# How to I write the quotient 5/i in standard form?

Nov 21, 2016

$\frac{5}{i} = - 5 i$

#### Explanation:

In general, a complex number multiplied by its conjugate is real, as $\left(a + b i\right) \left(a - b i\right) = {a}^{2} - {\left(b i\right)}^{2} = {a}^{2} + {b}^{2}$. Thus, to change an expression with a complex number for a denominator into one with a real denominator, one can multiply the numerator and the denominator by the conjugate of the denominator:

$\frac{z}{a + b i} = \frac{z \left(a - b i\right)}{\left(a + b i\right) \left(a - b i\right)} = \frac{z \left(a - b i\right)}{{a}^{2} + {b}^{2}}$

In the given example, the conjugate of the denominator $i$ is $\overline{i} = - i$. Thus, we can multiply the numerator and the denominator by $- i$ to simplify:

$\frac{5}{i} = \frac{5 \cdot \left(- i\right)}{i \cdot \left(- i\right)}$

$= \frac{5 i}{{i}^{2}}$

$= \frac{5 i}{- 1}$

$= - 5 i$