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4.31 Dividing complex numbers

Complex fractions

(i) Do not divide by $0$ under any circumstances!

(ii) To simplify a complex fraction $z/w$ , multiply by $\bar{w}/\bar{w}$ .

(iii) If $z\neq 0$ , then $z$ has an inverse for multiplication. Observe that $z=x+iy$ has conjugate $\bar{z}=x-iy$ , so

z\bar{z}=(x+iy)(x-iy)=x^{2}+y^{2}.

Suppose that $x^{2}+y^{2}\neq 0;$ then

z^{-1}={{1}\over{x+iy}}={{x-iy}\over{x^{2}+y^{2}}}

satisfies $z^{-1}z=1$ .

Example. To find $1/(2+3i)$ .