Imagining Numbers

peterwayner writes "One mathematician I know told me that the most important lesson he learned was how to read a math book. It did no good, he said, to just start plowing through the theorems because that brought confusion. The key was to skim the book five or six times to get an idea of what the writer was trying to do. Then, and only then, was it possible to figure out the equations. This is what Barry Mazur tries to do in his book Imagining Numbers . There are some equations, graphs and diagrams, but first and foremost he offers plenty of poetry, philosophy and history to lay a foundation for understanding imaginary numbers." Peter's review continues below -- despite its complicated, abstract subject matter, he says that it's "simple enough to be accessible to most who will be interested in it."

Imagining Numbers
author	Barry Mazur
pages	267
publisher	Farrar, Straus and Giroux
rating	8
reviewer	Peter Wayner
ISBN	0374174695
summary	How to imagine imaginary numbers like the square root of minus fifteen.

Much of modern mathematical literature is structured with crisp, scripted precision. First there is theorem one, then theorem two, which leads to theorem three, which could only be followed by theorem four, and so on until we reach theorem n. If you want to learn the mathematics of complex numbers (a +bi), then classic texts (this or this) will get you there.

Some may like this logical progression, but it leaves others cold in the same way that crisp, modern architecture by Mies van de Rohe leaves some craving a more layered, fractured, ornate, organic and just plain fun place to live and work. Less isn't more, as Robert Venturi said, less is a bore.

If you happen to feel a chill when churning through an assembly line of theorems, you might enjoy the treatment of Mazur, a professor at Harvard who seems to spend as much time reading poets like Rilke or Stevens as he does examining old mathematical texts. Mazur is not the kind of machine that turns coffee into theorems-- he's too busy stopping to smell the rhetorical flourishes.

The book isn't aimed at mathematicians per se. The publisher, Farrar, Strauss and Giroux specializes in mainstream literature and that's probably the best pigeonhole for this book. Mazur wants the reader to understand how to think about imaginary numbers, not evaluate some integrals -- and that reader could really be anyone with the desire to think about mathematical things. The book is simple enough to be accessible to most who will be interested in it.

In many ways, Mazur attempted a much harder task than just teaching complex analysis. It's one thing to learn how to find the roots of polynomials, but it's another thing to try to help people get a feeling or an intuition for the square root of minus fifteen. Integers are easy to understand and even feel by counting out things, but imaginary numbers don't seem to exist. Mathematicians have spent many years trying to find the best metaphors and structures to understand how to find answers for all polynomials and it's never been an easy struggle.

The best part of the book is, without doubt, the historical treatment of how other mathematicians confronted the question of irrational and complex numbers. These ideas have always been hard to grasp and it took time to evolve the most compact and consistent nomenclature.

If you're interested in mathematics as more than just a mechanism that churns out answers, you'll probably enjoy the book. It's a light, friendly, philosophical expedition looking for a way to make imaginary numbers work in our minds.

---

Peter Wayner is the author of Translucent Databases , a book on how to imagine databases that hold no information yet still do useful work. You can purchase Imagining Numbers from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.

did this author start nothing.net?

it seems as though he is making quite a bit of money off nothing.

a book on how to imagine databases that hold no information

How to imagine imaginary numbers

I wish I had nothing that could make me a lot of money as well.

One time, I thought of this number

I called it seight, it would be between seven and eight. Yes, that was me.

This is great....

...if you're a person who even understands higher math. But what about morons like me who still have to break out the calculator to do simple calculations. The ironic thing is I can code but probably will never get past a certain plateu thanks to my shortfalls. I never got past algebra 1 in HS...

Anybody have any good sources of help for the math-disabled

Re:This is great....

Try The Number Devil [amazon.com] by Hans Magnus Enzensberger. It is a very accessible introduction to mathematical thinking for those who are not necessarily already inclined to it. The book consists of a series of dreams of a young boy who hates math and is visited by "the number devil". Originally seen as a torturer, the number devil ultimately reveals the beauty and - most importantly - comprehensibility of mathematics

Re:This is great....

