Write Math

8/10/2019 Write Math

1/31

A Guide to Writing Mathematics

Dr. Kevin P. Lee

Introduction

This 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 now.

The Greek word mathemas , from which we derive the word mathematics, embodies thenotions of knowledge, cognition, understanding, and perception. In the end, mathematics isabout ideas . 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. Putting mathemas on paperwill require writing sentences and paragraphs in addition to the equations and formulas.

Mathematicians actually spend a great deal of time writing. If a mathematician wants

to 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 equations.Mastering 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 official), 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 career.

You will nd that writing good mathematical explanations will improve your knowledgeand understanding of the mathematical ideas you encounter. 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 class.

1


2/31


3/31

symbols can correspond to different 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.

3xy < 2.5z R .

9 s = t.

Can you identify the verbs? On the other hand, an expression like

5x 2 z 10y

is 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 equation

R = pq,

where p is the price at which each widget is sold and q isthe number of widgets sold. 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, if the price is increased by x dollars, then the revenue is

R = (15 + x)(2000 150x )

= 150x 2 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 bycommas.

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 andgrammatical errors better than your eyes. If you nd yourself saying a series of fragmentedsentences and equations, you should do some rewriting.

3


4/31

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 pluralwe, so dont 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 importantformula on a line of its own. Its hard to pick out the important formulas below:

If d is Bobs distance above the ground in feet, then d =100 16t 2 , where t is the number of seconds after BobsFlugelputz-Levitator is activated. Solving for t in theequation 100 16t 2 = 0, we nd that t = 2 .5. Bob hitsthe ground after 2 .5 seconds.

This is clearer:If d is Bobs distance above the ground in feet, then

d = 100 16t 2 ,

where t is the number of seconds after BobsFlugelputz-Levitator is activated. Solving for t in theequation

100 16t 2 = 0 ,

we nd that t = 2 .5. Bob hits the ground after 2 .5 sec-onds.

Symbols and words.

It 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 words.

Dont use mathematical symbols when you really mean something else. A commonmistake is to misuse the = symbol. For instance:

32 x 2(3x ) = 1 = (3 x )2 2(3x ) + 1 = 0 =(3x 1)2 = 0 = 3 x = 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:

4


5/31

32 x 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 line.

32 x 2(3x ) = 1

(3x )2 2(3x ) + 1 = 0(3x 1)2 = 0

3x = 1x = 0 .

For a difficult computation where the reader might not readily follow each step, you caninclude words to describe the steps you take.

We want to solve for x in the equation

32 x 2(3x ) = 1.

We can rewrite this equation in terms of 3 x :

(3x )2 2(3x ) + 1 = 0 .

After factoring, this becomes

(3x 1)2 = 1

and it follows that 3 x = 1, or x = 0.

However, make sure that your paper has a single ow. Dont explain a calculation usingthe two-column method.

32 x 2(3x ) = 1 Solve this equation.(3x )2 2(3x ) + 1 = 0 Collect the terms on one side.

(3x 1)2 = 0 Factor.3x = 1 Use the Zero Factor Property.x = 0 Solve for x.

5


6/31

This is hard to read through. Its also bad style.

Some things are best expressed with words. But other things are best expressed withmathematical notation. For instance, it hard to read:

It follows that x plus two is larger than zero.

Here, mathematical notation is more appropriate.

It follows that x + 2 > 0.

Miscellaneous comments.

Here are a couple of other pointers to help you get started with your mathematical writing.

Dont start a sentence with a formula. While it may be grammatically correct, itlooks strange.

t = 5 when w = 2000, so we can conclude that the newfactory will be completely overrun with cockroaches in 5years.

f is globberuxible at x = 3.

Adding just a word or two can x these examples.

Since t = 5 when w = 2000, we can conclude that thenew factory will be completely overrun with cockroachesin 5 years.

The function f is globberuxible at x = 3.

6


7/31

Dont turn in pages of unreadable scribbles to your professor. In college, papers are

typed. They are also usually double-spaced with large margins. Mathematics papersadhere to the same standards as papers written for other classes.

While it is a good idea to type your paper, you may have to leave out the formulasand insert them by hand later. It is perfectly acceptable to write formulas by hand ina math paper. Just make sure that your mathematical notation is legible. If you dodecide to type the equations, please be aware that variables in equations and formulasare usually italicized (to set them apart from the text). Many word processingprograms contain equation editors. In newer versions of Microsoft Word, the equationeditor is available under the Insert menu. Select Object... , and then Equation .1 If you are going to be writing a lot of technical documents, it might be worthwhile to

learn TEX or LA

TEX. These are professional mathematical typesetting languages. Thisdocument was written with LATEX. You may also nd satisfactory results typingpapers in Maple or some other mathematically oriented software program.

