A word becomes imaginary

Question

I'm important in geometry
If you add a letter, you get the German translation
If you re-order the letters of the English word and add brackets () and multiply, it becomes imaginary
If you re-order the letters of the German word and add brackets () and multiply, it becomes real

Find the word with the above properties.

Answer 1

A line!

This is because

1. Lines are quite important in geometry...

2. linie is the German translation of "line"

3. $i \cdot \ln(e)=i$ which is imaginary

4. $i \cdot i \cdot \ln(e)=-1$ which is real!