First things first. A number is not defined by its representation. What does that mean? All of the following objects represent the same number.
cat * dogfor
cat = 2and
dog = 1/4
A number retains its value regardless of how you write it. It is what it is.
I've got it on good authority that some old nerd hid a bunch of treasures all over the real number line, right on top of some numbers. Lucky for us, I've got a fancy number detector that'll help us track them down. Ok it's not that fancy. It's actually pretty cheap but it works so back off.
My fancyish number detector can't take us directly to a number in one go. What it will do is take us to the general area of the number, then we recalibrate it and have it search again. This will give us a much smaller area to search. If we repeat this process for long enough, we'll find our number! Probably. Well, maybe. It depends on your definition of probably.
Anyway! Some of these treasures are supposed to be pretty huge and we're going to have to make multiple trips. The bad news is that it takes about an hour for each scan. The good news is that my fancyish number detector has a way for us to map out the path to the number so we only have to figure it out once. After that, we can just follow our trail right back to the number!
The detector splits the number line up into 10 equal chunks and assigns a number to each chunk. These chunks are closed intervals. "Closed" just means that the endpoints of each interval are included in the interval. For now, we'll be searching between 0 and 1.
For example, on the first scan, interval number 1 is [1/10, 2/10], and interval number 2 is [2/10, 3/10]. This means that 2/10 is in both interval 1 and interval 2. Yeah it's a little inefficient but that's what happens when you buy a cheap number detector from a shady guy behind the gas station.
So all we have to do is run the scan, go to the region that that the scanner pops out, write down the number of the region, and repeat until we find our treasure. Piece of cake, yeah? Let's try one!
Since these are test runs, I've already found the numbers and marked them in red on the map so you can see where they are and how all of this works.
4/10 = 40/100and
5/10 = 50/100
We can look at the map as a whole, but it gets tiny pretty fast. We'll stick with the funky zooming thing. (I'm going to animate this eventually)
This one is a little trickier.
Wait a second. Why did we get region 5 on the first scan? Why didn't we get region 4 instead, since 1/2 is clearly in region 4 as well? Let's try starting with region 4 and see where it takes us.
Welp that was fun. I'm going to go take a nap. Feel free to take my detector and go find some more numbers on your own. There are a lot of them out there. Hundreds, at least.
Shout out to my real analysis professor Andrew Cotton-Clay for planting this lovely image in my brain