Software engineering rarely has anything to do with complex math. (Computer Science occasionally does). If you can do simple algebra, you can probably write 90% of all end user applications out there. There's no calculus in a web browser, there's no trignometry in an email client.

The only place I can think of that does involve some hard math, is in 3d engines for games, or highly technical/scientific applications that deal with math. (CAD programs, MAPLE, MathCad, etc.)

Re:This is great....

Bad news... *you* are the idiot.

For real.

You seem to have confused the SCIENCE (note that word... "science"...) of computer science with the TRADE SKILL of programming.

If you went to university just to learn how to code a front-end for accounts payable... then you are as seriously misguided as a plumber who enrolls in course on continuum mechanics to learn his trade.

Most plumbers I know are smart enough to understand the difference... on the other hand, a shocking number of CS students I see... aren't.

Bottom line: Please drop out of university immediately... you don't even understand what you are taking. You'd be much happier at the DeVry school of Football, Truckin' and Codin'.

Re:This is great....

Unfortunately for real world programing jobs a cs degree is required. Even for simple customer service jobs. This is the problem.

no math?

Computer programming doesn't involve math in the same sense that economics doesn't involve math. You can do both of them with only very simple math skills, but you're going to understand what you're doing a lot better if you do know some math.

I think people studying software in school (CS majors, that is) should continue to be required to take calculus. And this is coming from someone who failed second semester calculus four times in a row, took it at a community college, dropped it, then took it again, and got an "A".

So to get to my point: sure, a web browser doesn't require any math. But if the people who wrote them understood more about the mathematics of the efficiency of algorithms, perhaps there'd be a chance that they wouldn't be so damnably slow. I mean really, I have this computer that's multiple hundreds of megahertz, and the blasted thing should be able to render any web page (minus network delays) in tiny fractions of a second, but instead it sometimes takes several seconds. It's possible that it just has so many features that it's going to be that, but I think perhaps instead somebody out there just didn't understand the difference between O(n) and O(n^2), or they didn't care.

Basically, I think a software professional ought to have enough general math ability that when writing any algorithm, they're just automatically aware of what category it falls into (O(n), O(n^2), O(n log n), etc.) without really consciously thinking about it.

As an example, if I write code that dynamically resizes an array when it runs out of space, and it does this by adding 5 extra elements each time, I should be aware when doing this that it will take O(n^2) time to put n elements in that array (if I work from the beginning). Whereas if I do what Perl does and double the size each time, I will waste a little memory, but in return the running time becomes O(n) again. They didn't teach me that factoid in school, but they taught me enough math to figure it out on my own. And that's a good thing if software isn't going to be complete crap.

Having said that, many math textbooks and math courses are complete crap, because teaching math is about like anything else, which is to say that you can do it if you don't have any communication skills and don't even care about being able to communicate, but if you don't have those skills then you'll make lots of people miserable.

So, IMHO, computer science students should be required to take advanced math, and advanced math students should be required to take creative writing. :-)

Re:This is great....

The ability to do arithmatic quickly and accurately in one's head is fairly orthogonal to the ability to comprehend higher math.

Hell, my current math professor has to write out simple arithmatic that I can do easily in my head, but he's one of the most gifted math teachers I've ever had.

Imaginary Numbers

Here's an imaginary number for you:

The number of people who regularly visit Slashdot that have unbiased opinions on Microsoft. ;)

Reminds me of...

"We're sorry, but the number you have dialed is imaginary. Please rotate your phone ninety degrees and try again. Thank you."

Five or Six TIMES?!?!?!

I have trouble reading math books once! Who has enough time to read one five or six times?

Maybe...

...if this book was available during my school daze, I would have paid attention in class. Then maybe I would have gotten better than a C in math.

I have always been more of an abstract thinker (which is weird being a programmer.) As such, I have never gotten along very well with the subject. Maybe a book like this would have put me on a better track. Then again, probably not.

Just my opinion,
SirLantos

Re:Maybe...

I have always been more of an abstract thinker (which is weird being a programmer.) As such, I have never gotten along very well with the subject.

