# Math Help - Explain i^-5

1. ## Explain i^-5

Could somebody explain to me why i^-5= -i and not 1/i

Thanks

2. ## Re: Explain i^-5

$i^{-5} = i^{-4} \cdot i^{-1} = i^{-1}$

However, if you multiply by i/i (which happens to be the conjugate) you get $i^{-1} \cdot \dfrac{i}{i} = \dfrac{i}{i^2} = -i$

$\dfrac{1}{i}$ is a correct intermediate step but we tend to want a real denominator so we multiply through by the complex conjugate which is i/i in this case

3. ## Re: Explain i^-5

I don't understand how i/i²=-i
Shouldn't that also be i^-1 since i^1/i^2= i^(1-2)

4. ## Re: Explain i^-5

Originally Posted by NME
I don't understand how i/i²=-i
Shouldn't that also be i^-1 since i^1/i^2= i^(1-2)
Think back to the very basics of complex numbers and how we define $i^2$ (or i, depending on your textbook).

In other words you can simplify $i^2$ into a real number

5. ## Re: Explain i^-5

Oh that's right xD! How could i even miss that!

Well thank you very much for your help and patience !

6. ## Re: Explain i^-5

Originally Posted by NME
Could somebody explain to me why i^-5= -i and not 1/i

Thanks
$\displaystyle i^{-5} = \frac{1}{i^5} = \frac{1}{i^4\cdot i} = \frac{1}{i} = \frac{i}{i^2} = \frac{i}{-1} = -i$

7. ## Re: Explain i^-5

Another way of looking at it: $i+ \frac{1}{i}= \frac{i^2}{i}+ \frac{1}{i}= \frac{-1}{i}+ \frac{1}{i}= 0$

That's why $\frac{1}{i}$ is -i.

8. ## Re: Explain i^-5

Originally Posted by HallsofIvy
Another way of looking at it: $i+ \frac{1}{i}= \frac{i^2}{i}+ \frac{1}{i}= \frac{-1}{i}+ \frac{1}{i}= 0$

That's why $\frac{1}{i}$ is -i.
Thank you sir (:

9. ## Re: Explain i^-5

another way of looking at it is this:

suppose we had some number (which we don't know what it is), let's call it x, and -x = 1/x.

then (-x)(-x) = (1/x)(-x) (nothing funny going on here).

but (-x)(-x) = (-1)(x)(-1)(x) = (-1)(-1)(x)(x) = x^2, while (1/x)(-x) = -x/x = (-1)(x/x) = (-1)(1) = -1.

therefore, for such an x, x^2 = -1.

so x is a square root of -1. this tells us not only that -i = 1/i, but also that -(-i) = 1/(-i), or: i = 1/(-i).

in geometric terms: if you go 1/4 the way around a circle, the way back home is the same way you get 1/4 the way around the circle in the opposite direction.