http://lifehacker.com/5628396/the-basic-guide-to-troubleshooting-common-windows-pc-problems
http://naldzgraphics.net/tutorials/a-collection-of-psd-to-html-conversion-tutorials/
How to Read
Mathematics
by
Shai Simonson
and Fernando Gouvea
Mathematics is a language that
can neither
be read nor understood without initiation. 1
A reading protocol is a set of strategies
that a
reader must use in order to benefit fully from reading the text. Poetry
calls
for a different set of strategies than fiction, and fiction a different
set
than non-fiction. It would be ridiculous to read fiction and
ask oneself
what is the author's source for the assertion that the hero is blond
and
tanned; it would be wrong to read non-fiction and not ask such
a
question. This reading protocol extends to a viewing or listening
protocol in art and music. Indeed, much of the introductory course
material in
literature, music and art is spent teaching these protocols.
Mathematics has a reading protocol all its own,
and just as
we learn to read literature, we should learn to read mathematics. Students need to learn how to read
mathematics, in the same way they learn how to read a novel or a poem,
listen
to music, or view a painting. Ed
Rothsteins book, Emblems of Mind, a fascinating book
emphasizing the
relationship between mathematics and music, touches implicitly on the
reading
protocols for mathematics.
When we read a novel we become absorbed in the plot and characters. We try to follow the various plot lines and
how each affects the development of the characters.
We make sure that the characters become real
people to us, both those we admire and those we despise.
We do not stop at every word, but imagine the
words as brushstrokes in a painting.
Even if we are not familiar with a particular word, we can still
see the
whole picture. We rarely stop to think
about individual phrases and sentences. Instead, we let the novel sweep
us
along with its flow and carry us swiftly to the end.
The experience is rewarding, relaxing and
thought provoking.
Novelists frequently describe characters by involving them in
well-chosen
anecdotes, rather than by describing them by well-chosen adjectives. They portray one aspect, then another, then
the
first again in a new light and so on, as the whole picture grows and
comes more
and more into focus. This is the way to
communicate complex thoughts that defy precise definition.
Mathematical ideas are by nature precise and well defined, so that a
precise
description is possible in a very short space.
Both a mathematics article and a novel are telling a story and
developing complex ideas, but a math article does the job with a tiny
fraction
of the words and symbols of those used in a novel.
The beauty in a novel is in the aesthetic way
it uses language to evoke emotions and present themes which defy
precise
definition. The beauty in a mathematics
article is in the elegant efficient way it concisely describes precise
ideas of
great complexity.
What are the common mistakes people make in trying to read
mathematics? How can these mistakes be
corrected?
Dont
Miss the Big Picture
Reading Mathematics is not at all a
linear
experience ...Understanding the text requires cross references,
scanning,
pausing and revisiting 2
Dont assume that understanding each phrase, will enable you to
understand
the whole idea. This is like trying to
see a portrait painting by staring at each square inch of it from the
distance
of your nose. You will see the detail,
texture and color but miss the portrait completely.
A math article tells a story. Try
to see what the story is before you delve
into the details. You can go in for a closer look once you have built a
framework of understanding. Do this just
as you might reread a novel.
Dont
be a Passive Reader
A three-line proof of a subtle
theorem is the
distillation of years of activity. Reading
mathematics
involves a return to the thinking that went into the writing 3
Explore examples for patterns. Try special cases.
A math article usually tells only a small piece of a much larger and
longer
story. The author usually spends months
discovering things, and going down blind alleys. At
the end, he organizes it all into a story
that covers up all the mistakes (and related motivation), and presents
the
completed idea in clean neat flow. The
way to really understand the idea is to re-create what the author left
out. Read between the lines.
Mathematics says
a lot with a little.
The reader must participate. At
every stage, he/she must decide whether or not the idea being presented
is
clear. Ask yourself these questions:
Why is this idea true?
Do I really believe it?
Could I convince someone else that
it is
true?
Why didn't the author use a
different
argument?
Do I have a better argument or
method of
explaining the idea?
