plus two is larger than zero. x Here, mathematical notation is more appropriate. x +2 It follows that 0. > Miscellaneous comments. Here 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, it looks strange. t =5when w = 2000, so we can conclude that the new factory will be completely overrun with cockroaches in 5 years. f is globber uxible at x =3. Adding just a word or two can x these examples. Since t =5when w = 2000, we can conclude that the new factory will be completely overrun with cockroaches in 5 years. The function f is globber uxible at x =3. 6

11 In the last example, x is a place holder. It doesn't require a proper introduction. However, it would be better to write: 2 +1 for all real numbers )= x Let ( x . x f If describing all the variables gets tedious, try not assigning any variables at all. The following example clearly needs improvement. wh The volume is . The following example is adequate, but wordy. is wh ` is the length, The volume of the box is ,where w the width, and h is the height. We can write this most elegantly by removing the variables. The volume of the box is the product of the length, the ! width, and the height. You need to be especially careful with variables representing real-world quantities. Avoid describing them vaguely, as in: Let D ( ) be the distance at a time t . t Including units would make this clearer, but the description is still vague. Let D ( t ) be the distance in miles at t hours. Try to be as speci c as possible. ) be Agnes's distance from the arena in miles D ( t Let t hours after the riot began. Also, be careful that each symbol you use represents only one thing. This can actually be more subtle than it sounds. The following example seems to be rather clear. 11

12 Let P be the escaped wombat population (in thousands) years after 1990 and suppose that t t : 5(1 : 12) =0 P : The wombat population in 1992 is approximately 672. t = 2 and observing that We can see this by setting 2 : 5(1 : P =0 12) =0 6272 thousand wombats. : If we want to predict when the wombat population will reach 2000, we set = 2 and solve for t using P logarithms. t : 5(1 : 12) 2=0 log 2 = log 0 : 5+ t log 1 : 12 log 0 : 5 log 2 t = :  12 23 years. log 1 : 12 The wombat population will reach 2000 in the year 2002. I think that the above example would be considered unobjectionable by most readers. It looks very clear and understandable. The variable P is always standing for the wombat population. However, notice that in the rst paragraph, P is the wombat population in general. In the next paragraph, P =0 : 6272, the wombat population in 1992. And in the last paragraph, =2. Themeaningof P appears to be changing every time that it is P used. In the rst paragraph, P represents the population at any time. In the other instances, P represents the population at one particular time. The problem can be xed omitting some variables and adding others. 12

13 Let P be the escaped wombat population (in thousands) years after 1990 and suppose that t t : 5(1 : 12) =0 P : in the above equation, we can see By substituting 2 for t that in 1992, the wombat population is approximately 672. 2 5(1 : 0 : 12) =0 6272 thousand wombats. : t Let be the year when the wombat population 2000 reaches 2000. Then, t 2000 : 12) : 2=0 5(1 log 2 = log 0 : 5+ t 12 log 1 : 2000 5 log 2 log 0 : t =  12 : 23 years. 2000 : 12 log 1 The wombat population will reach 2000 in the year 2002. While in the above example, we can a ord a little bit of sloppiness with the variables, in more complex problems, this could be a source of potential trouble. When a symbol is used to represent two di erent things (even, or perhaps especially, if those things are similar), the reader (and the writer!) can become confused. A symbol used in two di erent ways is not only confusing, but often results in incorrect mathematics! Just as variables need to be introduced carefully, also be sure not to pull formulas out of thin air. Tell the reader how you get each formula or what each formula means. It's not very pleasant to get hit with formulas without any warning. Using pictures in mathematics. A picture can really be worth a thousand words. I strongly encourage you to use visual arguments in your mathematical writing. However, if you do include a picture, a diagram, a graph, or some other visual mathematical representation, make sure that you fully explain how it ts into your mathematical argument. Looking at the graph, we can see that the result is true. What should the reader look for in the graph? Why does the graph support the argument? Be more speci c. 13

14 The graph increases sharply at t = 3, con rming our earlier prediction that the robots will begin a homicidal rampage three years from now. A good graph should convey relevant and speci c information to the reader. The following graph is vague. Graphs and diagrams need to be neatly drawn and clearly labeled. Indicate the scale on the axes. You should point out signi cant graphical features. Cooties infections versus time No. of infections maximum number of infections 1000  900 800 700 600 I (in thousands) 500 400 300 200 100 10 20 30 40 50 60 70 80 90 100 t after epidemic begins (in days) Time If you draw a graph by hand, use a straight edge. You may want to generate your graphs using a computer. Be careful though. Programs like Excel or Microsoft Oce generally are not good at generating mathematical graphs. You will more likely have success using a math program like Maple. Any diagrams you draw should also be carefully labeled. Be sure to label everything that you refer to in your argument. 14

15 Epilogue Writing mathematics is not the easiest thing to do. Writing mathematics is a skill which takes practice and experience to learn. There are many resources here at Purdue Calumet which are available to you to help you with your mathematical writing. Among these are the Math Lab and the Writing Lab. If you have not written mathematics much before, it may feel frustrating at rst. But learning to write mathematics can only be done by actually doing it. It may be hard at rst, but it will get easier with time and you will get better at it. Do not get discouraged! Being able to write mathematics well is a good skill to learn, and one which you will keep for a lifetime. 15