<sarcasm>
Yes, somehow there is something concrete and real about programming, but math is just way out there and totally wierd, with no correlation at all with reality.
</sarcasm>

Dude, math, programming, physics, and almost any form of engineering are all abstract arts. We deal with invisible quantities that do magical things that have no correlation with reality. Heck, even music can fall into this arena of abstract arts.

Abstract thinkers make grade A programmers, mathematicians, physicists, chemists, engineers, etc...

Interesting...

Now that's really a new approach to understand maths... What's his reason for approaching math with poetry? I can see that feeling numbers is sometimes much faster than knowing numbers,but doing that sort of thing with imaginary numbers is certainly interesting.

Re:Interesting...

There is something deeply poetic about math. The theorems read like well-rhymed versus. To a guy who appreciates math, "The square of the hypotenuse is equal to the sum of the squares of the sides" stirs up a bit of emotion like a well-written poem.

To a beginner, who hasn't travelled through the wilderness of multi-variable calculus (IE, finding the volume of a hypersphere by taking the integral of it in several dimensions), and who hasn't even seen the simple and elegant Linear Algebra in its full glory, math is still mysterious, and is seemingly unknowable.

The beginner thinks of math as "2x7" and "4x = 3". They know only a few theorems that make any sense at all. The expert sees how all the theorems interrelate. He sees just how important the ones he learned in High School really were. He sees the grand scheme of things, and it looks like a giant, beautiful fractal, except it is much more complicated, and much more intelligent in design.

I applaud his efforts. He is taking a very abstract subject in math -- one which I find very enjoyable -- and exposing it to the rest of the world for its beauty.

This reminds me

... of an anecdote I came across in an essay about the difficulty of writing math books for the lay reader.

A statistician met his friend after a long time. After convincing the friend that statistics was not all about adding long columns of numbers, he proceeded to show him some interesting things like how to estimate the population based on a sample using the normal distribution. Pointing at the equation of the Gaussian distribution, the friend asks "what's this?" Statistician: "Oh that's pi, of course". Friend: "You mean the ratio of the diameter of a circle to the radius?" Statistician: "Sure". Friend (indignant): "Youre kidding me! The diameter of a circle can't have anything to do with the population of a country!"

An extreme example, perhaps, but shows how difficult it can be to write non-technical math books. Too often authors oversimplify things to increase readership. Mathematicians loath this and try to make their writing as stiff and formal as possible, "giving no indication that either the author or the intended reader is a human being". Yup, that's how one mathematician described "The Ideal Mathematician". Any honest effort that attempts to strike a balance needs to be applauded.

Re:This reminds me

I agree, approximating Pi by 2 is indeed oversimplification.

Some statistician!

Statistician: "Oh that's pi, of course". Friend: "You mean the ratio of the diameter of a circle to the radius?" Statistician: "Sure".


Where I come from, we call that value "two".

Only then?

"...Then, and only then, was it possible to figure out the equations..."

FACT: I got A's in calculus, and did nothing more than 'plow'. It wasn't the easiest thing to do but I learned, and I understand. (I'm a calc tutor now)

OPINION:
The above is B.S. Doing this with sections of a book, or even the whole book might be helpful, but only if the book is written to support it (ie a 'themed' book) and it certainly isn't a REQUIREMENT for understanding.

Re:Only then?

I got an A in Calc 1, and I've got a 94% halfway through Calc 2 right now, and I'd have to agree with you. I might add though, that even more important than plowing is to DO ALL THE HOMEWORK. There is a direct coorelation between the amount of homework/sample problems people do and how well they understand math. There is a good coorelation between understanding math and the grade you get.

There have been several topics I was confused about, but I plowed through, then did 50 sample problems (over 20+ hours) and found aftrwards that now I understood it, and it was actually easy. It's like a sport, you have to practice!

Hands on is the best for those who can

IMHO, assuming you have access in school to the resources: the best way to understand concepts like imaginary numbers is through hands on lab work. I would have never understood control systems just from books. But once you start playing around with tuning some circuits and watching response on an oscilloscope, 'imaginary' numbers in your system become very real. As I told someone (a lawyer) once who asked if 'i' made any sense (of course, I corrected him; to any electrical engineer, it's 'j'), "Sure it does, I've seen in on an oscilloscope.

