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Imagining Numbers 265

Posted by timothy from the 20030313-and-iv-for-instance dept.
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.
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Imagining Numbers

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  • did this author start nothing.net? (Score:5, Funny)

    by trmj ( 579410 ) on Tuesday March 18, 2003 @11:33AM (#5536582) Journal
    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.
    • I wish I had nothing that could make me a lot of money as well.

      Dot com, we miss you already.
    • Imaginary numbers are hardly "nothing"

      For every solution in two space or three space there are a number of complex solutions, and equations which have no solution in two or three space could have a complex solution. While its not as practical as two or three space math which we commonly use (like what is the area of this paper, what is the volume of this cup), complex numbers still have an important role in mathematics and engineering. It is hardly "nothing".
      • Part of the confusion is the misleading misnomer of "imaginary numbers." One of those bad choices in the history of science. Had they instead made a more obvious connection with vectors the issue wouldn't have been so confusing.

        I ran into this all the time trying to tutor lower division physics students. When they got to A/C circuits and the little bit of complex mathematics they required the students became rather hopelessly confused. Once you get the whole abstraction of "imaginary number" and the p

    • no he watched a lot of Seinfeld though... the show about nothing!

  • This is great.... (Score:5, Interesting)

    by xtermz ( 234073 ) on Tuesday March 18, 2003 @11:34AM (#5536590) Homepage Journal
    ...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.... (Score:5, Informative)

      by aborchers ( 471342 ) on Tuesday March 18, 2003 @11:45AM (#5536676) Homepage Journal
      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.... (Score:5, Informative)

      by WPIDalamar ( 122110 ) on Tuesday March 18, 2003 @11:54AM (#5536738) Homepage
      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.)

      • no math? (Score:5, Insightful)

        by Anonymous Coward on Tuesday March 18, 2003 @12:44PM (#5537205)

        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. :-)

      • Software Engineering may not have anything to do with complex math, but it has everything to do with being able to read and write complex technical specifications.

        A software design and analysis is a lot like a mathematical proof. You have a hypothesis (this design will solve the problem) and you document the steps necessary to get to the solution.

        If somewhere along the way you can't draw a clean line from one portion of the design to another, you've identified a flaw in the hypothesis. You throw the desig
      • > 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.
        >
        Not surprising at all when you consider that most software is just an automation of the thinking that we humans do for common daily activities. Such thinking involves processing of higher-level symbols by our brain. These symbols are high-level abstractions into "common-sense" symbols of the sen
      • Tell that to the software engineers. They seem to use lots of complex math very effectively. For example, suppose you are designing a web crawler for a search engine. How should you gather pages? The answer requires lots of thinking about graphs, recursive equations, and other complex mathematical things.

        In general, math doesn't help you write any actual code. But it is vital in designing applications, and design is really the interesting part of software engineering.

      • Hardly. Graphs, automata, sets. All (three among many I should add), that belong to both math and CompSci, and all play a heavy role in programming. If you don't want to understand those things, fine. You can get a job as a code monkey, probably not a secure one, but you can get a job. Just don't expect to be taken seriously, due to extreme lack of knowledge. One of the first (easy) questions I ask an applicant is to draw a state machine for a simple regex. If you can't do that, you have no business designi

      • Not so.

        Imagine having an e-mail application that handles 15,000 peices of e-mail a day. This sort of application would require algorithms to be used or created that are very low order. Anything that has to do with data processing had better be coded by an individual who has a higher education than just algebra. This individual should be able to reduce an algorithm to it's bottlenecking factor, optimize it, and recode it.

        That is something you learn in Computer Science. It is not something that uses sim

    • Re:This is great.... (Score:2, Informative)

      by Anonymous Coward
      Undoubtedly the best book to get started in thinking mathematically is Innumeracy [amazon.com], by John Allen Paulos.

      All you need is a rudimentary understanding of numbers (what it means to be bigger and smaller, and how the basic operations work) to follow along. Paulos is so lucid schools would do well to require this book for reading in math courses.
    • Much of math, especially easy math like in high school, is hard to learn if you do not have the right mindset for it. It is a type of common sense that some people just have less of than others.

