Suppose, the initial velocity of a car is 1m/s, acceleration 2m/s^2, distance travelled (-8)m. What is the time required to reach that distance? This is the first question.
The answer of the first question will be an imaginary number. But we know time can only flow in one direction. If we consider past time, then we will get negative value. Now what does an imaginary number mean in this case of time? This is the main question.
Sometimes when you get a complex result from a physics calculation, that means "it can't happen". Sometimes it means that something else happens, like an oscillation rather than an expected decay. Which is it? The math can't tell you, you must understand the physics.
Yes surely we know this problem will yield an imaginary number and what actually this mean, we are going to find that.
Basically, this problem does not have any answer. Mathematics is a perfect model, which can describe a physical system and it does not matter whether the solution is logical or not. It is tit for tat. The thing you input in your equation, the output will be the exact aftermath.
Here we have input the value of displacement as the negative 8 meter, which physically impossible considering the mentioned system. So, math result has also become an impossible one, the square root of a negative number, which is commonly known as imaginary number.
So does it mean, imaginary number mean an impossible solution? Exactly but it means something more. It means in your math, the information required to solve the math are not completed. As we lack in information about the system, we can not get a proper real solution.
So what was the lacking info in the given question? That lacking information can be anything. Such as if you say that, when the car started moving, another big car collided against the direction of the first one and so our first car started going to the opposite direction and eventually crossed (-8) meter. If you add this type of information in your math, then the system is comprehensible and then we can do the math without making a sweat about the imaginary number.
Ok nice answer. But does that mean all the math that we have done so far using imaginary number (or simply call it 'i') are wrong? The answer is not all of them. But how? Ok, if we solve cubic equation using Tartaglia or Cardano formula, then we will find a lot of cases where we can cancel out those i, and then what remains will be real number. I guess, here you will not find any problem with that. Of course, there will be many cases where you will not be able to cancel all the i, those all are imaginary, absurd and don't have any solution. But remember any cubic equation with real coefficients will always have at least one real solution.
Then what about Schrodinger equation and all other electrical problems? The specialists on those subjects can tell you about that. But I will just say, when we are doing current-voltage related math; remember they are just approximation. They are not any perfect value. The parallel circuit and other stuffs and formula are just a model to find an answer mathematically and they are not any practical real value like the math you do in velocity, time and in other classical mechanics.
Sometimes this i is useful for telling you a sequence. Such as sine graph. This thing can be nicely described in terms of i because of its rotational property. Here notice that you can understand the value of -1 and 1. But the meaning of i and i^3 refers something impossible, right? So this must be wrong somewhere. Is that what you are thinking? See this is a function. In case of function, what you input does not matter, you just do the math accurately and you will get the correct solution. Using impossible for getting possible, isn't it fascinating? Nobody will tell you this secret. But now you know all the strategy regarding i, that they use in math; be it probability amplitude or any kind of quantum physics.
Though we can do the math manipulating i by the use of function, we are loosing the understanding of math. But see nobody cares about understanding if you get your solution done. Use rigorous proof or trick, that does not matter as long as your solution is correct. Remember, math is a model. This model always being updated by discovering new ideas and as a physicist we always try to use those new ideas in solving our problem. What if our solution is wrong? We will continuously keep looking for new tricks to solve our problems at any cost. Because physicist is not about being honest but about being successful in explaining the universe.
