Complex numbers flashcards: i, conjugates, division

Question 1

What is the imaginary unit $i$?

Accepted Answer

$i$ is defined by $i^2=-1$ (so $i=\sqrt{-1}$ by definition).

Question 2

How do you multiply complex numbers in $a+bi$ form?

Accepted Answer

Use distribution (FOIL) and $i^2=-1$: $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$.

Question 3

How do you divide complex numbers: $\dfrac{a+bi}{c+di}$?

Accepted Answer

Multiply top and bottom by the conjugate $c-di$:
$$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(a+bi)(c-di)}{c^2+d^2}.$$

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