Question

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

Answer 1

`5/i=-5i`

Explanation:

In general, a complex number multiplied by its conjugate is real, as `(a+bi)(a-bi) = a^2-(bi)^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:

`z/(a+bi) = (z(a-bi))/((a+bi)(a-bi)) = (z(a-bi))/(a^2+b^2)`

In the given example, the conjugate of the denominator `i` is `bar(i) = -i`. Thus, we can multiply the numerator and the denominator by `-i` to simplify:

`5/i = (5*(-i))/(i*(-i))`

`=(5i)/(i^2)`

`=(5i)/(-1)`

`=-5i`