Why didn't the author explain it the
way that I
understand it?
Is my way wrong?
Do I really get the idea?
Am I missing some subtlety?
Did this author miss a subtlety?
If I can't understand the point,
perhaps I can
understand a similar but simpler idea?
Which simpler idea?
Is it really necessary to understand
this
idea?
Can I accept this point without
understanding
the details of why it is true?
Will my understanding of the whole
story suffer
from not understanding why the point is true?
Putting too little effort into this participation is like reading a
novel
without concentrating. After half an
hour, you wake up to realize the pages have turned, but you have been
daydreaming and dont remember a thing you read.
Dont
Read Too Fast
Reading
mathematics too quickly results in frustration.
A half hour of concentration in a novel might net the average
reader
20-60 pages with full comprehension, depending on the novel and the
experience
of the reader. The same half hour in a
math article buys you 0-10 lines depending on the article and how
experienced
you are at reading mathematics. There is no substitute for work and
time. You can speed up your math reading
skill by
practicing, but be careful. Like any
skill, trying too much too fast can set you back and kill your
motivation. Imagine trying to do an hour
of high-energy
aerobics if you have not worked out in two years. You
may make it through the first class, but
you are not likely to come back. The
frustration from seeing the experienced class members effortlessly do
twice as
much as you, while you moan the whole next day from soreness, is too
much to
take.
For example, consider the following theorem from Levi Ben Gershons
manuscript
Maaseh Hoshev (The Art of Calculation), written in 1321.
When you add consecutive numbers starting with 1, and the number of
numbers
you add is odd, the result is equal to the product of the middle number
among
them times the last number. It is
natural for modern day mathematicians to write this as:
A reader should take as much time to
unravel the two-inch version as
he
would to unravel the two-sentence version.
An example of Levis theorem is that 1 + 2 + 3 + 4 + 5 = 35.
Make
the Idea your Own
The best way to understand what you are reading is to make the idea
your
own. This means following the idea back to its origin, and
rediscovering it for
yourself. Mathematicians often say that to understand something you
must first read
it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a
different level of chunking up complicated ideas.
Make the idea fit in with your own
perspective and experience.
"When
I use a word, it means just what I choose it to
mean"
(Humpty
Dumpty to Alice
in Through the Looking Glass by Lewis Carroll)
The
meaning is rarely
completely transparent, because every symbol or word already represents
an
extraordinary condensation of concept and reference 4
A well-written math text will be careful
to use a
word in one sense only, making a distinction, say, between
combination
and permutation (or arrangement). A strict mathematical
definition might imply that "yellow rabid dog" and "rabid yellow
dog" are different arrangements of words but the same combination of
words. Most English speakers would
disagree. This extreme precision is utterly foreign to most fiction and
poetry
writing, where using multiple words, synonyms, and varying descriptions
is de
rigueur.
A reader is expected to know that an absolute
value is not about some value that happens to be absolute, nor is a
function
about anything functional.
A particular notorious example is the
use of It
follows easily that and equivalent constructs. It means something like
this:
One can now check that the next
statement is
true with a certain amount of essentially mechanical, though perhaps
laborious,
checking. I, the author, could do it,
but it would use up a large amount of space and perhaps not accomplish
much,
since it'd be best for you to go ahead and do the computation to
clarify for
yourself what's going on here. I promise
that no new ideas are involved, though of course you might need to
think a
little in order to find just the right combination of good ideas to
apply.
In other words, the construct, when used
correctly, is a signal to the reader that what's involved here is
perhaps
tedious and even difficult, but involves no deep insights.
The reader is then free to decide whether the
level of understanding he/she desires requires going through the
details or
warrants saying Okay, I'll accept your word for it.
Now, regardless of your opinion about
whether that
construct should be used in a particular situation, or whether authors
always
use it correctly, you should understand what it is supposed to mean. It follows easily that does not mean
if you
cant see this at once, youre a dope,
neither does it mean
this
shouldnt take more than two minutes,
but a person who doesnt know the lingo
might
interpret the phrase in the wrong way, and feel frustrated. This
is apart
from the issue that one persons tedious task is another
persons
challenge, so the author must correctly judge the audience.
