I remember my lecturers telling me that under no circumstances we may ever write sqrt(some negative) it does not exist, it is impossible. it is true that i²=-1 but that's something different

you can think of this as a plane, the x axis is the real axis (a+i*b -> a is the real part) and the y-axis is the imaginary axis (b is the imaginary part) so if you have i as a number, you end up on the imaginary axis. this can be seen as a rotation of the unity point ( 1+i*0 or 1) this is a rotation over 90°, if you do this again (i*i) you rotate again over 90°, so you end up at -1 (+i*0) so you can write i²=-1, but the square root of a negative number... that does not exist!

but this is rather advanced

This is very simple and I knew already knew it,i studied this 5 years ago and I still use imaginary numbers in my studies.But this topic is helpful for those who haven't studied imaginary numbers or know very little about it.
Well this is really good topic about imaginary numbers.

Informative post, unfortunately I pretty much knew things contained in there prior to reading it. I do however have two questions.

1) I once saw a problem in a precalculus textbook asking to find the square root of i. How would one go about figuring this out this?

2) I'm not sure if complex numbers are outside the scope of this topic, but what are they exactly? I remember once hearing someone talk about graphing 3 dimensional objects (e.g.: a cube) on the complex plane. He said that it might seem strange to do this at first, but it is sometimes done in certain sciences (like physics) not to measure the objects, but to measure degree of complexity ( ).

Complex Numbers are mathematical elements which are (somewhat) connected to Imaginary Numbers.
I'll let you discover the concepts yourself from this Wikipedia link: http://en.wikipedia.org/wiki/Complex_number

It has been a long time since I have dealt with any of this stuff, so I might make an error trying to explain what I know (or is that 'knew'?)

To me, Imaginary Numbers are the ones missing on your Pay Cheque...