## Imagining Numbers 265

*"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*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*. 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."Imagining Numbers | |

author | Barry Mazur |

pages | 267 |

publisher | Farrar, Straus and Giroux |

rating | 8 |

reviewer | Peter Wayner |

ISBN | 0374174695 |

summary | How to imagine imaginary numbers like the square root of minus fifteen. |

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

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

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

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

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

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

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

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

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

a book on how to imagine databases that hold no informationHow to imagine imaginary numbersI wish I had nothing that could make me a lot of money as well.

## Re:did this author start nothing.net? (Score:2, Funny)

I wish I had nothing that could make me a lot of money as well.Dot com, we miss you already.

## Re:did this author start nothing.net? (Score:2)

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".

## Re:did this author start nothing.net? (Score:2)

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

## Re:did this author start nothing.net? (Score:2)

the show about nothing!## This is great.... (Score:5, Interesting)

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

## Re:This is great.... (Score:5, Informative)

## Re:This is great.... (Score:2)

## Re:This is great.... (Score:5, Informative)

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)

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

cando it if you don't have any communication skills and don't even care about being able to communicate, but if you don't have those skills then you'll make lots of people miserable.So, IMHO, computer science students should be required to take advanced math, and advanced math students should be required to take creative writing. :-)

## Re:no math? (Score:2)

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).No, it's still O(n). You have demonstrated how the lack of Calculus can harm a programmer.

If the array doubles and gets cop

## Re:This is great.... (Score:2)

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

## Re:This is great.... (Score:2, Insightful)

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

## Re:This is great.... (Score:2)

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.

## Re:This is great.... (Score:2)

## Re:This is great.... (Score:2)

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:5, Insightful)

For real.

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

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

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

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

## Re:This is great.... (Score:4, Interesting)

## Re:This is great.... (Score:2)

A degree in busines with a specialty in MIS is also appropriate and to me is a more valuable degree. Most programs are internal projects in the office. Directors and Vp's who talk to the programmers use language like "ERP"

## Re:This is great.... (Score:2, Insightful)

Should you be comfortable with math? Absolutely! You should know Calc, and should be very much aware of exponents and iteration (another poster's comments about O(n) and O(n^2) are dead

## Re:This is great.... (Score:2)

science*and* softwareengineering. The latter definately has practical value. The former... well let's just say that I forgot everything I learned in Models of Computation about 3 days after the final exam.I personally believe that they should be completely different majors. Are there any colleges out there that make this split?

## Re:This is great.... (Score:2)

Ariane 5 failure was largely the result of a software reuse problem and hence in the domain of Software Engineering.

However, you could argue pedantically that putting a large number into fewer bits was a representation problem and hence in the realm of Computer Science...

STF

## Re:This is great.... (Score:2)

Please do not misuse the word "engineering". [...] To be called an "engineer" you must graduate from a school authorized to bestow the particular engineering degree.Authorized by whom? Do you realize that a school does not even need to be accredited to bestow an engineering degree? I've met so many engineers who didn't even know basic math, it's frightening.

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

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.

## Re:This is great.... (Score:2)

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)

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.

## Re:This is great.... (Score:2, Insightful)

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

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)

## Negative phone numbers?!! (Score:3, Funny)

## Re:Reminds me of... (Score:2)

That'll never work. You'll just end up dialling a negative number, and I don't know anyone with a negative phone number.$ echo 555-1212 | bc

-657

$

## This reminds me (Score:5, 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.

## Re:This reminds me (Score:4, Funny)

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

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

tellingthe 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 wheretheyare. 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)

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)

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

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.

## It's simpler to take a peek, then dive ... (Score:2)

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

Math F

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

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)

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."

## Re:Hands on is the best for those who can (Score:2)

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)

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)

## Re:Understanding the symbols (Score:4, Informative)

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)

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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"

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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

## Re:Understanding the symbols (Score:2)

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

Me too

## Re:Understanding the symbols (Score:2)

## Discovery of irrational numbers (Score:2, Interesting)

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

## Re:Discovery of irrational numbers (Score:2)

## Re:Discovery of irrational numbers (Score:2)

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

## Re:I got out of math (Score:2)

They are used to explain some phenomena that would seem trivial from an intuitive point of view (such as sound, resonance, pendulum motion etc).

Don't let the words frighten you. At least not these... if you are looking for a monster, go look at quaternions in physics.

## Discovery of imaginary numbers (Score:5, Interesting)

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)

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)

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!

## Re:How Math is Done vs. How Math is Presented (Score:2, Insightful)

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.