Use mathematical notation correctly. As you learn to write more complicatedformulas, it is all too easy to leave out symbols from formulas. Learn how to usesymbols properly!

Use language precisely and correctly. Make sure that the words you use really meanwhat you think they mean. Mathematics requires very precise use of language.Another thing to avoid is overuse of the word it. Mathematical papers with a lot of

pronouns like it and that tend to be hard to read. It is often hard for the readerto see what it is referring to. If you, the author, are also having difficulty seeingwhat it is referring to, then you may be having some difficulty with themathematical ideas; you may need to think more about the ideas you are writingabout.

Try to write as simply and directly as possible. No one likes to read ponderouspretentious prose.

1 In Microsoft Word, it is also possible to place a button on the tool bar which activates the equationeditor. Select Configure... beneath the Tools menu. In the window that pops up, select the Commandstab. Under the Insert category you will nd the Equation Editor command. Drag the equation editor

icon to the tool bar.

7


8/31

Mathematical Ideas into Writing

Organizing your paper.

A well-organized paper is easier to read than a disorganized one. Fortunately, there aresome standard ways to order a mathematics essay.

First, there is some type of introduction. Usually, the introduction states the problem.Even if you are answering a problem from a text book, you should not assume that thereader is familiar with the text book or even has a copy of the text book available to himor her. However, do not just copy the problem! You must rewrite the problem in your ownwords.

A good introduction should also discuss the signicance of the problem. The

introduction is where you will need to hook the reader.It is not a bad idea to also preview the rest of the paper in the introduction. Give thereader some idea of what to expect later.

We will analyze the revenue using a linear model and thenexamining the graphs generated by the model.

The production of fava beans will be modeled using a Cprogram.

First, we will analyze the population using numericalmethods. Then, we will analyze the population usingformulas. We will then compare the two different results.

Some papers then state the answer to the problem right after the introduction. Otherpapers place the answer at the end. This is a matter of taste. Sometimes, the end resultis the most important thing in the paper. You may need to place the end result at thebeginning to entice the reader. On the other hand, sometimes the method of arriving atthe end result is more important. In such a case, putting the result at the end may bemore sensible.

In any case, it is best to state the result in terms of the original problem usingreal-world terms.

The solution is t = 6.

8


9/31

The solution to the equation is t = 6. The population of

Utopia is at its smallest 6 years after the plague begins.

Make sure that the arguments you write are carefully organized. It may help you towrite an outline before you begin writing a mathematics paper. Writing an outline will alsohelp you think about the concepts more clearly and thus will help you learn the material.As you write about more advanced mathematical problems, organization will become evenmore important.

Writing for your audience.

For most papers that you write in your math class, you should assume that the reader hasabout the same mathematical knowledge that you have. When you write up the solution toa homework problem, it might be helpful to think that you are writing to a student inanother section of the same class or in a similar class at another school. Some of the papersyou will be writing will be directed toward a reader who may know less math. The purposeof a math paper is not just to show the professor that you know something. Your mathprofessor already knows the subject; you are not writing for his or her benet. You arewriting for someone who doesnt know the subject. (That someone may be you! You canuse your writing assignments to help review for exams.)

In your mathematics writing, you will be communicating to the reader why and how you arrived at a solution. You will also want to convince your reader that your particular

reasons and your particular means to the solution are correct. A good mathematical papernot only should provide clear explanations, but should also be able to persuade a skepticalreader.

Many times, if you can arrive at the same solution through alternate routes, you canmake your writing more persuasive. You may want to analyze a problem using bothcomputers and algebra. Or you might compare a graph with real-world information.Pictures and graphical depictions can be very helpful for your reader.

Specic examples will also help to make your writing more persuasive. You can help areader understand an abstract general argument by showing how the argument applies to aspecic case. You can also use extreme cases to show the limits of an argument.

