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Imagining Numbers
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.
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.
Work sucked, until it became unemployment, when it became slightly more tolerable. -Tet
...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
I lost my concept of community when my community lost all concept of me.
"We're sorry, but the number you have dialed is imaginary. Please rotate your phone ninety degrees and try again. Thank you."
"Dogs and cats, living together...it's mass hysteria!"
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.
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.
... 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.
War doesn't prove who's right, just who's left.
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.
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.
There is no reasonable defense against an idiot with an agenda
:wq
Abstract math (ring theory, group theory, etc.) is not directly related to imaginary numbers. Sure, imaginary numbers may exhibit ring or group properties, but that is more incidental that causal.
I *am* the geekest link!
<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...
The radical sect of Islam would either see you dead or "reverted" to Islam.
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.
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).
War doesn't prove who's right, just who's left.
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.
The radical sect of Islam would either see you dead or "reverted" to Islam.
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!
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!
...look at An Imaginary Tale: The Story of Sqrt(-1) 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.
People are never as simple as their stereotypes. This applies equally to Christians, Muslims, and Emacs-lovers.
What the hell do you call '555-1212' ?!! Looks like -657 to me!
the preceding comment is my own and in no way reflects the opinion of the Joint Chiefs of Staff
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
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").
I just want to take over the world...Why does that automatically make me EVIL?
Yes, but programming, math, etc. is extremely structrured. A plus B always equals C, it is all strict rules that MUST be followed or your answer is wrong.
When I say I am an abstract thinker, I mean that I understand things that aren't required to have structure more than things that absolutly must be a certain way.
Yes, programming requires a bit of imagination. But it is all logic, it must be constructed a certain way or it will fail. Peotry, on the other hand, requires nothing. There is nobody that can say that one persons peotry is absolutly wrong, yes it can be structured, but if the artist decides they don't want it to be structured then that makes the poem more special to the poet.
Before you throw your sarcasm around, be sure you understand what the person is trying to say. Change your perspective to their own and then, respond with intelligence, not with sarcasm.
Just my opinion,
SirLantos
The flying hamster of DOOM rains coconuts on your pitiful city.
http://www.PacificT.com/ComplexFunctions.html ,
http://www.PacificT.com/Exponential.html.
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.
War doesn't prove who's right, just who's left.
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. :-)
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
watch this
"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.
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".
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!
Pretty much any technical book I pick up I instantly measure it against
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?
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.
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
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.