Granted, if you never get to something like control systems, the above won't make sense. But once you're to a point where you have to deal with imgainary numbers, doing it hands on is best.

Re:Hands on is the best for those who can

"IMHO, assuming you have access in school to the resources: the best way to understand concepts like imaginary numbers is through hands on lab work."

Spoken like a physicist. To a mathematican, the best way to understand imaginary numbers is to say something like, "It annoyed people that the equation 'x^2 = -1' didn't have a solution. They just made up an answer to give them something to play with. Oh it also turns out that this models real world stuff for some reason, but that's not very important."

Imagining numbers, eh?

So what's the connection to Beowulf clusters? What kind of computational power do these "numbers" things have?

Barry Mazur ...

... is a very famous number theorist.
His results have had a key role in Wiles's proof of Fermat's last theorem.
He's at Harvard - see his homepage [harvard.edu].

Understanding the symbols

Does anyone have a good reference sheet of commonly used symbols in advanced math texts. I've been trying to learn stuff on my own but it is hard when you can't even verbalize what you are reading.

Re:Understanding the symbols

Part of the problem of mathemtics is that there is only a finite symbol set available to us (at least with TEX), so we tend to use the same symbol to mean different things in different fields. I'd try to pick up a book that has an index of notation. (Most have them, you just have to remember to look.) Otherwise, start with an introductory advanced math text (Eggen, Smith, and St. Andre, A Transition to Advanced Mathematics [amazon.com] comes to mind), and that should give you the foundations to move onto other books, as any good book will introduce any specialized notation. Another good resource is MathWorld [wolfram.com]. You can't exactly type in the symbols that you want, but you can search on terms that are appearing around the symbol to try to get a topic, and then things are well cross-referenced, so you can back up to a lower level of understanding if needed.

Re:Understanding the symbols

I highly recommend http://mathworld.wolfram.com for all your math reference needs. You may be referring to the greek alphabet, which is used extensively in math as a source of extra variable names. You can google for that. And I assume you're familiar with the differential symbol (a backwards 6 or a d), the integral symbol (a stretched out S), and the sum symbol (a capital sigma). If you don't know what those are, check mathworld.

Discovery of irrational numbers

There is an intriguing story about the discovery of irrational numbers.

In 1539 the mathematician Tartaglia won a contest involving solving cubic equations. His method used complex numbers, though he did not understand them as such. The mathematician Girolamo Cardano learned the method from him, promising him to keep it secret. However Tartaglia soon died, and Cardano published "Ars Magna" in 1545, in which he described the solution of cubics using imaginary numbers.

But it would be long before complex numbers would be properly understood and not looked upon with awe and mystery.

I got out of math

when I started to hear about "imaginary numbers". It's bad enough that we already have as many as we do, now they feel the need to invent some more.

Discovery of imaginary numbers

I posted this a while ago, but mistyped the subject as "discovery of irrational numbers". Braino :(

There is an intriguing story about the discovery of imaginary numbers.

In 1539 the mathematician Tartaglia won a contest involving solving cubic equations. His method used complex numbers, though he did not understand them as such. The mathematician Girolamo Cardano learned the method from him, promising him to keep it secret. However Tartaglia soon died, and Cardano published "Ars Magna" in 1545, in which he described the solution of cubics using imaginary numbers.

But it would be long before complex numbers would be properly understood and not looked upon with awe and mystery.

recommended books

have you seen what books they recommend to 'learn maths of complex numbers' ? Ahlfors and Cartan ! Caution, these are books on complex analysis, not on complex numbers. Don't buy them unless you've got already a good acquaintance on complex numbers ! Moreover, there are other prerequisites for Cartan, like point-set topology and real analysis (don't know for Ahlfors).

and anyway, these are dated books. Cartan dates back to the 60's and Ahlfors is (imo) even older. The presentation is a bit heavy. I'm sure you can find better and cheaper books. (personnally I learned from Cartan but I didn't find it easy to read).

How Math is Done vs. How Math is Presented

Much of modern mathematical literature is structured with crisp, scripted precision. First there is theorem one, then theorem two, which leads to theorem three, which could only be followed by theorem four, and so on until we reach theorem n.