      Not a diss at all, but it is just far harder for some to learn math than others, bceause of how they think.

      A good source of help for the "math-disabled" is some textbooks with answers and patience. Or start taking some math courses, several times if necessary until you can grasp the whole concepts, not just mem

    • Re:This is great.... (Score:4, Insightful)

      by jkujawa ( 56195 ) on Tuesday March 18, 2003 @12:18PM (#5536946) Homepage
      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.
    • I'd recommend taking some discreet math or logic courses and a junior college. It will ease you into more complex math and most likely allow you to avoid calculus, which tends to turn off people who aren't "into math".

    • Re:This is great.... (Score:2, Informative)

      by norite ( 552330 )
      There's a book by David Berlinsky, called "A Tour of the Calculus, The Philosophy of Mathematics" It's the best damn book on calculus I ever read! (Get it! ISBN # 0 434 09844 2) here's the kicker - it's not a text book; it's a novel - he teaches you calculus, but it's also a history lesson, a story of the people who developed the calculus. He takes you back to europe in the 1700's...

      Also, he actually explains terms like functions - and what a function is - in plain english. I went through high school not

  • Reminds me of... (Score:5, Funny)

    by wynlyndd ( 5732 ) <wynlyndd.gmail@com> on Tuesday March 18, 2003 @11:38AM (#5536613) Homepage
    "We're sorry, but the number you have dialed is imaginary. Please rotate your phone ninety degrees and try again. Thank you."

  • This reminds me (Score:5, Interesting)

    by arvindn ( 542080 ) on Tuesday March 18, 2003 @11:43AM (#5536655) Homepage Journal
    ... 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 (Score:4, Funny)

      by Anonymous Coward on Tuesday March 18, 2003 @12:10PM (#5536862)
      I agree, approximating Pi by 2 is indeed oversimplification.

    • Re:This reminds me (Score:2, Insightful)

      by Anonymous Coward

      I don't understand. How does that anecdote demonstrate that it's hard to write a book about mathematics that the lay reader? In that anecdote, the friend is telling the statistician what he doesn't understand! And that is the essential information that you need when writing expository prose. If you are telling someone how to get to where you are, the first step is to know where they are. Think about it: would you give someone directions to your house without first asking them what part of town they live

    • Some statistician! (Score:4, Funny)

      by volpe ( 58112 ) on Tuesday March 18, 2003 @01:35PM (#5537622)

      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".

    • Re:This reminds me (Score:2, Interesting)

      by kcelery ( 410487 )
      Very interesting read:

      http://pauli.uni-muenster.de/~munsteg/arnold.htm l

      Mathematics came as a mental tool on studies of real life problems. Over abstraction (unnecessary) creates tormented readers, and I was among one of them.
    • I found out that in order to learn math, you must know BEFOREHND, with an intuitive example, what the freak is going on. Nothing is better than an explanation meant for kids but writen by profesionals. And when the teacher is not clear, then the KID let's him know (so he has to explain it again!). And these answers try to be intuitive and fun. It's been a godsend to me, because answers like that are very handy. And I don't remember having fun while I learned...

      Dr. Math - http://mathforum.org/dr.math/
      Math F

  • Hands on is the best for those who can (Score:4, Insightful)

    by YetAnotherAnonymousC ( 594097 ) on Tuesday March 18, 2003 @11:45AM (#5536674)
    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 (Score:5, Interesting)

      by zzyzx ( 15139 ) on Tuesday March 18, 2003 @12:02PM (#5536802) Homepage
      "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."
    • I am not sure about the example you picked.I have'nt done too much control systems, but AFAIK almost all the effects you can see in the the lab are the real part of the equation, right? All the equations might be of the form e^ik but what you see on the scope is only sin(k) or cos?(k).
      In this case I think you still have to get back to maths to explain the physics.
      Easiest way to teach somebody to learn imag. numbers is to ask him to code a Fast fourier transform, MHO only.

