Equality of Two Complex Numbers

Equality of Two Complex Numbers

Definition. Two complex numbers $z_1=a+ib$ and $z_2=c+id$ are said to be equal if their corresponding real and imaginary parts are equal:

Example. If $x+iy=4+3i$, then $x=4$ and $y=3$ — the real parts must match, and separately the imaginary-part coefficients must match.

Worked example. If $7a+i(3a-b)=21-3i$, find $a$ and $b$.

Solution. Rewrite the left side with real and imaginary parts grouped: $7a+(3a-b)i=21-3i$. By equality of complex numbers, the real parts must match: $7a=21$, so $a=3$. The imaginary parts must match: $3a-b=-3$, so $3(3)-b=-3\Rightarrow9-b=-3\Rightarrow b=12$.

This equality rule is the single most-used tool in the whole chapter: whenever a complex equation needs to be solved for one or more real unknowns, the standard first move is to write both sides in $a+ib$ form and split the one complex equation into two ordinary real equations — one from matching real parts, one from matching imaginary parts. …