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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
I called it seight, it would be between seven and eight. Yes, that was me.
...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.
Here's an imaginary number for you:
;)
The number of people who regularly visit Slashdot that have unbiased opinions on Microsoft.
"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!"
I have trouble reading math books once! Who has enough time to read one five or six times?
...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
The flying hamster of DOOM rains coconuts on your pitiful city.
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.
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.
"...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.
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.
So what's the connection to Beowulf clusters? What kind of computational power do these "numbers" things have?
You see? You see? Your stupid minds! Stupid! Stupid!
... 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.
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.
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
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.
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!
It only took me 4 times in my second Calculus class to get a pass grade and get the hell out!!!
I only look human.
My mother is a halfling and my dad is an ogre, so that makes me an Ogreling
...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?
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. :-)
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.
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.
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
This is just pitiful. Maybe you should have made an effort to pay attention in class even without this book as a crutch! And maybe you could have still gotten a higher grade than C even if you didn't pay attention in class by taking the time to study on your own! Man, I hate when people claim that the reason they failed at something is due to outside circumstances. Dude, you got a C because you didn't try hard enough. Don't give me this "I have always been more of an abstract thinker and The Evil System just isn't set up to teach misunderstood geniuses like me mathematics" crap.
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
"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 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
"We are the music makers, and we are the dreamers of dreams."
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!
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..
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?
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"?
There is no such thing as luck. Luck is nothing but an absence of bad luck.
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.
The technique he uses to preview the material 5-6 times is known as photoreading. A technique taught by a company Learning Strategies
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."
Good security is based upon reality and common sense. Common sense is a function of having common knowledge.
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.
If you are interested in books of this sort, I highly recommend e: the Story of a Number" 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.
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...
SIGBUS @ NO-07.308
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.
FlatLand!
Flatland is a great way to visualize geometric shapes and concepts in 0d, 1d, 2d, 3d, and it begins to talk about 4d. Of course, 4d wasn't really understood when this book was written, but its a great and fun start.
~~~
Click here, you know you wanna!
char buffer[VERY_BIG_NUMBER];
like everyone else and the problem just goes away.
VBG
Rich
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...
Chris Mesterharm
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 Forum - http://mathforum.org/
unfinished: (adj.)
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 ...
unfinished: (adj.)
And I clicked on this story link because I thought it was about corporate accounting... darn...
It's remarkable that even after inverting huge vectors of hundreds or thousands of dimensions (the equivalent of flipping a pancake in 7900-dimensional-space), complex numbers still awe and terrify and lend mathematical insight.
Why is it that we can gain mathematical insight by using a 1000-dimensional matrix with complex entities, as opposed to a 2000-dimensional matrix with only real numbers?
Human intuition is illogical even in field of mathematics, the mainstay of logic. Look at the great mathematical tools and paradigms that people have opened up over the past centuries. Many of these are not about opening up mathematics, but opening up the minds of mathematicians.
Oh mind, what did I ever do to you to deserve this?!