  • Barry Mazur ... (Score:5, Informative)

    by glMatrixMode ( 631669 ) on Tuesday March 18, 2003 @11:50AM (#5536709)
    ... 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 (Score:4, Interesting)

    by baywulf ( 214371 ) on Tuesday March 18, 2003 @11:51AM (#5536717)
    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 (Score:4, Informative)

      by mitchkeller ( 208117 ) <justice.gogeek@org> on Tuesday March 18, 2003 @12:27PM (#5537044) Homepage
      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 (Score:5, Informative)

      by kurtkilgor ( 99389 ) on Tuesday March 18, 2003 @12:46PM (#5537222)
      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.
    • 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.

      My answer probably isn't what you'll want to hear: Don't do math this way. It's backwards.

      Math is not at all about symbols. Math is about quantities abstracted from real things (e.g., numbers and lines). Symbols may make it easier to represent these numbers and lines in a systematic, structured, and complex

      • Math is about quantities abstracted from real things


        i.e. symbols. Everything is about symbols. Everything you do, know, and learn. I think you meant "Math is not about notation"

        • i.e. symbols. Everything is about symbols. Everything you do, know, and learn. I think you meant "Math is not about notation"

          Oh, no, I didn't mean that at all. When I say that math is not *about* symbols, I simply mean that the symbols themselves are not essential to the science of mathematics. In other words, take the symbols away, and you can still have mathematics. Read Apollonius' On Conic Sections, for example. He gives the definition of a parabola in a very long paragraph of words only, not mode
          • Words are symbols, numbers are symbols. Everything is symbols. The difference I was pointing out between symbols and notation is that the particular notation (the coice in symbols used to convey semantic meaning) is what you meant. For example, using a longer list of symbols, (i.e. the long paragraph of words) is simply a different choice of symbols (notation) to describe the concept. The idea that you are getting at, I believe, is that not knowing the conventional notation used for math shouldn't preclude
            • Words are symbols, numbers are symbols. Everything is symbols.

              I agree wholeheartedly with the first two. I did not mean to imply that Apollonius *wasn't* using symbols, I just meant that he was using a different method of signifying.

              However, I disagree with the third. Everything is not symbols. The things signified by symbols, are not themselves symbols. When I say "five" or "cinco" I signify something that is not itself a symbol. When I say "parabola" or "y=x^2", I signify something that is not it

              • My point is that there is a reality that underlies our mathematical symbology or notation that is completely independent of how we choose to signify it. So, my advice to the original poster was this: begin your study of mathematics by understanding the reality, not by trying to decipher the symbols. The symbols are there to *aid* understanding and signification. If they don't help, don't use them.

                That is all syubject to debate. I contend that everything is symbols. You have no way of relating to the wo

                • That is all subject to debate. I contend that everything is symbols. You have no way of relating to the world except through symbols, therefore you can't prove that there is any reality apart from them. Additionaly there is no real thing that is 5, or red.

                  Yes, it's subject to debate. But its a debate that I've had literally hundreds of times, and I'm convinced that over the course of time that I've developed an understanding of its nuances. Oversimplifying things: modern philosophers (beginning with Lo
                  • Actually I did:


                    The difference I was pointing out between symbols and notation is that the particular notation (the coice in symbols used to convey semantic meaning) is what you meant

                    In case you need it in other words notation is the set of symbols that are used. Using one notation over another is what you describe.
                    This is what it boils to:


                    Math is not at all about symbols. Math is about quantities abstracted from real things (e.g., numbers and lines

                    These statements are false. For starters yo

                    • In case you need it in other words notation is the set of symbols that are used. Using one notation over another is what you describe.

                      Okay. If that's what you mean, then the rest of what you've said makes sense. I still don't agree, but at least it makes sense.

                      For starters your example is bogus lines and numbers aren't real things.