I was a math PhD student some years back (but bailed with my Masters), so this review held particular interest for me. One professor I had at some point, probably in college, once compared doing math to cooking. The kitchen might be a mess afterwards, but the finished product looks great.

He was trying to make the point to us that as we sought to prove the various exercises, we shouldn't expect to go from point A (the hypothesis) to point B (the conclusion) but should instead expect to make several wrong turns and, in effect, make a mess along the way. When we finally got there, though, we should clean things up to make a better presentation. Hence the "crisp, structured precision" of most math texts. A good instructor will, while going over such a proof, offer insight into what thought processes led to each decision along the way.

These were relatively difficult, but still low-level exercises, since they had both hypothesis and conclusion. One (humbling) thing to remember about reading math is that someone was the first to prove these theorems. Not only did this person not know the direction the proof would take in advance, but he/she didn't know either the hypothesis or conclusion either!

For more history...

...look at An Imaginary Tale: The Story of Sqrt(-1) [amazon.com] by Paul Nahin. I thought the history behind the development of complex numbers was very fascinating; the people involved were very human, not noble god-like geniuses with no failings. A friend of mine bought this for me for my birthday, as I create fractal art and most of the mathematics I use involve complex numbers.

how to read/understand math ...

Given that much of the business of creative mathematicians amounts to inventing new patterns of provable relations between objects and properties, probably there are more ways to understand math than there are branches of math --

Spatial models just happen to appeal to me -- and the posts here indicate that is probably pretty common. Many of us just live with the convenience of that (and with its limitations, because many math concepts are hard to geometrize). But it's not the only way, and a few folks seem to find other and non-spatial thought patterns more natural.

In the end, the advice to look over the whole of some new math thing before diving into the detail sounds good, and probably that is because it actively encourages trying to pick out the kinds of relationships and features that the individual reader finds intuitive or meaningful. Those things, whatever they are for the individual reader, will not only stick best in the mind, but also they may in turn provoke further thought and maybe new invention.

Terry

Ugh

Frankly, I'm about halfway though this book and at times, it's all I can do to keep from tossing it in the trash bin in disgust.

The author seems to be incapable just getting to the subject and explaining himself in a clear and consise manner. Instead, he embarks on these long, florid poetry-filled diatribes about the imagination, and a yellow tulip.

In the few places where he's actually able to keep himself on topic for more than a page, the historical description of the search for imaginary numbers is actually an interesting story in and of itself.

Why he feels the need to expound on it with inapropriate references to poetry and half-baked philosophies on the nature of imagination is beyond me. I'm not against the poetry per se, it's just that there are many occasions where I'll read a passage, hit the poetry, sit back and think, "What the hell does that have to do with the subject?" Even when there is a conceptual link, most of the time, it's very weak. (Of the I'm talking about imagination, and the word imagine is in the poem level)

Frankly, it's been a very dissapointing read. If you're looking for an interesting math book (some people would consider that an oxymoron), I'd recommend David Berlinski's "A Tour of the Calculus" or either of Simon Singh's excellent books ("Fermat's Enigma" and "The Code Book").

Visualizing functions of a complex variable

As a shameless self-plug, here's pages on visualizing complex variables with the software I write:

http://www.PacificT.com/ComplexFunctions.html [pacifict.com],

http://www.PacificT.com/Exponential.html [pacifict.com].

Imagining Imaginaries

Here's a hint to imagine the complex number i. (the mathematicians here will recognize that it's nothing more than a linear-algebraic interpretation of i ).

First let's reinterpret ordinary numbers. There are many ways to interprete them; here's one which can be (see below) generalized to complex numbers.

Take an ordinary number n. For example you may choose n=0 or -3 or 150 or sqrt(2)=1.4142... or pi=3.14159265... This is what's called a real number. Here's the interpretation of this number n that I'd like to propose to you :

You can think of n as multiplying everything by n. For example imagine you've got $10 and n=2. Then, after n has "acted" on your $10, you've got $20. On the other hand, if n=-1, you've got $-10, so you've got a debt.

