Why are "imaginary numbers" imaginary? 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?  
 A: 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").
A: 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.
A: 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.