Make sure that what you write is relevant to the problem. Including extraneous

comments or information demonstrates a lack of understanding of the ideas and concepts,and reduces the overall effectiveness of your mathematical writing. Thinking about thereader will help you to decide which details you need to include and which details youshould leave out. Calculations which are tedious and uninteresting to the reader can bereadily omitted. (Again, mathematics writing is not the same as showing work. You dont

9


10/31

need to show everything.) The reader of a college mathematics paper will probably not be

interested in reading how to multiply 5 and 74. Leave out what is unimportant. On theother hand, dont leave out anything which is critical to the key ideas you are trying toexplain. Learning what is important and what is unimportant will help you understandmathematics better.

You should not assume that the reader is familiar with the problem you are solving.While you do not need to restate the problem in its entirety, be sure to give an overview of all important details in the problem. You also should not assume that the reader is in thesame mind set as you. In your writing, state any assumptions which you have made. Forinstance, in physics problems, it is often assumed that everything is frictionless. But justbecause this assumption is made nearly all the time doesnt mean that your reader willautomatically make this assumption; your reader may not be familiar with physics. Justbecause you assume something is true doesnt mean that your reader will. So write it down!

Dening variables and formulas.

Quantities and functions can be, and often should be, represented with letters. However,the letters which are chosen are arbitrary. You should explicitly state what all letters inyour formulas represent in as precise a manner as possible. For instance:

Either n or n + 1 is even.

What is n? If n = 8 .5 is the above statement true? A better way of stating this is:

For any whole number n , either n or n + 1 is even.

A common phrase used in mathematics is Let....

Let x be any real number.

Let P be the population of Los Angeles in 2010.

Let f (x ) = x 2 + 1.

10


11/31

In the last example, x is a place holder. It doesnt require a proper introduction. However,

it would be better to write:

Let f (x ) = x 2 + 1 for all real numbers x .

If describing all the variables gets tedious, try not assigning any variables at all. Thefollowing example clearly needs improvement.

The volume is wh.

The following example is adequate, but wordy.The volume of the box is wh, where is the length, w isthe 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, thewidth, and the height. !

You need to be especially careful with variables representing real-world quantities.Avoid describing them vaguely, as in:

Let D (t ) be the distance at a time 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 specic as possible.

Let D (t ) be Agness distance from the arena in miles thours after the riot began.

Also, be careful that each symbol you use represents only one thing. This can actuallybe more subtle than it sounds. The following example seems to be rather clear.

11


12/31

Let P be the escaped wombat population (in thousands)t years after 1990 and suppose that

P = 0 .5(1.12)t .

The wombat population in 1992 is approximately 672.We can see this by setting t = 2 and observing that

P = 0 .5(1.12)2 = 0 .6272 thousand wombats.

If we want to predict when the wombat population willreach 2000, we set P = 2 and solve for t usinglogarithms.

2 = 0 .5(1.12)t

log2 = log 0.5 + t log1.12

t = log2 log0.5

log1.12 12.23 years.

The wombat population will reach 2000 in the year 2002.

I think that the above example would be considered unobjectionable by most readers. Itlooks very clear and understandable. The variable P is always standing for the wombatpopulation. However, notice that in the rst paragraph, P is the wombat population ingeneral. In the next paragraph, P = 0 .6272, the wombat population in 1992. And in the

last paragraph, P = 2. The meaning of P appears to be changing every time that it isused. In the rst paragraph, P represents the population at any time. In the otherinstances, P represents the population at one particular time. The problem can be xedomitting some variables and adding others.

12


13/31

Let P be the escaped wombat population (in thousands)t years after 1990 and suppose that

P = 0 .5(1.12)t .

By substituting 2 for t in the above equation, we can seethat in 1992, the wombat population is approximately672.

0.5(1.12)2 = 0 .6272 thousand wombats.

Let t2000 be the year when the wombat populationreaches 2000. Then,

2 = 0 .5(1.12)t 2000

log2 = log 0.5 + t2000 log 1.12

t 2000 = log2 log 0.5

log1.12 12.23 years.

The wombat population will reach 2000 in the year 2002.

While in the above example, we can afford a little bit of sloppiness with the variables, inmore complex problems, this could be a source of potential trouble. When a symbol is usedto represent two different things (even, or perhaps especially, if those things are similar),the reader (and the writer!) can become confused. A symbol used in two different ways isnot only confusing, but often results in incorrect mathematics!

Just as variables need to be introduced carefully, also be sure not to pull formulas outof thin air. Tell the reader how you get each formula or what each formula means. Its notvery 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 visualarguments 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 fullyexplain 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 specic.

13


14/31

The graph increases sharply at t = 3, conrming ourearlier prediction that the robots will begin a homicidalrampage three years from now.

