This question already has an answer here:
- What are imaginary numbers? 21 answers
I know three questions (that gained momentum) that have been posted asking a question which seems the same, but answers to none of them answer the following very well.
Please jump to point 2 & 3 for immediate addressing to the problem.
Knowledge that I currently have: I was introduced to imaginary numbers a while back where just the square root of -1 seemed to be called iota. This made a whole lot of 'things' definable/analyze-able. I have currently studied the two dimensional complex plane, the rotation, Euler form, defining locus/area which is indefinitely better than an algebraic equation/expression, etc.
My Question: However, I have failed to grasp that how imaginary numbers even make sense. Let's take an example, let there be a quadratic polynomial not intersecting the x-axis. However, it does with two imaginary numbers. How does it intersect the x-axis when the equation of the quadratic dictates that it doesn't intersect the x-axis (discriminant<0)?
My current understanding:Several sites led me to the conclusion that imaginary-axis acts as the 'z' axis and gets the quadratic to intersect with the x-axis somehow. This pushes the curve form a 2-D to a 3-D curve existing in the complex space when we start feeding complex values into the function with the x-axis as the intersection the the two, Cartesian and Complex plane. This 3-D structure now intersects somewhere on the complex plane with the x-axis giving us the complex roots of the quadratic.
I however fail to understand that how this curve transitions from the Cartesian plane to the complex space while being continuous. Does it transform to a surface instead of a curve? If it transforms to a surface then it would have infinite roots with the x-axis(as seen in the video mentioned in the footnote)
I understand that the complex numbers were 'invented' rather than discovered (I'm saying this because other numbers; irrational, rational, negatives actually mean something, hence discovered).
Thank you very much for your valuable time. I really appreciate it.
FOOTNOTE : I have been to the following questions/sites :
5 This video here is very good
EDIT: The question was marked as a duplicate though it did not in a way adress the same question as the question that was marked as 'original'. I have therefore, changed the title and content to make the question as precise and understandable as possible.