What is an imaginary number anyway?

Imaginary numbers are based on the mathematical number $$i$$.

$$i ext is defined to be sqrt-1$$

From this 1 fact, we can derive a general formula for powers of $$i$$ by looking at some examples.

Table 1

$ext Table 1\eginarraychline Expression & & Work & Result \hline edi^ extbf2 & = & i cdot i = sqrt-1 cdot sqrt-1 & ed extbf -1 \hline edi^ extbf3 & = & i^2 cdot i = -1 cdot i & ed extbf-i \hline ed i^ extbf4 & = & i^2 cdot i^2 -1 cdot -1 = & ed1\hlineendarray$

You should understand Table 1 above .

Table 1 above boils down to the 4 conversions that you can see in Table 2 below. You should memorize Table 2 below because once you start actually solving problems, you"ll see you use table 2 over and over again!

Table 2

$ext Table 2$

How to use the pythagorean Theorem

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