Now, let's carry on the example when n=2. The question i'm asking is : is there another number x such that x does half the job of n ? That is, to let x act twice is the same as to let n act once ? Answer : yes, such a number x exists and can even be choosed to be positive - it's called the square root of n. In the case n=2, we have x=1.4142...

At last, let's carry on the example where n=-1. Can we find a number i such that "to let i act twice is the same as to let n act once" ? In other words, is there any number i which does half of the job of -1 ? Well no real number does, but one introduces the new number i, which does the trick.

Personnally, this is as I think of i. These examples, with dollars, may seem oversimplified but it's a very deep interpretation of numbers, it's the main idea behind Linear Algebra. For example, in Algebraic Number Theory, the linear algebraic formalism is used to introduce concepts as fundamental as the degree, norm and trace of a field extension.

You have to know math thoroughly to appreciate it.

What bothers me about books at this level is that they tend to give an impression of being something more than an extremely superficial (albiet fundamental) approach to the material.

You really have to know math thoroughly to appreciate it. All this rhetoric about mathematical beauty refers to something quite alien from ordinary human experience. Typically, math nonfiction just gives people terms to throw around that they don't really understand. (like Godel incompleteness)

If you just want to "get a feel" for advanced mathematical concepts, don't bother. It's a waste of time. On the other hand if you're fairly young and interested in math, it's a fine book to... um... "inspire" you I guess.

Math for dummies?

Books trying to sell math for dummies suck. Just read real math books. It takes time and is hard to understand but it's the only way to really understand.

Innumeracy

The best book about mathematics is Innumeracy: Mathematical Illiteracy and Its Consequences [amazon.com].

An excellent read for anyone with a grasp of mathematics, it is also an easy read for people who don't quite get it. The writing is entertaining and gives the mathematically challenged a better handle on basic statistics and how to handle really large numbers correctly.

It was required reading in our quantitative analysis class during my MBA and I have loaned it out to a number of people to enlighten them.

--Keith

Most math writers are terrible writers.

"It did no good, he said, to just start plowing through the theorems because that brought confusion. The key was to skim the book five or six times to get an idea of what the writer was trying to do."

I agree with this advice. However, it wouldn't be this way if math writers were good writers. I have never seen a math book in which the author did all that could be done to make the subject clear. Maybe subconsciously they don't really want you to know what they know. Mathemeticians did not get into the field because they like people.

New ways to teach math...

I think new ways of teaching math like this are great. Having a math degree myself, I was recently asked to speak at a career day at a local school. The number one things the kids wanted to know is *why* they needed to learn math...

This book is a great step towards teaching/giving interest to a larger 'math-challenged' audience.

Besides, if it wasn't for math guys, we wouldn't have computers... >:) (Interestingly enough, Alan Turing killed himself with an APPLE. hehehe ok bad joke)-

"The reason that every major university maintains a department of mathematics is that it's cheaper than institutionalizing all those people"

LosT

Relating to complex numbers

I once had a EE professor who explained complex power (i.e., the complex number component of AC power) with a beer analogy:

Complex power is like the head on your beer. You can't do anything useful with it (e.g.: drink it, or use it to power your PS2), but you have to carry it around with you, consuming resources. And, of course, you try to minimize it, where possible.

Worked for me!

Measure

Pretty much any technical book I pick up I instantly measure it against [amazon.com]
Expert C Programming just based on the fact that I have never come accross a book as clear, informative, and entertaining in any field. Looking at my bookshelf here at work I have math books, programming books, general documentation - and most of them are dry as hell and were a pain to get through. Has anyone found a good math book that can match Expert C Programming in its writing?

Math is easy

As a Math PhD student,my opinion is that math is in fact very easy. It all follows from simple logical thinking.

However, most books try to impress with lots of formulae without explaining the basic math behind them. They focus on being able to do the calculations, but not on actually understanding what is going on.

I would compare that to writing programming code without adding any comments. When following the code you'll see you get the right result, but if you have to find out how it exactly works, it takes a LOT of work, because you don't have the whole picture.

If you really want to understand math, don't take a book on complex numbers, but take something even simpler than that, then try to really understand what is going on.