Know
Thyself
Texts are written with a specific
audience in
mind. Make sure that you are the
intended audience, or be willing to do what it takes to become the
intended
audience.
T.S.Eliots
A Song for Simeon:
Lord, the Roman hyacinths
are blooming in bowls and
The winter sun creeps by the snow hills;
The stubborn season has made stand.
My life is light, waiting for the death
wind,
Like a feather on the back of my hand.
Dust in sunlight and memory in corners
Wait for the wind that chills towards
the dead
land.
For example, Eliots
poem pretty
much assumes that its readers are going to either know who Simeon was
or be
willing to find out. It also assumes
that its reader will be somewhat experienced in reading poetry and/or
is
willing to work to gain such experience.
He assumes that they will either know or investigate the
allusions
here. This goes beyond knowledge of
things like who Simeon was. For example,
why are the hyacinths Roman? Why is that important?
Elliot assumes that the reader will read
slowly
and pay attention to the images: he juxtaposes dust and memory, relates
old age
to winter, compares waiting for death with a feather on the back of the
hand,
etc. He assumes that the reader will
recognize this as poetry; in a way, he's assuming that the reader is
familiar with
a whole poetic tradition. The reader is supposed to notice that
alternate lines
rhyme, but that the others do not, and so on.
Most of all, he assumes that the reader
will read
not only with the mind, but also with his/her emotions and imagination,
allowing
the images to summon up this old man, tired of life but hanging on,
waiting
expectantly for some crucial event, for something to happen.
Most math books are written with
assumptions about
the audience: that they know certain things, that they have a certain
level of
mathematical maturity, etc. Before you
start to read, make sure you know what the author expects you to know.
An
Example of Mathematical Writing
To allow an opportunity to experiment with the guidelines presented
here, I
am including a small piece of mathematics often called the birthday
paradox. The first part is a concise
mathematical article explaining the problem and solving it. The second is an imaginary Reader's attempt
to understand the article by using the appropriate reading protocol. This articles topic is probability and is
accessible to a bright and motivated reader with no background at all.
The Birthday Paradox
A professor in a class of 30 random students offers to bet that
there are at
least two people in the class with the same birthday (month and day,
but not
necessarily year). Do you accept the
bet? What if there were fewer people in
the class? Would you bet then?
Assume that the birthdays of n
people are uniformly
distributed among
365 days of the year (assume no leap years for simplicity). We prove that, the probability that at least
two of them have the same birthday (month and day) is equal to:
What is the chance that among 30 random people in a room, there are
at least
two or more with the same birthday? For
n = 30, the probability of at least one matching birthday is
about 71%.
This means that with 30 people in your class, the professor should win
the bet
71 times out of 100 in the long run. It turns out that with 23 people,
she
should win about 50% of the time.
Here is the proof: Let P(n) be the probability in question. Let Q(n) = 1 P(n)
be the probability that no two people have a common birthday. Now calculate Q(n) by calculating the
number of n birthdays without any duplicates and divide by the
total
number of n possible birthdays.
Then solve for P(n).
The total number of n birthdays without duplicates is:
365 364 363 ...
(365 n +
1).
This is because there are 365 choices for
the first birthday, 364
for the
next and so on for n birthdays. The total number of n
birthdays
without any restriction is just 365n because there
are 365
choices for each of n birthdays.
Therefore, Q(n) equals
Solving for P(n) gives P(n) = 1 Q(n)
and hence our result.
Our
Reader Attempts to Understand the Birthday Paradox
In this section, a naive Reader tries to make sense out of the last
few
paragraphs. The Readers part is a
metaphor for the Reader thinking out loud, and the Professionals
comments represent
research on the Readers part. The
appropriate protocols are centered and bold at various points in the
narrative.
My Reader may seem to catch on to things relatively quickly. However, be assured that in reality a great
deal of time passes between each of my Readers comments, and that I
have left
out many of the Readers remarks that explore dead-end ideas. To experience what the Reader experiences
requires much more than just reading through his/her lines. Think of
his/her
part as an outline for your own efforts.
