Divide Complex Numbers – step‑by‑step (LaTeX)

given two complex numbers \( w = a + ib \) and \( z = A + iB \):

\[ \frac{w}{z} = \frac{a + ib}{A + iB} \]

multiply numerator and denominator by the conjugate of the denominator \((A - iB)\):

\[ = \frac{ (a + ib)(A - iB) }{ (A + iB)(A - iB) } \]

expand numerator and denominator:

\[ = \frac{ aA + bB + i(bA - aB) }{ A^2 + B^2 } \]

write in standard form \(x + iy\):

\[ \boxed{ \frac{w}{z} = \frac{ aA + bB }{ A^2 + B^2 } + i \frac{ bA - aB }{ A^2 + B^2 } } \]

real part \(\dfrac{aA+bB}{A^2+B^2} \quad \) , imaginary part \(\dfrac{bA-aB}{A^2+B^2}\).

\(\dfrac{a+ib}{A+iB}\)

conjugate method: multiply numerator & denominator by \(A-iB\)

Enter complex numbers \(w\) and \(z\)
\(w =\) \(+ i\)
\(z =\) \(+ i\)
result \(\displaystyle \frac{w}{z} = x + i y\)

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