Imaging = photoreading.

I don't see any of this work as particularly groundbreaking.

The technique he uses to preview the material 5-6 times is known as photoreading. A technique taught by a company Learning Strategies [learningstrategies.com]

I am a certified Photo Reader, I can cruise through a 400 page technical book in one night, and recall it all the next day and every day thereafter.

The remainder of the techniques he talks about are "Mind Mapping" which are also taught by Learning Strategies.

Sounds to me like a book that teaches you a different perspective on mathematics, but doesn't teach you any new knowledge.

--
"Give a man fire, and he'll be warm for a day; set a man on fire, and he'll be warm for the rest of his life."

A short imaginary joke

If you think about it over history you can see how people got less and less confortable with number systems as they got more complicated.

We started with natural numbers
then added fractional numbers
then added negative numbers
then added irrational numbers
then added imaginary numbers

Need to explain begets the need for higher math.

I think that the reason that most people do not know mathematics is that they do not care about mathematics. When you are reading about abstract concepts that have no correspondence to your own experience, you are justifiably frustrated. Just as the desire to learn the subtleties of one's natural language can come only from the need to explain new experiences, so the desire for higher mathematics can come only from the need to express new abstractions that vaguely coalesce in your mind as you tackle some unusual programming task. My recent programming adventures provide an example of this happening. For the last few months I've been struggling with using dataflow graphs as a generic programming tool, and the need to describe the entities I was creating pushed me into rereading mathematical texts that lay dormant on my shelves for quite some time. And I found consolation in multivalued functions, and operators, and some abstruse terminology from group theory. And then my ideas suddenly seemed a little clearer and cleaner and I think I could explain them better now than before.

Book about the exponential function

If you are interested in books of this sort, I highly recommend e: the Story of a Number" [amazon.com] by Eli Maor. He strikes a wonderful balance between history and mathematics. He has also written other books (that I have not yet read) about infinity and trig functions.

Numbers at right angles to each other

As part of a theory course I held for some prospective Radio Amateurs, I mentioned how the complex numbers turned out useful when dealing with electrical components at AC.

The argument went as follows: "We have a series connection of a resistor and inductor, with some AC current going thru them. This is drawn as a set of rotating pointers, with the current and voltage of the resistor to the right, the voltage of the inductor 90 degrees ahead of the current, pointing upwards.

"The ratio for the voltage to current for the inductor is w*L, but note that these voltages are 90 degrees out of phase. We use the label j to indicate this, so multiply with j means turn the phase 90 degrees. So the voltage for the inductor becomes j*w*L."

To emphasize this, the same argument was repeated again for a capacitor, ending with the formula V = -j / w * C, and again it was noted that we can turn things around by multiplying with j.

"Now, look at what happens if we multiply twice by j; we end up with the pointer going the other way around. Evidently j*j = -1."

Thus the meaning of the complex numbers was imparted, avoiding the gee-whiz effect of the expression "square root of -1".

On a much lighter note, when I went to University, they would offer so-called "Thousand Island Dressing" which appeared to be a 50/50 mix of mayonnaise and ketchup. We called it "600+j800 islands" indicating that we would have to imagine some of them to make the full 1000...

The best math book I ever read

The best math book I read while getting my degree, and the most unique math book I've ever seen, was/is "A Pathway into Number Theory," by R. P. Burn (Cambridge: Cambridge University Press, 1982, ISBN 0521241189).

Burn covers the main points of an introduction to number theory with what I can only describe as a combined experimentalist/Socratic approach--the book has no prose text in the conventional sense, and no formal proofs. Rather, the book is a series of questions that build upon each other, starting with the simple (e.g., "What is the relation between each number in table 1.1 and the number below it?") and building to the powerful (e.g., the fundamental theorem of arithmetic). Burns works through special cases of fundamental results, then leads the reader to speculate on the underlying principle, then helps him prove that it is true in general.

In the introduction he states that the book was put together "by keeping a record of how I actually resolved the blocks which I encountered as I read a number of standard texts. Time and again, it was the exploration of special cases which illuminated the generalities for me. This collection of explorations was then organised into a sequence in such a way that the 'pathway' would climb towards the standard theorems which occur here as problems for the student at the end of each section." It was a marvelous way to learn.