Know Thyself
Reader (R): I dont know anything about probability, can I
still make
it through?
Professional (P): Lets give it a try. We may have to
backtrack a lot
at each step.
R: What does the phrase 30 random students mean?
"When I use a word, it means
just
what I choose it to mean"
P: Good question. It doesnt
mean that we have 30 spacy or scatter-brained people.
It means we should assume that the birthdays
of these 30 people are independent of one another and that every
birthday is
equally likely for each person. The author writes this more technically
a
little further on: Assume that the
birthdays of n people are uniformly distributed among 365 days
of the
year.
R: Isn't that obvious? Why bother saying that?
P: Yes the assumption is kind of obvious.
The author is just setting the
groundwork. The sentence guarantees that
everything is normal and the solution does not involve some imaginitive
fanciful science-fiction.
R: What do you mean?
P: For example, the author is not
looking for a solution like this:
everyone lives in Independence
Land and is born
on the 4th
of July, so the chance of two or more people with the same birthday is
100%. That is not the kind of solution
mathematicians enjoy. Incidentally, the
assumption
also implies that we do not count leap years.
In particular, nobody in this problem is born on
February
29. Continue reading.
R: I dont understand that
long formula, whats n?
P: The author is solving the
problem for any number of people, not just for 30. The author, from now
on, is
going to call the number of people n.
R: I still don't get it. So
what's the answer?
Don't Be a Passive Reader
- Try
Some Examples
P: Well, if you want the
answer for 30, just set n = 30.
R: Ok, but that looks
complicated to compute. Wheres my
calculator? Lets see: 365 364 363
... 336. Thats tedious, and the final
exact value wont even fit on my calculator.
It reads:
2.1710301835085570660575334772481e+76
If I cant even calculate the answer once I know the formula, how
can I
possibly understand where the formula comes from?
P: You are right that this answer is inexact, but if you
actually go
on and do the division, your answer wont be too far off.
R: The whole thing makes me
uncomfortable. I would prefer to be able
to calculate it more exactly. Is there
another way to do the calculation?
P: How many terms in your
product? How many terms in the product on the bottom?
R: You mean 365 is the first term and 364 is the second? Then there are 30 terms. There are also 30
terms on the bottom, (30 copies of 365).
P: Can you calculate the answer now?
R: Oh, I see. I can pair up
each top term with each bottom term, and do 365/365 as the first term,
then
multiply by 364/365, and so on for 30 terms.
This way the product never gets too big for my calculator.
(After a few
minutes)... Okay, I got 0.29368, rounded to 5 places.
P: What does this number mean?
Don't Miss the Big Picture
R: I forgot what I was doing. Lets see. I was calculating
the answer
for n = 30. The 0.29368 is
everything except for subtracting from 1.
If I keep going I get 0.70632. Now what does that mean?
P: Knowing more about probability would help, but this simply
means
that the chance that two or more out of the 30 people have the same
birthday is
70,632 out of 100,000 or about 71%.
R: Thats interesting. I wouldnt have guessed that. You mean that in my class with 30 students,
theres a pretty good chance that at least two students have the same
birthday?
P: Yes thats right. You might want to take bets before you ask
everyone their birthday. Many people dont think that a duplicate will
occur. Thats why some authors call this
the birthday paradox.
R: So thats why I should read mathematics, to make a few
extra
bucks?
P: I see how that might give you some incentive, but I hope
the
mathematics also inspires you without the monetary prospects.
R: I wonder what the answer is for other values of n. I will try some more calculations.
P: Thats a good idea. We can even make a picture out of all
your
calculations. We could plot a graph of the number of people versus the
chance
that a duplicate birthday occurs, but maybe this can be left for
another time.
R: Oh look, the author did some calculations for me. He says
that for
n = 30 the answer is about 71%;
thats what I calculated too.
And, for n = 23 its about 50%.