A good graph should convey relevant and specic information to the reader. Thefollowing graph is vague.

Graphs and diagrams need to be neatly drawn and clearly labeled. Indicate the scale onthe axes. You should point out signicant graphical features.

Cooties infections versus time

10 20 30 40 50 60 70 80 90 100

100

200

300

400

500

600

700

800

900

1000

N o. of i nf e c t i on s I

( i n t h o u s an d s )

Time t after epidemic begins (in days)

maximum number of infections

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 Office generally arenot good at generating mathematical graphs. You will more likely have success using amath program like Maple.

Any diagrams you draw should also be carefully labeled. Be sure to label everythingthat you refer to in your argument.

14


15/31

Epilogue

Writing mathematics is not the easiest thing to do. Writing mathematics is a skill whichtakes practice and experience to learn. There are many resources here at Purdue Calumetwhich are available to you to help you with your mathematical writing. Among these arethe Math Lab and the Writing Lab.

If you have not written mathematics much before, it may feel frustrating at rst. Butlearning to write mathematics can only be done by actually doing it. It may be hard atrst, 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 keepfor a lifetime.

15


16/31

A mathematical writing checklist

Below is a checklist which will help you follow the guidelines outlined above in yourmathematical writing.

Is your paper neatly typed?

If you write the equations by hand, make sure that you have written in all of theequations. Also make sure that you have included all of the diagrams and graphs youintended to. Make sure that the paper is double-spaced and has wide enough margins.

Has the paper been proofread?

In college, sloppy work is not appreciated. Do check over everything.

Is there an introduction?

Make sure that you explain the problem to the reader. Assume that the reader isunfamiliar with the problem. The introduction should also try to indicate to thereader why the problem is interesting and give some indication of what will follow inthe paper.

Did you state all of your assumptions?

Write down any physical assumptions that you made. (Did you assume that there wasno friction? That the population grew with unlimited resources? That interest ratesremained steady?) Write down any mathematical assumptions that you made. (Did

you assume that the function was continuous? Linear? That x was a real number?)Are the grammar, spelling, and punctuation correct? Is the writing clearand easy to understand?

Make sure that there are no sentence fragments. The formulas and equations tooneed to be contained in complete sentences. Equations and formulas (and the wordstoo) should have correct punctuation as well. Make sure that your paper owssmoothly and reads well. And please, dont be careless! Check your spelling!

Are all of the variables dened and described adequately?

Make sure that you introduce each variable that you use. Describe each variable asprecisely as possible. Dont forget any units!Are the mathematical symbols used correctly?

Dont use an = sign outside of a formula. Make sure that the symbols are notmisused. Use equations and formulas where they are appropriate.

16


17/31

Are the words used correctly and precisely?

Avoid using vague language and too many pronouns. Use words where they areappropriate.

Are the diagrams, tables, graphs, and any other pictures you includeclearly labeled?

Graphs should be drawn with a straight edge (or computer-generated) with axesclearly labeled (with units if appropriate) and the scale indicated. Diagrams shouldbe neatly drawn with relevant labels.

Is the mathematics correct?

This should be obvious.Did you solve the problem?

Sometimes in all of the fuss, people forget to answer the problem. Do answer thequestion! Also, see if you can write the solution in real-world terms.

17


18/31

A Mathematical Writing Checklist

Below is a list of guidelines you should follow for mathematical papers. For more details, please consultA Guide to Writing Mathematics .

Is your paper neatly typed?If you write the equations by hand, make sure that you have written in al l of the equations. Also make surethat you have included all of the diagrams and graphs you intended to. Make sure that the paper isdouble-spaced and has wide enough margins.

Has the paper been proofread?

In college, sloppy work is not appreciated. Do check over everything.

Is there an introduction?

Make sure that you explain the problem to the reader. Assume that the reader is unfamiliar with theproblem. The introduction should also try to indicate to the reader why the problem is interesting and givesome indication of what will follow in the paper.

Did you state all of your assumptions?Write down any physical assumptions that you made. (Did you assume that there was no friction? That thepopulation grew with unlimited resources? That interest rates remained steady?) Write down anymathematical assumptions that you made. (Did you assume that the function was continuous? Linear? Thatx was a real number?)

Are the grammar, spelling, and punctuation correct? Is the writing clear and easy tounderstand?

Make sure that there are no sentence fragments. The formulas and equations too need to be contained incomplete sentences. Equations and formulas (and the words too) should have correct punctuation as well.Make sure that your paper ows smoothly and reads well. And please, dont be careless! Check your spelling!

