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I understand the definition that each member of $\Bbb{C}$ is of the form $a+bi$. I also recognize that $i^2 = -1$. What about this is imaginary? Like it has become a "real" number in the sense that we use it and analyze it and study it. We have the complex numbers $\Bbb{C}$ and the real numbers $\Bbb{R}$, but both of these are real things. They are not fictitious or imaginary in the sense of being untrue or false or somehow unreal So why are members of $i\Bbb{R}\subseteq \Bbb{C}$ called imaginary numbers?

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    $\begingroup$ Related: physics.stackexchange.com/q/11396/2451 and links therein. Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Dec 28 '14 at 15:24
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    $\begingroup$ On that note, what's so irrational about the irrationals? $\endgroup$ – GFauxPas Dec 28 '14 at 17:25
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    $\begingroup$ @GFauxPas I think that's because people used to think that every length could be represented by a fraction of natural numbers. An exception to that rule would be considered "irrational". $\endgroup$ – Wood Dec 28 '14 at 17:31
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    $\begingroup$ "So why are members of $i ℝ⊆ℂ$ called imaginary numbers?" Because it is old left-over terminology from the days before we actually knew anything. $\endgroup$ – goblin Dec 28 '14 at 17:35
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    $\begingroup$ @GFauxPas: To the best of my knowledge, the original meaning of "irrational" is "not expressible as a ratio of integers". The modern English meaning is ultimately derived from ancient attitudes towards such numbers. $\endgroup$ – Hurkyl Dec 29 '14 at 7:48
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The words "imaginary" and "real" when applied to numbers are names, not literal descriptions. The so-called "natural numbers" are simply the names and shorthand symbols that we assign to different quantities. A quantity is a real property of a group of objects, as real as the objects and the group. When we finally realized that a quantity subtracted from itself results in another quantity, we named the resulting quantity "zero" and assigned it a symbol. When we realized that the natural numbers were capable of representing a direction as well as a quantity, we gave the resulting one-dimensional quantities a clever notation and called them "signed numbers." When we realized that measurements required units we broadened our view of quantity to include units and parts of a unit, and we called a part of a unit a "fraction" and assigned it an ingenious notation. When we realized that the square root of a non-square number cannot be represented by a fraction, we called it an "irrational" number since we called the fractions "rational." When we realized that there could be two-dimensional numbers, we called them "imaginary" and "complex" because we did not fully understand what we were doing. The smartest among us are only human; we only gradually come to understand things in spite of our hubris. It is also perfectly sensible to have three dimensional numbers but they do not have nice properties so we do not talk about them. Four and eight dimensional quantities have nicer properties so we name and talk about them. The ancient absurd notion that the entities of mathematics are unreal and that mathematics is about reasoning is the cause that most of us never see the wonder and utility of the perfectly real science of mathematics, which is as real as all the other sciences. Physicists are wiser; they don't delight in saying that mass and force and energy and momentum and fusion, for example, are "abstract" because you can't put them in your pocket. You can't put most real things in your pocket. The only things that are not real are fictitious; and mathematics is not fictitious.

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  • $\begingroup$ I didn't really think there was much sense behind the term. I just wanted to make sure I wasn't missing something. Thanks! $\endgroup$ – Stan Shunpike Dec 29 '14 at 7:33
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No number is "real" in the sense that they exist physically in the real world. Natural numbers are useful for counting, so they have a direct correspondence to discrete stuff, like "9 apples". Real numbers are useful for things like length. Complex numbers are useful for other more complicated things. When people say they're not "real", I guess they're usually thinking something like "there is no $2-5i$ length". But that's like saying that real numbers are not "real" because there is no "$\sqrt{2}$ apples".

In physics, I'd say no number is "real". But in math, anything you can rigorously define and can't find any contradiction (in other words, seems to be well defined), I'd be ok with calling it "real" (in contrast with, for example, a "square circle").

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  • $\begingroup$ But ... I can definitely have sqrt(2) apples. It's an actual number that exists in our world, even though I can't "write it down" conveniently. (?) $\endgroup$ – Dogweather Aug 15 at 18:01
  • $\begingroup$ @Dogweather I was trying to give an example of something that can only be measured in integers. Some better examples are: bits (binary digits), how many times you press a button, number of people, etc. Even those can be measured in non-integers in the context of probability, for example. $\endgroup$ – Wood Aug 16 at 14:38
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I do not want to contradict Wood saying in physics nothing is real. Nobody can affirm reality for sure, certainly.

But it might be a good and straightforward guideline for entering the issue by simply following the common terminology, considering "real" as real and "imaginary" as imaginary! Nearly the whole truth is therein.

In a general manner, imaginary numbers are used where beyond real numbers a category is needed which is "less real".

One important example are quantum observables which are treating with complex (and thus with imaginary) numbers. Only the measurable values of an observable, called eigenvalues, are real numbers, and inversely only real numbers may be measurable.

Another interesting application is the representation of time in Minkowski spacetime. Time may be considered as less perceivable than space. So it was proposed to consider time periods as imaginary numbers. This form of representation has been abandoned for practical reasons, and by this you can see that physics has some play to opt for or against the use of imaginary numbers - it is not more and not less than a tool.

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  • $\begingroup$ One can describe decaying states with a complex energy eigenvalue. As long as it's a good approximation, it's just as "real" as any other approximation in physics. To a mathematician complex numbers and quaternions are completely identical in "value" to real numbers and that's really the only place where these human constructs live. That nature can be described by them is a consequence of symmetry properties. If something has the properties of a rotation, the complex numbers fit, if it's a rotation in three dimensions, the quaternions will do. $\endgroup$ – CuriousOne Dec 28 '14 at 16:42
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    $\begingroup$ I didn't say that in physics nothing is real, I just said that numbers are not "real". What I mean is that you can't touch it, measure it, point to it and say something like "that's the number 73"; and it has no effect on anything, like forces, charges, or energy do. It's just an abstraction used to represent aspects of the world, like quantity. The same goes for equations, triangles, proofs, variables, words... none of them are "real" in the sense I just described. $\endgroup$ – Wood Dec 28 '14 at 17:19
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    $\begingroup$ Richard Feynman summed this up nicely in the Feynman Lectures when he said he didnt believe mathematics was a natural science. I think his statement hints at the notion that numbers have no "reality". $\endgroup$ – Stan Shunpike Dec 29 '14 at 7:35

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