## Re:How Math is Done vs. How Math is Presented (Score:2)

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

## Re:How Math is Done vs. How Math is Presented (Score:2)

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.

## Re:How Math is Done vs. How Math is Presented (Score:2)

triedproving things at random ? It's actually quite fun.## For more history... (Score:2, Informative)

## how to read/understand math ... (Score:2, Insightful)

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)

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 poemlevel)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").## Fritjof Capra (Score:2)

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"?

## Re:Ugh (Score:2)

I agree I can't understand some of his theorems, but I can't understand poetry at all.

## Visualizing functions of a complex variable (Score:2, Informative)

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

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

## Imagining Imaginaries (Score:4, Informative)

i. (the mathematicians here will recognize that it's nothing more than a linear-algebraic interpretation ofi).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 choosen=0 or -3 or 150 or sqrt(2)=1.4142... or pi=3.14159265... This is what's called areal number. Here's the interpretation of this numbernthat I'd like to propose to youYou can think of

nasmultiplyingeverything byn. For example imagine you've got $10 andn=2. Then, afternhas "acted" on your $10, you've got $20. On the other hand, ifn=-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 numberxsuch thatxdoes half the job ofn? That is, to letxact twice is the same as to letnact once ? Answer : yes, such a numberxexists and can even be choosed to be positive - it's called thesquare rootofn. In the casen=2, we havex=1.4142...At last, let's carry on the example where

n=-1. Can we find a numberisuch that "to letiact twice is the same as to letnact once" ? In other words, is there any numberiwhich does half of the job of -1 ? Well no real number does, but one introduces the new numberi, 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.## Re:Imagining Imaginaries (Score:2)

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

## Re:Imagining Imaginaries (Score:2)

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.

## Re:Imagining Imaginaries (Score:2)

## Re:Imagining Imaginaries (Score:2)

my point was that numbers could be thought of as "multiplication operators" - the part on sqare roots did not pretend to any originality.

## You have to know math thoroughly to appreciate it. (Score:2)

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)

"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.

## Re:Most math writers are terrible writers. (Score:2)

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.

## Re:Most math writers are terrible writers. (Score:2)

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)

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)

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)

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)

We started with

naturalnumbersthen added

fractionalnumbersthen added

negativenumbersthen added

irrationalnumbersthen added

imaginarynumbers## Need to explain begets the need for higher math. (Score:3, Interesting)

## The best math book I ever read (Score:2)

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

## Simple answers to all "complex" questions ... (Score:2)

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 ...## Re:Maybe... (Score:2, Informative)

I *am* the geekest link!

## Re:Maybe... (Score:5, Insightful)

<sarcasm>

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

</sarcasm>

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

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

## Re:Maybe... (Score:2, Interesting)

When I say I am an abstract thinker, I mean that I understand things that aren't required to have structure more than things that absolutly must be a certain way.

Yes, programming requires a bit of imagination. But it is all logic, it must be constructed a certain way or it will fail. Peotry, on the other hand, requires nothing. There is nobody that can s

## Re:Sure, blame the book (Score:2)

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.This try-harder I-can-do-anything attitude is a myth. It's partly true and it's the right attitude to have, but again, it's not completely accurate. Your environment will affect you to some extent. As a kid, if you have a clueless teacher and a single parent with no education, or a clueless teacher with a set of parents with one or two Masters under

## Re:Interesting... (Score:4, Interesting)

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

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

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

## Re:Imaginary Numbers (Score:2)

The square root of your increase in sex appeal from posting that comment is 0, since the increase is 0 and the square root of that is 0. Now, the square root of my score (assuming it will be negative due to the fact im making fun of your terrible math skills) WILL be a complex/imaginary number

Remember the assumption. The scapegoat of math.

## Re:Imaginary Numbers (Score:2)

If your life wasnt so damn

complexI could have got it right the last time.And my sex appeal is not for you to be considering, or to even

imagine## Re:Only then? (Score:5, Insightful)

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

## Random-access reading (Score:3, Interesting)

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 g

## Re:Complex numbers shouldn't make physical sense (Score:2)

## Re:Complex numbers shouldn't make physical sense (Score:2)

If this was the case, one would just use a two dimensional vectorOf course, a complex number

isa two dimensional vector.The coordinates on a map provide a real world example, which was my point. I can give definitions of multiplication and addition in my latitude and longitude example that result in a

vector space.The solutions we care about deal with real values.oh no, if you study the physical sciences in even a rudimentary manner, you will find there are measurable quanties in th