A deeper understanding of the imaginary number

[TackyJan]

I know what a complex number is. Learned it way back when I took college classes. I know it is a number that has a real and imaginary part of the form a + bi. What I have always failed to understand is what conceptually does it mean. I know what i is , it's the square root of -1. I just could never understand what it means to say that a number has a real and imaginary part. And that the imaginary part lies perpendicular to the real part. So an imaginary number seems like a two-dimensional vector . Beyond this however I fail to understand how imaginary number came to be and what its underlying meaning is.

Note, I accidentally put imaginary number on this post I tried to go back and rename the post but I could not figure out how to do it in Tapatalk.

 

[Pythagorean]

Not quite. A complex number is one that has a real part and an imaginary part. Beyond that, I don't particularly understand the "nature" of complex numbers myself. The 2D vector concept only works up to a point.

For example, if we assume:

$$i + 1 = \hat{i}+\hat{j}$$

then let's square each side. Left:

$$(i+1)^2 = i^2 + 2i +1 = 2i$$

But what do we do for the right to analogize squaring a complex number for a vector? We could take a dot product and end up with 2. Is that comparable to 2i? Then 2i would be a scalar, not a vector with a 0 component. I don't know what would happen if we generalized this comparison or if any comparisons could be drawn. Is there an analogy to tensor ranking in imaginary numbers? Or are vectors and complex numbers just fundamentally different.