Talk:Imaginary number
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Despite their name, imaginary numbers are just as real as real numbers.
Um. How's that? A number with a square that's negative sounds decidedly unreal to me... Evercat 22:01, 21 Aug 2003 (UTC)
It is math, after all. All numbers are real. Perhaps a reword is nessesary. Vancouverguy 22:04, 21 Aug 2003 (UTC)
- I tried to reword it satisfactorally. --Alex S 03:48, 20 Feb 2004 (UTC)
Can one of you maths experts tell me what useful purpose imaginary numbers serve? It's something they never taught (or at least I don't recall being taught) at school. What are the practical applications?
- Most of them are in differential equations and analysis, which are subjects studied after calculus; maybe that's why you haven't seen them. Michael Hardy 19:52, 17 Oct 2004 (UTC)
- Well for practical applications you'd be better off asking an engineer or a physicist. But I'll take a stab at it, though consequently I'll have to be a little vague.
- First, what you really want to ask is about the utility of the complex numbers, which are constructed from the imaginaries and the reals.
- The complex numbers are (in a sense I won't define here) a completion of the real numbers. In a way looking at real functions is like using blinders. Often the whole situation becomes clearer if you take the blinders off and look at the complex function which extends it, even if in the end you only care about the real function.
- Complex numbers have a simple geometric interpretation, and conversely some simple geometric operations have simple interpretations as complex functions. A non-trivial practical example is a conformal map, that is, a function which preserves angles. This is important in cartography.
- A number of easily defined complex functions are periodic. Periodic functions arise in studying electromagnetism, for example, and it turns out that formulating them in terms of complex functions can be very useful. Electrical engineers use them all the time.
- Complex numbers also arise in quantum mechanics, though how and why is somewhat harder to explain.
- It's interesting to note that many, in fact probably most, applications outside math utilize the geometry of the complex numbers, and don't have much to do with "the square root of minus one" as such, at least not in any direct way.
- Complex numbers, or just imaginary numbers are an extra way of accounting, or just counting. It's for working with two axes and dimensions. i is like a second variable: ax + by => a + bi, but it "intermultiplies" into the first variable as a tool. As for the above comments, imaginary and real numbers are not real; they're abstract. lysdexia 13:56, 16 Oct 2004 (UTC)
<math>-i = (-1)i \,\!<math>
replace i with the square root of -1
<math>(-1)i = (-1)\sqrt{-1}<math>
bring -1 inside the radical
<math>(-1)\sqrt{-1} = \sqrt{(-1)^{2}(-1)}<math>
square -1
<math>\sqrt{(-1)^{2}(-1)} = \sqrt{1(-1)}<math>
simplify
<math>\sqrt{1(-1)} = \sqrt{-1} = i<math>
refer back to first line
<math>-i = i \,\!<math>
add i to both sides
<math>0 = 2i \,\!<math>
divde by 2
<math>0 = i \,\!<math>
square both sides
<math>0^2 = i^2 \,\!<math>
simplify
<math>0 = -1 \,\!<math>
Is something wrong with this argument? Something about real numbers that does not hold for imaginary numbers?
- The problem is here:
- <math>(-1)\sqrt{-1} = \sqrt{(-1)^{2}(-1)}<math>
- <math>\sqrt{(-1)^{2}}<math> is 1, not -1. Ashibaka ✎ 19:59, 5 May 2004 (UTC)
- That's funny. It's an order of operations mistake, evaluating multiplication with exponentiation first instead of exponentiation with rooting(?): ((-1)^2)/2 => (-1)^2/2. lysdexia 13:56, 16 Oct 2004 (UTC)
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Imaginary & Complex Numbers
The way I read them before is the simpler way: we take up the current definition of complex numbers according to this site, and we make both "imaginary number" and "complex number" mean that.
[Haven't read the definitions properly but I think that the system described above matches with what I read before]
Brianjd 12:00, 2004 Jun 18 (UTC)
Complex Number Identies
I wasn't sure to post this under imaginary numbers or complex numbers: It would really be useful to have a page of identities for imaginary numbers similar to Trigonometric_identity. For example it could have how to calculate complex exponents, trig functions, log function, and other useful knowledge about trig functions. Ok just a thought.
Horndude77
A rose by any other name
I wonder if the "reality" of "imaginary" numbers would be questioned at all if Decartes had not choosen such a misleading name. He's probably responsible for turning more people off math than anyone else. If he weren't dead, I'd say it was a deliberate ploy to obtain job security by mystification of his art :-)
Maybe "quadrature" or "orthogonal" numbers would have been better, but to late to change now. As Elaine Benes on Seinfeld might say "They're only *called* imaginary! Get over it!"
I heartily agree
I heartily agree that Descartes has done a great disservice to Math by naming imaginary numbers "imaginary". I don't understand, why we simply can't use this notation, as shown above by someone:
Instead of 5 + i4, just write 5x + 4y.
Simple as that! What's all the fuss about. All you are saying is that this is a two dimensional number. It is 5 units on the positive x-axis and 4 units on the positive y-axis. End of story. Why complicate matters and needlessly spin people's brains by using an aburd name as "imaginary" for something which is really quite simple?
- using x and y would really screw up maths since those letters are basically the default names for variables.
- also complex numbers are supposed to be a superset of real numbers so its 5 (the real part which stands alone in its normal form) and i4 the imaginary part (which is a real number times i). i do agree that the name imaginary was probablly not the worlds best choice of words but its what we are stuck with, its a peice of important jargon that if changed would cause huge confusion for no real gain. Plugwash 21:39, 13 Jun 2005 (UTC)
Hi! I would like to know what's the difference between a complex number and a 2D vector! I work with computer graphics (but i'm not very good at math) and they look the same... With the disadvantage that complex numbers aren't 3D :-P