Are all of the variables dened and described adequately?

Make sure that you introduce each variable that you use. Describe each variable as precisely as possible.Dont forget any units!

Are the mathematical symbols used correctly?

Dont use an = sign outside of a formula. Make sure that the symbols are not misused. Use equations andformulas where they are appropriate.

Are the words used correctly and precisely?

Avoid using vague language and too many pronouns. Use words where they are appropriate.

Are the diagrams, tables, graphs, and any other pictures you include clearly labeled?

Graphs should be drawn with a straight edge (or computer-generated) with axes clearly labeled (with units if

appropriate) and the scale indicated. Diagrams should be neatly drawn with relevant labels.Is the mathematics correct?

This should be obvious.

Did you solve the problem?

Sometimes in all of the fuss, people forget to answer the problem. Do answer the question! Also, see if youcan write the solution in real-world terms.
http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingman.pdfhttp://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingman.pdf


19/31

Common Errors in Writing Mathematics

The following things are often confused:

equations, expressions, functions.

An equation is a statement that two things are equal, like

x3 + 3 x 2 = 0 or y = x2 .

An expression is some mathematical symbolism without a verb in it, for instance

x + 1 or (x + 1)( y 2) or f (x+ a) f (x)

a .

By verb I dont necessarily mean something written as a word; more likely it will be an equalsign, or an inequality sign.

A function is a process that takes inputs and makes outputs. The symbol f denotes afunction, but f (3) does not; f (3) is a result or value of a function. So, do not refer to thefunction f (3). You mean the value f (3) or the output f (3).

Exception. You may refer to the function f (x) instead of the value f (x). This usage istechnically incorrect, but accepted. The point is, x is a dummy variable here. You are merelysaying that f is a function because it applies to inputs, rather than referring to the specicoutput when the specic letter x is input. Similarly, if z is your dummy variable, you may referto the function g(z) instead of the value g(z). However, in both cases it would be betterto say the function f or the function g. Or, leave out function and just say f (x).

Examples . Consider

A function is an equation. For instance, f (x) = x2 3.

No! A function is a process. A function is dened by giving an equation that provides a formula in terms of a dummy variable, but the equation is not itself the function.

Example 2 . What is wrong with:

Let f (x) = x2 . Inf (x) = f (2x 1) f (3x+2) (1)

for the rst function f you replace the dummy variable by 2 x 1 in the equation of f , getting (2 x 1)2 .

Answer: Two things. There is only one function f in (1). It is evaluated twice. Second, thereis no equation of f ; rather, there is a denition of f , the statement that starts with Let.

Correct rewrite: Let f (x) = x2

. Inf (x) = f (2x 1) f (3x+2)

for the rst evaluation of f you substitute 2 x 1 for the dummy variable x in thedenition of f , getting (2 x 1)2 .

But why use awkward word expressions like rst evaluation of f when you can use brief symbols:

S Maurer


20/31

Common Errors in Writing Mathematics S Maurer

Better rewrite: Let f (x) = x2 . In

f (x) = f (2x 1) f (3x+2)

to evaluate f (2x 1) you substitute 2 x 1 for the dummy variable x in the denitionof f , getting (2 x 1)2 .

Example 3 . What is wrong with writing

Given the function x + 10, evaluate the function for f (4).

Answer: x + 10 is not a function; it is an expression. Apparently this expression is meant tobe the output of a function named f , since the letter f is used later. Also (but minor), there isredundancy later in the sentence.

Corrected: Given the function f (x) = x + 10, evaluate f (4).

Even better (because it avoids the exceptional wording function f (x)):

Given the function f dened by f (x) = x + 10, evaluate f (4),or

Let f (x) = x + 10. Evaluate f (4).

This last version avoids the issue of what is rightly called a function by avoiding the word. Thenotation carries the day. Notice that the best version is also the shortest.

Example 4. What is wrong with writing

Let f (x) = x2 2x. To nd f (2/x ) just plug in 2 /x everywhere for x and completethe function.

Answer: The difficulty is the phrase complete the function. The author means: do anyobvious simplications to the expression (2/x )2 2(2/x ). So complete the simplicationwould be all right, though not as good as the single word simplify. Because x2 2x is notthe function itself, but rather the output, complete the function is incorrect.

Example 5. What is wrong with writing

Let f (x) = 2 x / (x + 1). Solve for f (3).

