Why can Imaginary Numbers, which are called that because they don't exist, be used in mathematical proofs?

Update: Anonymous dude, I can't quite follow the reasoning in your second paragraph. You seem to be stating that if a field (rational numbers in this case) is proven to exist, this proves that a larger field, of which this is a subset, exists. It seems to me that any such field extension is postulated, not proven,... show more Anonymous dude,
I can't quite follow the reasoning in your second paragraph. You seem to be stating that if a field (rational numbers in this case) is proven to exist, this proves that a larger field, of which this is a subset, exists. It seems to me that any such field extension is postulated, not proven, to exist. Could you clarify this please? I seem to accept complex numbers in the way your first paragraph describes, as very useful tools in calculations but nothing more. Negative natural numbers I think of as positive natural numbers going in the opposite direction, so they cause me no problem. With rational numbers, (e.g two and three quarters) I can think of the 2 as a natural number, and the 3 in 3/4 as another natural number counting units for which the "/4" simply defines a smaller size, so they aren't a problem either. Internal consistency is not proof. Neither is precision or logical meaning.
Update 2: Manipulating complex numbers is usually not a problem, but believing that something HAS been proven to me, when the proof includes a value which I can use, but not comprehend, is a problem. It isn't that I feel hypocritical accepting such a proof, I just question whether such things have REALLY been proven to... show more Manipulating complex numbers is usually not a problem, but believing that something HAS been proven to me, when the proof includes a value which I can use, but not comprehend, is a problem. It isn't that I feel hypocritical accepting such a proof, I just question whether such things have REALLY been proven to anyone.
Update 3: Buri, I looked up the link you gave, and followed it to some others, but their reasoning seems to be, "Complex numbers form part of a coherent system, and are very useful in solving many problems, therefore they must exist." Thanks for the effort, anyway. (If anyone's interested, it was Andrew... show more Buri,
I looked up the link you gave, and followed it to some others, but their reasoning seems to be, "Complex numbers form part of a coherent system, and are very useful in solving many problems, therefore they must exist." Thanks for the effort, anyway.

(If anyone's interested, it was Andrew Wiles so-called proof of Fermat's last theorem that brought this matter to my mind. Specifically all the complex numbers in the modular groups. I still consider the Taniyama-Shimura Conjecture to be a conjecture. Maybe I just have a closed mind as far as "i" is concerned. If you can convince me about root minus 1, then I will be unhappy, but at least then I can move on, and THAT I will appreciate, thanks)
Update 4: anonymous dude, The definition of (x_1, y_1)*(x_2,y_2) to be (x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2), appears (to me) to be neither reasonable nor consistent BECAUSE an apparent proof of the existence of root -1 results from it. I don't feel comfortable with thinking of ANY numbers as being perpendicular to R. ... show more anonymous dude, The definition of (x_1, y_1)*(x_2,y_2) to be (x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2), appears (to me) to be neither reasonable nor consistent BECAUSE an apparent proof of the existence of root -1 results from it. I don't feel comfortable with thinking of ANY numbers as being perpendicular to R. I can't make an analogy between this, and "+3-2 = 3 steps forward and 2 steps back". In the physical world, I know a third direction exists because I can observe it, but in mathematics, I can't see that just because two axes exist, it follows as the night, the day, that of necessity a third axis must also exist.
Update 5: "This polynomial cannot be factored, precisely because there is no real number r with the property that r^2 + 1 = 0." The difficulty I have with this line of reasoning is in observing that because no real number r, satisfies this equation, r must exist, but be imaginary. Why must it exist? I am not... show more "This polynomial cannot be factored, precisely because there is no real number r with the property that r^2 + 1 = 0." The difficulty I have with this line of reasoning is in observing that because no real number r, satisfies this equation, r must exist, but be imaginary. Why must it exist? I am not questioning the usefulness of complex numbers in many areas. I am questioning whether we are justified in using an incomprehensible, yet definable and manipulable concept in proofs. (perhaps I should have said incomprehensible to me.)
Update 6: In the fundamental theorem of Calculus, there is a caveat, "Provided the limit exists". Perhaps I am being overly skeptical in feeling that more branches of mathematics, including polynomials should include "Provided the value exists". I just don't know.
Update 7: zpconn, I have to admit that I have a lot of problems with the way maths is done. Mostly along the lines of "How reasonable and sensible does a postulate have to be before it is considered axiomatic, and therefore decisive in proving a point. I'm not questioning the self-consistency of complex number... show more zpconn, I have to admit that I have a lot of problems with the way maths is done. Mostly along the lines of "How reasonable and sensible does a postulate have to be before it is considered axiomatic, and therefore decisive in proving a point. I'm not questioning the self-consistency of complex number construction.
Update 8: "If you don't understand how complex numbers can be used to prove facts about real numbers, then you clearly just don't understand the proofs and have a lot more education to do in mathematics." I don't understand WHY complex numbers are trusted in proving facts about real numbers, Which is... show more "If you don't understand how complex numbers can be used to prove facts about real numbers, then you clearly just don't understand the proofs and have a lot more education to do in mathematics." I don't understand WHY complex numbers are trusted in proving facts about real numbers, Which is why I asked the question.
Update 9: Steiner. Irrational numbers do not upset me because they help to make the "number line" continuous. When mathematicians say "If This is equal to That" or "If we define This as That", and a few years later other mathematicians decide to use these statements to PROVE something, and... show more Steiner. Irrational numbers do not upset me because they help to make the "number line" continuous.
When mathematicians say "If This is equal to That" or "If we define This as That", and a few years later other mathematicians decide to use these statements to PROVE something, and forget about all the "If's", THAT causes me to question the validity of the proofs.
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