It's still in print.

Complex numbers shouldn't make physical sense

The reason complex numbers are so hard to understand is because they are rarely used to model the real world. Real numbers are intuitive because they are generally used to represent a magnitude. The variables in a problem often represent real numbers. However, for some problems, it becomes very difficult to work with real numbers. This is where complex numbers come to the rescue.

Complex numbers have extra properties that make it easier to solve problems, and they are a superset of the real numbers. To solve a problem, just assume the variables are complex and generate all the solutions. Any real solution to the original real problem must be a solution to the complex complex problem, and any real solution to the complex problem must be a solution to the original real problem. Therefore, you just need to generate all the solutions and throw away any complex solutions.

This is how complex numbers are used in practice. They are just a mathematical tool. Without this burden of giving complex numbers a physical interpretation, (Though this is still possible for some types of problems) it makes more sense to view them as abstract two dimensional objects. Addition is just vector addition and multiplication is scalar multiplication along with rotation.

This is one of the main ways math is generalized. By adding extra properties to an object, it makes it easier to work with the object. This can be seen in the historical changes in the concept of a number. From natural, integer, rational, real, and complex. By adding more structure, the object actually becomes easier to use.

Of course, the another way to generalize is to take a result and strip away all the unnecessary details. For example, one starts with calculus on intervals and then proceeds to metric spaces and then topologies...

Simple answers to all "complex" questions ...

... right here [mathforum.org] .

Sample questions:

Can you explain complex numbers simply?

How do you graph imaginary numbers?

Imaginary Numbers in Real Life

Is it possible to find the square root of a negative number and, if so, to what number system do these square roots belong?

How is the square root of -1 possible?

What are imaginary numbers, what is their purpose, and how are they used?

What is i?

What exactly is the complex number system comprised of? ... and many more ...

Darn!

And I clicked on this story link because I thought it was about corporate accounting... darn...

Re:President Bush

>God bless our great leader President Bush!!

in 1939, people would have said "God bless our great Fuhrer G.W Bush!!"

it's sad to see people forget History! and don't that stupid Saddam=Hitler thing, Saddam didn't ever had the biggest army in the world, Hitler in 1939 did!

Random-access reading

Perhaps I (and anyone else who has experienced) would do well to revisit these books using this prescanning approach.

Actually, I've found this approach useful for many books. In fact, one of the secrets that Evelyn Wood Reading Dynamics uses to improve reading comprehension at fast reading speeds is to skim the intro and the conclusion before tackling the meat of the chapter. It's also useful to skim a section in your textbook before the lecture on the same material. The idea is that you've at least got a vague notion about what the lecture is supposed to be about. This reduces the possibility that you will get so lost during the lecture that you spend the hour fantasizing about the blond with the nice-smelling hair sitting in front of you.

This approach is also implicit in most briefings that you present or attend when you enter the work world. The first few charts should explain what the purpose of the briefing is and present an outline. This helps the audience see the bigger picture before you get into the nitty-gritty.

I urge you to try the approach of 'prescanning' or 'random-access' reading if you have some technical material to read. Of course, if the book you're reading does not have a 'conclusions' or 'summary' section, then you have to be a bit more inventive. For example, you may want to skim the chapter and jot down the section headings. Then close the book and spend five minutes thinking about what YOU think the summary is going to be.

GMD

Re:Let me get this straight...

The destinction commonly used to define a terrorist act as opposed to a really really bad attack during wartime is the following: A political and diplomatic state of "War". The bombings of dresden and Tokyo involved a horrible loss of life. If the ARMY AIR CORE(not the USAF) had the technology to only destroy military targets they would have (much as the USAF currenlty does with laser guided weapons and cruise missiles). Furthermore, due to the attack on Pearl Harbor by the Japanese when the US and Japan WERE NOT AT WAR (i.e. an act of terrorism) a formal war was begun. As such, attacks on industrial centers should be expected and civilian populations should have evacuated much as the English city of Coventry was evacuated when it was learned that it was going to be bombed durring WWII..