The holiday hangout will go online tomorrow! If there's anything in the regular subforums that you're going to want to access over the holidays, copy it now while it's still accessible.

Don't like the snow? You can make a bookmark with the following text instead of a url: javascript:snowStorm.toggleSnow(). Clicking it will toggle the snow on and off.

Our new Indie Games subforum is now open for business in G&T. Go and check it out, you might land a code for a free game. If you're developing an indie game and want to post about it, follow these directions. If you don't, he'll break your legs! Hahaha! Seriously though.

Our rules have been updated and given their own forum. Go and look at them! They are nice, and there may be new ones that you didn't know about! Hooray for rules! Hooray for The System! Hooray for Conforming!

Richy
Registered User regular

Where is the complex number, the square root of -1.

i = SQRT(-1)

1/i = -i is correct, as far as I know. But I can't work out a demonstration for it. Worse, when I try, I get the wrong result and I can't figure out why:

1/i = 1/SQRT(-1) = SQRT(1)/SQRT(-1) = SQRT(1/-1) = SQRT(-1/1) = SQRT(-1)/SQRT(1) = i/1 = i

I'd appreciate it if someone could point out where I'm going wrong there. And also prove that 1/i = -i. Thanks!

i = SQRT(-1)

1/i = -i is correct, as far as I know. But I can't work out a demonstration for it. Worse, when I try, I get the wrong result and I can't figure out why:

1/i = 1/SQRT(-1) = SQRT(1)/SQRT(-1) = SQRT(1/-1) = SQRT(-1/1) = SQRT(-1)/SQRT(1) = i/1 = i

I'd appreciate it if someone could point out where I'm going wrong there. And also prove that 1/i = -i. Thanks!

## Posts

So

As i/i =1, you can say

1/i = 1/i * i/i

= i/(-1)

=-i

1/(a + bi) = (a - bi)/(a^2 + b^2).

You can see that the above forumla works by multiplying both sides by (a + bi).

1 = (a + bi)(a - bi)/(a^2 + b^2) = (a^2 + b^2)/(a^2 + b^2) = 1.

In your specific example,

1/i = 1/(0 + 1i) = (0 - 1i)/(0^2 + 1^2) = -i/1 = -i.

Hirocon, your reply also answered another equivalence that was bugging me that I hadn't asked, that I was going to work on after lunch. Awesome!

i * i = -1 (by definition)

=> i = -1 / i

=> -i = 1 / i

It is correct, because 1/i = (i)^-1. This means "the multiplicative inverse of 'i'" - so what number do you have to multiply "i" by to get 1, - i.

A nitpick (especially if this is asked on an exam): you would not show that 1/z = z/(a^2+b^2) in the above manner, since you are assuming what you are trying to show in the first place. Both sides are of course going to be equal since you already assumed they were equal to begin with: so multiplying 1/z by z to get 1 on the LHS will trivially give you 1 on the RHS.