\(\displaystyle \sqrt{-4} \ = \ ?, \ \sqrt{-4} \ = \ \sqrt{(4)(-1)} \ = \ \sqrt4 \sqrt{-1} \ = \ 2\sqrt{-1}\)
\(\displaystyle Now, \ let \ \sqrt{-1} \ = \ i. \ hence \ \sqrt{-4} \ = \ 2i\)
\(\displaystyle Observe \ that \ it \ is \ easier \ to \ write \ 2i \ than \ 2\sqrt{-1}, \ so \ by \ definition \ (no \ proof \ required),\)
\(\displaystyle we \ let \ \sqrt{-1} \ = \ i, \ i \ stands \ for \ imaginary.\)
\(\displaystyle Question: \ What \ does \ (\sqrt{-1})(\sqrt{-1}) \ = \ ?\)