Answer: There is nothing to solve for. You are merely evaluating . If the task had been

Find x if f (x) = 3,

then there would be something to solve for, because you would start with 3 = 2 x / (x + 1) andhave to isolate x.

Example 6. Whats not right here?

Let g(x) = ( x+1)( x+2). To evaluate g(y 1) we must plug y 1 into g.

Answer: y 1 was already plugged into g when we wrote g(y 1). That already is a name forthe output we want. To evaluate it, that is, nd a more familiar name for it, we must plugy 1 into the the right side of the dening equation g(x) = ( x+1)( x+2), obtaining y(y+1).

2


21/31

Common Errors in Writing Mathematics S Maurer

Final Remark. Please refer to function notation, not functional notation. The adjectiveform of (mathematical) function is function, not functional. This is because functionalalready has another meaning in ordinary English. Functional notation means any notationthat works well, whether or not it has anything to do with functions, or even with math.

For more information, see the handouts from my manuscript Short Guide to Writing Math-ematics .

3


22/31

Lab #1

Do experiment 3.6 " The Computer May Lie" on page 25 of your textbook, preferablyusing the Mathematica notebook with the same title. (Alternatively, you may Use theFunction Iterator available at the Dynamical Systems and Technology Project web site).

Choose "generic" seeds, such as 0.836, not whole numbers. Write an essay describingyour findings, answering the questions posed in the Results section, and questions 1 and2 on page 26. Please do not list or print out the entire orbit for each of your chosenseeds. Rather, simply give the seed and a short description of the outcome (tends to afixed point, goes to infinity, etc.) In each case, the computer is not telling the whole story.Do you see what else is happening? Comment on this in your essay after explaining whatthe computer shows.

Please be sure to review the Lab Report Expectations.

The Dynamical Systems and Technology Project web site is the work of Professor BobDevaney at Boston University, and the Mathematica notebooks were written bySebastian Marotta.
http://math.bu.edu/people/bob/MA471/mathematica.htmlhttp://math.bu.edu/DYSYS/http://math.bu.edu/DYSYS/http://math.bu.edu/people/bob/MA471/mathematica.html


23/31

18.091 Lab 1The Computer May Lie

Alejandro Ochoa

May 19, 2005

1 Introduction

In this experiment we investigated the orbits of several seeds under iterationof three functions. We will discuss how our data compares to what we expectfrom theory, which will lead to the discussion of computation as performed bya computer. The functions chosen are

1. F [x] = x2 2 on the interval ( 2, 2),2. F [x] = x2 2.1 on the interval ( 2, 2),3. D [x] = Frac [2

x], the doubling function on the inverval [0 , 1).

2 Methods

Some of the seeds were chosen by me, while others were randomly generatedby Mathematica using a function. Note that Mathematica is software capableof doing rational arithmetics, which implies the format in which we type thenumber matters. For example, typing 0 .1 for a seed will make Mathematicado all the computations with 1 / 10 as a oating point number, with limitedprecision, while typing 1 / 10 will force Mathematica to make exact calculationswith that rational number.

3 Data and Observations

For each function, on the table we listed in the rst column the seeds used, andin the second column the observed behaviour of the orbits of the correspondingseed. Seeds with ve or six signicant digits were generated by Mathematica.Double quotation marks indicate the behaviour is the same as described above.Plots of sample orbits are also included for the rst and third functions to betterillustrate the behavior of such orbits.

1


24/31

2Figure 1: The orbit of a seed under 100 iterations of F [x] = x 2

1. F [x] = x2 2,Seed Behavior

1 Fixed point.0.1 Apparently chaotic, bounded in [ 2, 2].0.13 0.0468936 0.510909

0.66461

0.706763 0.12377

0.626387 0.374533 0.499365

2. F [x] = x2 2.1,Seed Behavior

11Diverges to

0 0.25

1.985 0.1850490.2834940.6431120.9872010.252688

2


25/31

4

Figure 2: The orbit of the seed x0 = 0 .1 under 100 iterations of D

3. D [x] = 2

x Floor [2 x],Seed Behavior

0.1 Eventually xed, converges to 00.0628138 0.860487 0.53821 0.762382

0.495847 0.392426 0.142857 0.33 1/ 7 Periodic33/ 100 Eventually periodic

Discussion

For the function F [x] = x2 2 the experiment with the computer suggests thatall orbits are chaotic (see Figure 1), except we happened to select the pointx0 = 1 which is xed under iteration by F. However, we can algebraicallyverify that the equation F [x] = x has another solution, namely x = 2, so thereare two xed points for this function that the computer would not have foundfor us, for example, if it only sampled random numbers. We did not identifyany periodic points either, while we know there are innitely many from theory.

