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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.
Try The Number Devil 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
Trouble making decisions? Just flip for it.
... 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.
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.)
Undoubtedly the best book to get started in thinking mathematically is Innumeracy, 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.
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!
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.
...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.
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 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. 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.
"You will only be remembered for two things: the problems you solve or the ones you create." Mike Murdock
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.
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.
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?
Regarding the following segment:
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.
How is resizing an array O(n^2)? It's linear with respect to the number of elements, which is to say O(n)! It doesn't matter whether you increment by 5 or double the size each time you reallocate. The allocation is constant, then the copy of elements from the old array to the new array is linear.
If your collection were continuosly growing, then you could say that the complexity of continuously incrementing your dynamic array would have a complexity of O(n^2), but that's not really the way the efficiency of an algorithm is calculated.
But if you were to think of it that way, then doubling the size of the array with each reallocation does yield a complexity of O(n). It would be O(n log n).
Further more, when you say you might "waste a little memory", you're dramatically understating the cost. If you remember any of the math you learned in school ( and so far, it sounds like you've forgotten most of it), you'd recognize that doubling is exponential growth. You can use up a whole lot of memory in a hurry that way.
As far as the specific example goes, you have to choose a reallocation strategy that is appropriate for the use case. Frankly, it sounds like niether your math nor your programming skills are particularly strong.
Also, he actually explains terms like functions - and what a function is - in plain english. I went through high school not actually knowing what a function was, because nobody bothered explaining what it was! I could vaguely see what it did, but not understanding what it was - big difference. I wish mathematics was taught in this style more, using creative language, and plain english. maybe the purists will see this as unecessary fluff, but if you can't get through to your audience, and get them to understand and enjoy the subject then you're totally wasting your time - pack up and go home. Math is actually a surprisingly simple subject; moreover, it's FUN, too! It's a real pity it isn't taught in plain simple terms :(
-- Fuck Beta