Surfing with the aliens

So, I was playing with my old radio and by some strange stroke of fate, I stumbled across some sort of alien communication channel. Apparently they look like what we know as unicorns: four legs and a horn on their heads.

By tweaking the frequency knob just a bit, I was somehow able to access and understand their fantastic intergalactic Extertube(tm) network, where apparently millions of these aliens can share information at the speed of thought.

I listened avidly. Some of things I learned about them is that their numbering system counts the number of legs plus the horn, kinda like how we humans count the 10 fingers in our hands. Counting for them goes like this: 0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, and so on.

I came across a particularly lively discussion involving their strange numbering system. The question: Is 0.444... == 1?

So basically, is a zero followed by infinite fours the same as one? Well, not in my planet.

But wait, if their 10 is equal to our 5, their 11 is equal to our 6, their 12 is equal to 7, and so on, what the hell is their 0.444... equal to anyways?

I scribble some math in my trusty notepad. Is 3 / 2 equal to 0.444...?

 3 | 2  I can fit one 2 in a 3
    1

 3 | 2  but I still have a remainder of 1
-2  1   let's put a dot and add a zero
 1      to the remainder like my 1st grade teacher taught

 3 | 2  There.
-2  1.
 10

 3 | 2  Ok, so 5 now, right?
-2  1.5 Wait. They don't have a 5.
 10     Their 10 is worth 5.
        Argh. I'm getting confused.

 3 | 2  I can fit two 2's in a 4.
-2  1.2 Two 3's would only fit in 11.
 10     So 2 it is. I get 10 - 4 = 1. Right?
- 4     But 1.2 is definitely not 0.444...
  1     This can't be right.

I scratch that and try a new fraction.

Is 1 / 3 equal to 0.444...? Scribble, scribble. No. Is 23 / 34 equal to 0.444...? Scribble, scribble. No. Is 34110 / 21300 equal to 0.444...? Scribble, scribble. No. After a bit of fumbling, I tried this:

 1 | 4  I can fit zero 4's in a one 
    0

 10 | 4  so I put a dot and add a 0
     0.

 10 | 4  now I can fit two 4's.
     0.1  Oh wait, no, I can't, because their
          one-followed-by-a-zero is actually
          only worth 5. So I can only fit one
          4 in a 10 and I get 1 left.

 10 | 4  so I have 0.1 and 1 left. Right?
- 4 0.1
  1

 10 | 4  repeat...
- 4 0.11
  10
 - 4
   1

 10 | 4  repeat...
- 4 0.111
  10
 - 4
   10
  - 4
    1

Ok, great, if I keep calculating this forever I will have 0.111... (with infinite ones). So 1/4 = 0.111... for our alien friends.

Awesome, since I know 0.111... will be an infinite number of ones (and no other number), I think I can do this then:

 0.1  0.11  0.111  0.1111...
 0.1  0.11  0.111  0.1111...
 0.1  0.11  0.111  0.1111...
+0.1 +0.11 +0.111 +0.1111...
 0.4  0.44  0.444  0.4444...

Alright! I figured out that 1 divided by 4, times 4 is 0.444...

Wait.

Anything divided by 4, times 4 is itself. Right? I try it with my super powerful Windows calculator to make sure I'm not getting confused by their weird number system.

1 / 4 = 0.25
0.25 * 4 = 1

Yeah. I'm not crazy.

Their 0.111... is the same as our 0.25
Their 0.222... is the same as our 0.50
Their 0.333... is the same as our 0.75
Their 0.444... is the same as our 1

And our 1 is the same as their 1.

Excited, I decide to share my discovery with the aliens. I tune my old walkie-talkie in the same frequency and start spelling out my calculation. 0.444... is equal to 1!

I disconnect my radio, feeling accomplished that I helped extraterrestrial beings in a intergalactic discussion. I grab some juice and decide to surf the 'net a bit to relax.

Hey, what is this? A new forum post. Apparently it's also a math question. Hah. I'm good at those. I just helped some aliens!

I rub my hands anxiously, pick my pencil and paper and start reading the question: is 0.999... == 1?