Does that make sense? I guess it
does. The more people there are, the
greater the chance of a common birthday.
Hey, I am anticipating the author. Pretty
good.
Okay, lets go on.
P: Good, now youre telling me when to continue.
Dont Read Too Fast
R: It seems that we are up to the proof.
This must explain why that formula
works. Whats this Q(n)? I guess that P stands for probability but
what does Q stand for?
P: The author is defining something new. He is using Q
just
because its the next letter after P, but Q(n) is also a
probability,
and closely related to P(n). Its time to take a minute to
think.
What is Q(n) and why is it equal to 1 P(n)?
R: Q(n) is the probability that no two people have the
same
birthday. Why does the author care about
that? Dont we want the probability that
at least two have the same birthday?
P: Good point. The author
doesnt tell you this explicitly, but between the lines, you can infer
that he
has no clue how to calculate P(n) directly.
Instead, he introduces Q(n) which
supposedly equals 1 P(n).
Presumably, the author will proceed next to tell us how to
compute Q(n). By the way, when you
finish this article, you
may want to deal with the problem of calculating P(n) directly. Thats a perfect follow up to the ideas
presented here.
R: First things first.
P: Ok. So once we know Q(n), then what?
R: Then we can get P(n).
Because if Q(n) = 1 P(n), then P(n)
= 1 Q(n). Fine, but
why is Q(n) = 1 P(n)?
Does the author assume this is obvious?
P: Yes, he does, but whats worse, he doesnt even tell us
that it is
obvious. Heres a rule of thumb: when an
author says clearly this is true or this is obvious,
then take 15
minutes to convince yourself it is true.
If an author doesnt even bother to say this, but just implies
it, take
a little longer.
R: How will I know when I should stop and think?
P: Just be honest with yourself. When in doubt, stop and
think. When
too tired, go watch television.
R: So why is Q(n) = 1 P(n)?
P: Lets imagine a special case. If the chance of getting two
or more
of the same birthdays is 1/3, then what's the chance of not getting two
or
more?
R: Its 2/3, because the chance of something not happening is
the
opposite of the chance of it happening.
Make the Idea Your Own
P: Well, you should be careful when you say things like opposite,
but you are right. In fact, you have
discovered one of the first rules taught in a course on probability. Namely, that the probability that something
will not occur is 1 minus the probability that it will occur. Now go on to the next paragraph.
R: It seems to be explaining why Q(n) is equal to long
complex-looking formula shown. I will
never
understand this.
P: The formula for Q(n) is tough to understand and the
author
is counting on your diligence, persistence, and/or background here to
get you
through.
R: He seems to be counting all possibilities of something and
dividing by the total possibilities, whatever that means.
I have no idea why.
P: Maybe I can fill you in here on some background before you
try to
check out any more details. The
probability of the occurrence of a particular type of outcome is
defined in
mathematics to be: the total number of possible ways that type of
outcome can
occur divided by the total number of possible outcomes.
For example, the probability that you throw a
four when throwing a die is 1/6. Because there is one possible 4, and
there are
six possible outcomes. What's the probability you throw a four or a
three?
R: Well I guess 2/6 (or 1/3) because the total number of
outcomes is
still six but I have two possible outcomes that work.
P: Good. Heres a harder example. What about the chance of
throwing a
sum of four when you roll two dice?
There are three ways to get a four (1-3, 2-2, 3-1) while the
total
number of possible outcomes is 36. That
is 3/36 or 1/12. Look at the following 6
by 6 table and convince yourself.
1-1,
1-2, 1-3, 1-4, 1-5, 1-6
2-1, 2-2, 2-3, 2-4, 2-5, 2-6
3-1, 3-2, 3-3, 3-4, 3-5, 3-6
4-1, 4-2, 4-3, 4-4, 4-5, 4-6
5-1, 5-2, 5-3, 5-4, 5-5, 5-6
6-1, 6-2, 6-3, 6-4, 6-5, 6-6
What about the probability of throwing a 7?
R: Wait. What
does 1-1 mean? Doesnt that equal 0?