                      You say this because you think that by "things", I mean "substances". Of course quantities and qualities are not "things" in that sense, but they certainly have *some* be
                    • You're right, this is not the place for a long out discussion on this. I'm assuming we can agree to disagree.



                      to recommend some books on the philosophy of science for you to read


                      Me too ;)

    • Another good way to learn math, math symbols, etc. is to take a couple of physics courses. Math can be kind of abstract, but since physics builds upon itself, and always has a physical analog for the little symbols, it can give you a good handle on basic calculus & other mathematical principles. I never got good grades in math classes till I got through two semesters of physics in college.
  • 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 numb

  • I got out of math (Score:3, Funny)

    by Lxy ( 80823 ) on Tuesday March 18, 2003 @11:54AM (#5536735) Journal
    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 (Score:5, Interesting)

    by arvindn ( 542080 ) on Tuesday March 18, 2003 @11:58AM (#5536773) Homepage Journal
    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 (Score:4, Informative)

    by glMatrixMode ( 631669 ) on Tuesday March 18, 2003 @11:59AM (#5536785)
    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 (Score:5, Interesting)

    by Mignon ( 34109 ) <satan@programmer.net> on Tuesday March 18, 2003 @12:06PM (#5536828)
    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!

    • 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!

      That's not always entirely true. In theory, you're right, but in both math and science intuition is always a factor. Of course, you can't use your intuition as your proof, but it is often useful to carry you in the right direction.

    • 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!


      Funny, I wouldn't have found that humbling, more like ... encouraging.

    • "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."

      It's not like people prove things at random. At least in my experience, they notice some patterns and start to wonder if that's always true and work on a proof (or disproof) of it.

  • For more history... (Score:2, Informative)

    by fractalus ( 322043 )
    ...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.
  • 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 s

  • Ugh (Score:5, Interesting)

    by hal200 ( 181875 ) <{slashdot} {at} {jdk.ca}> on Tuesday March 18, 2003 @12:32PM (#5537085) Journal
    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").

    • Deja vu. This is exactly what I experienced when I read "The Tao of Physics". Couldnt he have just talked about physics instead of sounding like John Edward from "Crossing over with John Edward"?

    • I'll anyday take Fermat over Chaucer.
      I agree I can't understand some of his theorems, but I can't understand poetry at all.
  • 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 (Score:4, Informative)

    by glMatrixMode ( 631669 ) on Tuesday March 18, 2003 @12:41PM (#5537166)
    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 left out the best part of the story: geometry. Multiplying with -1, geometrically on the number line, means mirroring a point at the origin, flipping it to the other side. Now what geometrical operation does half the job of mirroring?

      A 90 degree rotation, that's right. And that's what multiplication with i is, nothing but a (counterclockwise) 90 degree rotation. Rather than a number line, you now have a number plane.

      So if you multiply 1 by i, on the one hand you get i, and on the other hand you ge

      • AxelBoldt,

        since the celebrated geometric interpretation does not seem to have convinced the masses, I thought it's be appropriate to present another interpretation. Would have been useless to repeat an interpretation they did already know.
    • Your explanation might work as an explanation of square roots, but it really doesn't provide an explanation of i beyound saying that i=sqrt(-1) or i*i=-1, which of course, is its definition. You haven't added any meaning or insight into the nature of i, beyond its definition.
  • 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

  • Most math writers are terrible writers. (Score:3, Insightful)

    by Futurepower(R) ( 558542 ) on Tuesday March 18, 2003 @01:34PM (#5537617) Homepage

    "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.

    • Math writers are very good writers for what they write. Graduate school in mathematics is nothing more math writing training.

      It's true they have to use only a very limited aspect of the English language to convery their ideas. It's like looking at computer code and trying to figure out what it does, instead of looking at what something does and then figuring out where the code for it is and how it work.

    • 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.

      I think that's unfair to mathematicians; a lot of them may not be "people people", but that doesn't mean they dislike people; it means they don't understand the social behavior of people, and don't

  • Relating to complex numbers (Score:3, Interesting)

    by mahler3 ( 577336 ) on Tuesday March 18, 2003 @01:51PM (#5537769)

    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 (Score:2, Informative)

    by mrcparker ( 469158 )
    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 (Score:3, Insightful)

    by +P'ther ( 659978 ) on Tuesday March 18, 2003 @02:21PM (#5538018)
    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.

  • A short imaginary joke (Score:4, Funny)

    by jbolden ( 176878 ) on Tuesday March 18, 2003 @03:18PM (#5538485) Homepage
    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. (Score:3, Interesting)

    by Chemisor ( 97276 ) on Tuesday March 18, 2003 @04:15PM (#5538924)
    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.
  • 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, startin
  • ... 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 ...

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