As for the rest of your post, I was actually about to start a thread similar to this and was happy to see someone had already started! I agree with the idea that basically everything can be explained by numbers. However, one thing I don't quite agree with is that you say that with the proper technology everything can be explained with infinite precision. The problem with that is once you get small enough, as in smaller than quarks, things begin to be expressed in probabilities. This means that you can't ever quite define the location of anything quantum sized, only where it might be.
That doesn't really seem to be a problem except that some people believe if you can completely define everything, then that could mean that if you had the equation for everything you could essentially have a "life" calculator that you could input a problem and have it tell you exactly what would happen. This would include things like human decisions. Basically, there is no such thing as free will, just your brain doing what the universal code says it will do under the right forces. However, since our brains may take advantage of quantum mechanics, if we do operate on probabilities then we could never properly define what a person would do in a given situation. I feel like I'm kind of going off on a tangent though...
Back to your idea that our brain is just taking in numbers and simulating a world around us. I believe that that is exactly what is happening. All your brain is is a computer that takes in input from your senses, and provides output to your body. Really, it doesn't matter what the makeup of the universe is. What we feel is just how our brain interprets it. This idea is what has had me really excited about virtual realities for a long time. The companies at the forefront of virtual reality are creating ridiculous helmets that spray scents into your nose, or full body suits that press your nerves to simulate touch. All that is really needed is a way to stop your brain from receiving input from your senses, and start receiving input from a computer. With a computer fast enough to simulate the universe, your brain wouldnt be able to tell the difference. It is receiving input in the exact format that it was evolved to receive it in.
I'm sure this isn't a unique idea, but it is one that I have had for a long time. The problem is that the equations needed to define everything would be massive. There are some problems out there with equations so large that it takes teams of hundreds of mathmeticians just to create them. Having enough equations for every single possible situation in the universe would be a ridiculously daunting task. One nice solution to that is what are called "simple programs". It is the idea that a simple program, with a small set of rules, when run creates staggeringly complex patterns as a result. These are programs that can be made with a few lines of code, and can be explained with a few sentences in plain english. It sounds kind of ridiculous that a simple program could have such complex results, but this article explains some of the details better than I ever could.
http://en.wikipedia.org/wiki/A_New_Kind_of_Science
An implication of simple programs would be that the universe is run with just a simple program. All of the equations that we know as physics, as well as constants that seem to pop up everywhere like pi and the golden ratio, are effects of the simple program. When run these are patterns that emerge from the programming. This is in contrast with the idea that they are all part of the program itself. If you were to look at all of the equations in physics you would see what situations are possible in the world. With the simple equation you would have no idea until you ran it. It also means that you could never predict what is going to happen next in a simple program. Since it gives you no clues as to what patterns will emerge, you have to just run it and find out. This theory also lends itself pretty well to free will. Since extremely complex behavior, like that of the universe, is all said to have these simple programs as their framework, the brain can be said to follow the same principles. If its behavior is complex enough, then it may have a simple program as its framework. If so, then it is impossible to predict what the program will do. All that can be done is to let the program run and see what happens.