How do you divide 7/(4i)?

Sep 5, 2017

$\frac{7}{4 i} = - \frac{7}{4} i$

Explanation:

In complex numbers, when we have to divide a number by a complex number, say $a + b i$, we do it by multiplying numerator and denominator by its complex conjugate, which for $a + b i$ is $a - b i$

Observe that we only change the sign of imaginary part, but not that of real part.

Here we have to find $\frac{7}{4 i}$, i.e. divide $7$ by $4 i$ or $0 + 4 i$.

As complex conjugate of $0 + 4 i$ is $0 - 4 i$ i.e. $- 4 i$, we multiply numerator and denominator by $- 4 i$

Hence, $\frac{7}{4 i} = \frac{7 \times \left(- 4 i\right)}{4 i \times \left(- 4 i\right)}$

= $\frac{- 28 i}{- 16 {i}^{2}}$

but in complex numbers ${i}^{2} = - 1$

Hence, $\frac{7}{4 i} = \frac{- 28 i}{- 16 {i}^{2}} = \frac{- 28 i}{16} = \frac{- 7 i}{4} = - \frac{7}{4} i$