Imaginary numbers do exist, in reality they are called complex numbers, and are for example the square roots of negative numbers. Thus √-1 has two answers, i and -i. i is the imaginary unit, it represents (0,1). Also in the complex realm each number has n nth roots (for example, 1 has 4 fourth roots: 1, -1, i and -i).
The best way to imagine them is to represent yourself a set of cartesian axis. Now on the x axis are the real numbers like 1, √2, -3, etc. All the other numbers are covered by complex numbers. In fact, the coordinates of the complex numbers represent their x and y coordinates.
You will study them, and then you'll realize that they make sense, and in fact they are pretty useful. They aren't even that difficult, and they are easy to manipulate. And depending of your teacher, they'll tell you how "mathematically beautiful" (I can't believe this term exists) Euler's Identity is: e^(i*pi) + 1 = 0, because it uses the most important numbers (1 and 0), constants (e, i, pi) and operations (+, *, ^) exactly once

Imaginary numbers were created so you can take the square root of negative numbers and still have it make sense. Taking the square root of negative numbers is critical in engineering to solve many problems.

Do you have the same problem with negative numbers? They don't describe anything real. How can you have less than nothing? If I have 4 sheep, how can I take away 5?? Yet no one questions that negative numbers are useful nor has a hard time understanding them.

Complex numbers are actually a very poorly taught section in mathematics ... To really start to understand complex numbers you need a good understanding of what a Group, Ring and Field are; I won't go into details but these are (essentially) ways to define sets of numbers that have certain properties.

Now, the field of complex numbers is equlivalent to the field of 2 dimensional rotational matricies which (essentially) means that for every complex number there is one (and only one) rotation matrix that it can be paired up with, and the properties (addition, subtraction, multiplication and division) are all maintained.

Once again, without going into details, the imaginary number on a complex number is related to the cosine of the angle of rotation and the real number is related to the scaling of the rotational matrix; the odd behavior of the imaginary number is because the cosine function is a periodic function.

Limits, baby :D

 Pretty much this. You look at the numbers as they get closer and closer to 0 and see where it converges and assume that must be what the number actually is at that point. This is one of the basic ideas behind derivatives.

limits.
not exactly getting close to 0 but any number
is the
lim x->2 , x will go at 1.9999999999999999999999999999999999999999999 but never touch 2

All math is highly interwoven with logic and brain development. Understanding complex and creative ideas is part of your growth as a human being so just accept it and move on.
If you can't get past imaginary numbers you will never be able to comprehend anything to do with frontier science.

MY ADVICE: forget what you have learned about math.
1 apple plus one apple was just a way for you to understand with real objects.
numbers are not real. They are ideas. They are representations of quantifiers. Do not fight it, there is no reason to.
If you are taking a class on existentialism, and your teacher says an orange is black, then that orange is fuckin black. get it? Just instead of existentialism, it's math....

Have you taken a class on existentialism?

hehe no, it was a writing class where we talked about nietzsche and some other writers. But i think where you are going with this is that in existentialism; Find your own path. correct? maybe i should fix it?

my point is, if you take a class on something that you have no idea where to even begin, then just accept what you hear and go from there.

EDIT: now that i think about it, taking something for granted is like the opposite of existentialism.... o well.
In science, just believe what you are told, until you know it all. Then feel free to question stuff.

Its not division by zero, its division close to zero. There is a fundamental difference between dividing between an infinitely small number and division by zero. Thats the reason why you always have to express it with limit notation (lim x-> 0 of 1/x is infinity) rather than 1/x is infinity.