# How do you find the conjugate of -2/i^3?

• Aug 15th 2012, 02:01 AM
viccal
How do you find the conjugate of -2/i^3?
What is the conjugate of $\displaystyle \frac{\ -2}{i^3}$?
• Aug 15th 2012, 02:14 AM
a tutor
Re: How do you find the conjugate of -2/i^3?
$\displaystyle \frac{-2}{i^3}=\frac{-2i}{i^4}=-2i$.
• Aug 15th 2012, 05:39 AM
HallsofIvy
Re: How do you find the conjugate of -2/i^3?
To find the conjugate of any complex number, replace i with -i. You could "rationalize the denominator" first, as "a tutor" did:
$\displaystyle \frac{-2}{i^3}= -2i$ and then its conjugate is $\displaystyle -2(-i)= 2i$. Or you can immediately write that the conjugate is $\displaystyle \frac{-2}{(-i)^3}= \frac{2}{i^3}$. If you were then to multiply both numerator and denominator by i you would again get $\displaystyle \frac{2i}{i^4}= \frac{2i}{(i^2)^2}= \frac{2i}{(-1)^2}= 2i$.
• Aug 15th 2012, 05:12 PM
Deveno
Re: How do you find the conjugate of -2/i^3?
aw...i was gonna say find its minimal real polynomial, and grab the other root :P
• Aug 18th 2012, 06:11 PM
viccal
Re: How do you find the conjugate of -2/i^3?
What's the reason you multiplied the numerator and denominator by i?
• Aug 18th 2012, 07:09 PM
skeeter
Re: How do you find the conjugate of -2/i^3?
Quote:

Originally Posted by viccal
What's the reason you multiplied the numerator and denominator by i?

tutor used the fact that $\displaystyle i^4 = 1$ in order to get $\displaystyle \frac{-2}{i^3}$ in the form $\displaystyle a \pm bi$ in an efficient manner.

he ended up with $\displaystyle -2i = 0 - 2i$ ... note that the conjugate is $\displaystyle 0 + 2i = 2i$