For the second function F [x] = x2 2.1 the computer strongly suggeststhat all seeds in ( 2, 2) diverge to innity, but this is not the case because the

3


26/31

2equation F [x] = x, or x x 2.1 = 0, has two solutions, namely

x0 = 1 9.4 ,2and one of them, x0

1.0330 is in the interval we sampled. Hence, amongall of the points that diverge to innity, there is one that remains xed and was

not detected with the computer.The function D is by far the most interesting case. Consider x0 = 0 .1, which

we know has an exact representation as a rational number, namely x0 = 1 / 10.The point 1/10 is eventually periodic, entering the 4-cycle

1 2 4 3 1...

5,

5,

5,

5,

5,

after the rst iteration. However, the computer shows that x0 = 0 .1 is eventuallyxed, converging to zero. Looking at Figure 2, we can observe that 0 .1s orbitstarts with a seemingly regular orbit as we expect it because it enters a 4-cycle. However, something happens around the 50th iteration, the orbit becomesirregular and then drops to zero where it remains xed.

It is plausible that the difference in the outcomes of the orbits of 0 .1 and 1 / 10is due to the fact that the result of a computation on a oating-point number(the logical representation of decimals on a computer) is bound to lose precisionbecause oating-point numbers only store a nite amount of digits, while thesoftware that deals with rational arithmetics only needs to store integers andhas an unlimited capacity to do so, always giving an exact answer. Althoughthe loss of precision is very small, after 50 iterations of the function on 0 .1 wecan observe the loss of precision built up to give a wrong answer which is notreasonably close to the right answer.

For the rst two examples, we were led to an incorrect hypothesis becausewe only sampled a nite amount of points. There are a dense and uncountableamount of points which share the behaviour of the points we sampled in thoseintervals. However, there are two and one points of interest respectively for eachF on the given interval, which display the very appealing property of remainingxed under iterations of F, and the probability that the computer would ran-domly choose these points is close to zero. The third example showed us thatthe loss of precision generated by storing numbers as oating-point numbers canbecome signicant when analyzing orbits of seeds of chaotic systems.

Whenever we use computers to model dynamical systems, we must alwayskeep in mind the two caveats we discovered, namely, that we should not gener-alize the behavior of an interval of seeds by analyzing just a few seeds, and that

we should be careful when considering results from a large amount of computa-tions on oating-point numbers, given that these results are not guaranteed tobe neither correct nor approximately correct.

4


27/31

Lab 1

Jeremy Hurwitz

February 14, 2005

1 Introduction

The goal of this experiment was to discover the behavior of various functions under iterationunder different seeds using Mathematica. The rst function studied was the doubling function,which was dened as F (x) = 2 x (mod 2). The rest of the functions were the family of functionsF (x) = x2 c, where c is any positive constant.

In each case, I attempted to classify the orbits as xed, periodic, tending towards a certainlimit or chaotic. To this end, I ran twenty iterations of each function suing Mathematica to see ifa pattern seemed to be appearing. If unsure, I ran more iterations.

2 The Doubling Function

For testing the doubling function, I divided possible seeds into three groups. The rst group wasrational numbers, written in the form p/q, p, qZ + . The second was rational numbers written as adecimal expansion. The last group was irrational numbers, such as the square root of 2.

2.1 Rational Numbers p/q

In the case of rational numbers written in fractional form, one of three results always showed up.Zero returned the xed point orbit F (x) = 0. If q = xn , for some n, the orbit became eventuallyxed at 0. All other orbits were periodic or eventually periodic.

2.2 Decimal Expansions of Rational Numbers

Decimal inputs returned incorrect results due to rounding errors. Because the computer storesthese numbers in the form a1 a2 a3 a n+ + + . . . + (1)

2 22 23 2n

after k iterations, the kth term becomes equivalent to 0 (mod 1). Therefore, the computer falsely

stated that all decimal inputs became eventually xed at 0. For example, x0 = 2 / 5 returned aperiodic orbit, whereas x0 = 0 .2 became xed.

2.3 Irrational Numbers

Irrational numbers appeared to be chaotic, at least until the rounding errors compounded enough tosend the orbit to zero. When I calculated the orbit without rounding, the orbits remained chaotic.