P: Sorry, my bad.
I was using the minus sign as a dash, just to
mean a pair of numbers, so 1-1 means a roll of one on each die - snake
eyes.
R: Couldnt you have come up
with a better notation?
P: Well maybe I could/should
have, but commas would look worse, a slash would look like division,
and
anything else might be just as confusing.
We arent going to publish this transcript anyway.
R: Thats a relief. Well, I know what you mean now. To answer your question, I can get a seven in
six ways via 1-6, 2-5, 3-4, 4-3, 5-2, or 6-1.
The total number of outcomes is still 36, so I get 6/36 or 1/6. Thats weird, why isnt the chance of rolling
a 4 the same as for rolling a 7?
P: Because not every sum is equally likely.
The situation would be very different if we
were simply spinning a wheel with the sums 2 through 12 listed in
equally
spaced intervals. In that case, each one
of the 11 sums would have probability 1/11.
R: Okay, now I am an expert. Is probability just about
counting?
P: Sometimes, yes. But counting
things is not always so easy.
R: I see, lets go on. By the
way, did the author really expect me to know all this?
My friend took Probability and Statistics and
I am not sure he knows all this stuff.
P: Theres a lot of information implied in a small bit of
mathematics. Yes, the author expected you to know all this, or to
discover it
yourself just as we have done. If I
hadnt been here, you would have had to ask yourself these questions
and answer
them by thinking, looking in a reference book, or consulting a friend.
R: So the chance that there are no two people with the same
birthday
is the number of possible sets of n birthdays without a
duplicate
divided by the total number of possible sets of n birthdays.
P: Excellent summary.
R: I dont like using n, so let me use 30. Perhaps
the
generalization to n will be easy to see.
P: Great idea. It is often
helpful to look at a special case before understanding the general
case.
R: So how many sets of 30 birthdays are there total? I cant do it. I guess I need to restrict my
view even more. Lets pretend there are only two people.
P: Fine. Now youre thinking like a mathematician. Lets try n = 2. How
many sets of two birthdays are there
total?
R: I number the birthdays from 1 to 365 and forget about leap
years.
Then these are the all the possibilities:
1-1,
1-2, 1-3, ... , 1-365,
2-1, 2-2, 2-3, ... , 2-365,
...
365-1, 365-2, 365-3, ... , 365-365.
P: When you write 1-1, do you mean 1-1 = 0, as in subtraction?
R: Stop teasing me. You
know exactly what I mean.
P: Yes I do, and nice
choice of notation I might add. Now how
many pairs of birthdays are there?
R: There are 365 365 total possibilities for two people.
P: And how many are there when there are no duplicate
birthdays?
R: I cant use 1-1, or 2-2, or 3-3 or ... 365-365, so I get
1-2,
1-3, ... , 1-365,
2-1, 2-3, ... , 2-365,
...
365-1, 365-2, ... , 365-364
The total number here is 365 364 since each row now has 364 pairs
instead
of 365.
P: Good. You are going a little quickly here, but youre 100%
right.
Can you generalize now to 30? What is
the total number of possible sets of 30 birthdays?
Take a guess. Youre getting good at this.
R: Well if I had to guess, (its not really a guess, after
all, I
already know the formula), I would say that for 30 people you get 365
365
... 365, 30 times, for the total number of possible sets of
birthdays.
P: Exactly. Mathematicians write 36530. And what is the number of possible sets of 30
birthdays without any duplicates?
R: I know the answer should be 365 364 363 362
...
336, (that is, start at 365 and multiply by one less for 30 times), but
I am
not sure I really see why this is true. Perhaps I should do the case
with three
people first, and work my way up to 30?
P: Splendid idea. Lets quit
for today. The whole picture is there
for you. When you are rested and you have more time, you can come back
and fill
in that last bit of understanding.
R: Thanks a lot; its been an experience.
Later.
1.
Emblems
of Mind, Edward Rothstein, Avon Books, page 15.
2.
ibid,
page 16.
3.
ibid,
page 38
4.
ibid, page 16.
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