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astrololo
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I was wondering, why is the set of complex numbers needed to solve problems that the set of reals doesn't permit to ? I mean, in relation to the fundamental theorem of algebra, that is.
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Yeah, I'm asking why by using complexs we can obtain solutions while if we use only the reals we can't.Geofleur said:Just to clarify, are you asking why we bother to enlarge the domain of admissible numbers, instead of just saying "This equation has no solutions"?
astrololo said:Yeah, I'm asking why by using complexs we can obtain solutions while if we use only the reals we can't.
jssamp said:Personally I feel the choice of wording is unfortunate. Complex numbers are no more unreal than integers. Because they are called imaginary doesn't make Euler's any less fundamental. It's sad that somebody had to use real and imaginary to describe two sets of one fundamental tool of knowledge and so lead so many of us to think half of the world is "not real" and "too complex" to understand.
epenguin said:To say this further, Arab mathematicians I think if not earlier recognised that many quadratic equations like
x2 + 1 = 0, or
x2 - x + 1 = 0 needed a square root of -1 in a solution, and this really seemed to make no sense so these solutions were called not real in the ordinary everyday, not the modern mathematical technical sense of real. They could just be dismissed for that reason, they were not a real solution to any problem. But then in surely one of the most brilliant mathematical steps ever, it was discovered in Renaissance Italy that they could be used for solving something quite concrete and real, as explained in posts above. This is the first example, another I have seen is to work out in now many different ways can you give change for a dollar?
These are minor things one could live without but on the applied side they open the mathematical treatment of e.g.oscillations and vibrations and circuits to engineers and physicists in a truly eye-opening and convenient way. And quantum mechanics is thought to need complex numbers inherently, not just for convenience of calculations. And there is much else.
One landmark theorem is the 'fundamental theorem of algebra' (which oddly is not algebraic) which says that, as per previous examples, every algebraic equation with real coefficients has a solution in the field of complex numbers (which includes the real numbers as special case). (From that you can deduce that if it is nth degree it has n solutions.) Then if the coefficients are complex numbers, real or non-real, they still have just complex-number solutions. Because of their 'nonrealness' It was hoped that their uses like above would be temporary and a way to solve the problems like the cubic without them would be found. I have read that this has been proved impossible. (I do not know how advanced the proof is.)
It is found that the real number system is essentially incomplete, and with complex numbers you see "behind the scene". The things most magic and mind-blowing in math seem mostly to come from complex numbers. Some mathematicians (I am not one) might agree with my impression that without much exaggeration, math is divided into two parts, glorified accountancy with no really big surprises on the one hand, and stuff depending on complex numbers on the other.
jssamp said:It's not possible using integers. The bias that led to the name is inherent in that statement. Why are integers your measuring stick for "realness". Is pi not a real value. what about the distance across my square room corner to corner, is that not a real distance? It isn't rational.
If complex numbers are in a sense not real, then what do you make of the realness or imaginary-ness of matrices, for example?aikismos said:I think I'd have to take issue with the notion that imaginary numbers are just as "real" as real numbers, since given it's readily accepted that it's not possible using integers to to multiply a number by itself and obtain a negative result. Integers are intuitive in the sense that the negative has an interpretation of direction relative to a point in a geometrical interpretation. Even the products of integers can have a geometric interpretation on the real number plane in regards to the slope of the main diagonal swept out by the area of the vectors that represent them. I don't know that the same can be said for the products of complex numbers on the Argand plane.
jssamp said:Good answer. I can't think of any interesting technical endeavor today that would be possible without complex numbers. Take communications and digital signals processing. Our electronic lives rely on the Fourier transforms and how fun would that be without using complex exponentials in
the integration.
sshai45 said:@aikismos: I'd want to say that with regards to complex numbers and "reality" thereof, a difficulty with your "length" example is that it would also "defeat" negative numbers as well, since you cannot have negative lengths, either.
sshai45 said:@aikismos:Yet most people have less trouble with negative numbers than with complex numbers. The question remains: why?
sshai45 said:@aikismos:Negative numbers... Then you can actually write complexes as a single string of digits with no "I"s, making them really feel like singular entities given our usual way of relating to numbers as strings of digits.
sshai45 said:@aikismos:With regards to "reality" of numbers, I'd say that numbers are "real" in the same sense that words are "real", and "imaginary" in the same sense that words are "imaginary". They are mental constructs we have that allow us to make sense of reality. As I read somewhere, or maybe I even thought of it myself but inspired by something else, you cannot stub your toe on a 1, so clearly, 1 is not "real" in a material sense. All numbers are invented by our minds. All numbers are "imaginary" in this sense.
PeroK said:If complex numbers are in a sense not real, then what do you make of the realness or imaginary-ness of matrices, for example?
Geofleur said:I'd like to elaborate a bit on the point that complex numbers do have a nice graphical interpretation. Set ## a + ib = (a,b) ##, a point in the Argand plane. Multiplying by ## i ## rotates the position vector for this point counterclockwise by 90##^{\circ}##. More generally, using the polar forms, multiplying two complex numbers gives
## z_1z_2 = r_1e^{i\theta_1}r_2e^{i\theta_2}= r_1r_2e^{i(\theta_1+\theta_2)}##,
So ## z_1 ## stretches ## z_2 ## by amount ## r_1 ## and rotates it by an amount ## \theta_1 ##. There's a delightful book that shows the unfolding of complex analysis from this geometrical perspective, Needham's Visual Complex Analysis.
Edwin McCravy said:The term "imaginary" is a misnomer. As has been pointed out, x+yi is simply the ordered pair of real numbers, or the vector, or the point (x,y). The operations are addition, (x,y)+(w,z) = (x+w,y+z) and multiplication (x,y)(w,z) = (xw-yz,yw+xz) . It's easy to remember the multiplication operation if you know the double angle formulas for cosine and sine. That's because for (cos(θ),sin(θ)) (cos(Φ),sin(Φ)) on the unit circle, their product is (cos(θ+Φ),sin(θ+Φ)).
The imaginary unit, i, is not an integer. It is an "imaginary" number.aikismos said:Tell me, @Edwin McCravy, which integer multiplied by itself has a product of -1? :D
symbolipoint said:The imaginary unit, i, is not an integer. It is an "imaginary" number.
I looked, but I guess I did not look closely enough.aikismos said:Precisely! Which is exactly why it's not a misnomer as submitted by McCravy.
However, i is a Gaussian integer. I'm not saying that i is an integer in the usual sense, though.symbolipoint said:The imaginary unit, i, is not an integer. It is an "imaginary" number.