1


28/31

3 F (x ) = x 2 c, c > 0I started by testing c = 2. Once I had a feel for how the orbit behaved for different seeds, I startedvarying c, attempting to nd a pattern that owed between all values of c. As with the doublingfunction, I looked for orbits that tended towards periodicity or towards a specic orbit.

3.1 Conditions For Divergence

The rst obvious characteristic of the entire family of functions was that if the seed was outsidesome specic interval, then it would tend towards innity. For c=2, the interval of convergence was[-2,2]. For c=1, the range was [ 1+ 2

5 , 1 5 ].2

Theorem 1. Let F (x) = x2 c, c > 0. Let p =1+ 4 c +1 and q = 1 4 c +1 . Then if2 2

i x0 > p , F (x0 ) diverges under iteration.

ii x0 < q 1, F (x0 ) diverges under iteration.Proof (i). Since x0 > p,

> 0, = x0

p.

2F (x0 ) = F ( p + ) = ( p + )2 c = p2 + 2 p + c

However, since p 2 c = p, this simplies to2F (x0 ) = p + 2 p + > p + = x0

Therefor, F (x0 ) is monotonically increasing under iteration. Furthermore, since F (x0 ) x0 > and gets bigger with every iteration, F (x) is not bounded. Therefore, F (x0 ) diverges.(ii). Since x0 < q 1andq < 0, > 1, = q x0 .

2 2

F (x0

) = F (q ) = ( q )2

c = q 2q + c2Since q = p 4c + 1 and q c = q ,2F (x0 ) = q 2q + = p 4c + 1 2q + = p + ( 1)( 4c + 1 + )

Since > 1, F (x0 ) > p , so by (i), F (x0 ) diverges.

If c > 2, then all orbits tended towards innity, unless they were exactly periodic. Because ofthis, all remaining discussion will be limited to c < 2 and values of x0 such that F (x)

3.2 Behavior Under Various Values of c

In general, there always existed k-cycles which could be found by solving F k (x) = x. For valuesnot lying on one of these cycles, there were three cases.

When 0 < c < 1, the orbit tended to converge to the xed point p = 1 4 c +1 , an attracting2xed point. Note that the other xed point is repelling, so any point within the convergent intervalhas to go towards this attracting point.

When c = 1, all orbits tend towards the 2-cycle (0,1,0,1,...). As long as x0 | < 1.68, the computer|could not tell the difference. The last case was 1 < c < 2. Within this interval, all orbits appeared to become chaotic.

2


29/31

4

2Table 1: Orbits of F (x) = x c, x0 not on a k-cycleRange Behavior0 < c < 1 towards a xed point

1 towards a cycle1 < c < 2 Chaotic

c > 2 towards innity

Summary

The most important result of this experiment was the rounding issue that arose when a decimalwas used as the seed of the doubling function. Unless very careful analysis is used as to how thecomputer is treating the number and how errors are propagated through the iterations, computerresults must be taken with a large grain of salt.

From a stand-point of generating and studying chaos, the main result was how easily very sim-ple, predictable systems became chaotic. The doubling function, which was completely predictablefor all rational numbers, showed no patterns for irrational numbers. For the family of functionsF (x) = x2 c, orbits were predictable (excluding individual k-cycles) for all values of c outside therange (1,2]. Within that range, unless the orbit was periodic, it was always chaotic.

3


30/31

Table 2: Doubling Function F (x) = 2 x (mod 2) Input Correct Orbit Computers Output0.2 periodic goes to zero1/5 periodic periodic1/9 periodic periodic0.23 periodic goes to zero5/17 periodic periodic0 xed point xed point1/2 eventually xed eventually xed1/10 eventually periodic eventually periodic1/4 eventually xed eventually xedx

R Q (10 trials) no pattern no pattern

2Table 3: x 2Input Eventual Behavior2 Fixed0 Eventually Fixed-2 Eventually Fixed-1 Fixed1 Eventually Fixed1/ 2 Chaotic 2 eventually xed 3 eventually xed8 other tests chaotic

Data

4


31/31

2Table 4: x 3Input Eventual Behavior 2 Periodic 3 to innity0 to innity2 periodic1+ 13

2 xed1 periodic

Table 5: Summary of Data for Other Values of c

c behavior4 2 xed points, all other tests to innity

1.5 chaotic or towards innity1 towards 2-cycle or towards innity

.5 towards .366025 or towards innity

Documents

Write Math