God created the real numbers

The reals aren't algebraically under multiplication; a simple equation like x*x=1 can't be handled in real numbers. The complex numbers are algebraically closed. So I suspect that God created the complex numbers.

God certainly had a fondness for the real subset. Measurements are real scalars -- so much so that it really does look like God created the reals. That's what's important to us. But the fundamental laws seem to require the complex numbers (or their equivalent, like matrices), and closure under arithmetic operations really does feel like it should be a requirement for the reality of the universe.

I'm more troubled by the fact that almost all real numbers are uncomputable (same goes for complex numbers, of course). It's very straightforward to see that this is the case, but the mathematics involved to even begin to ponder questions like "under which operations is the set of computable reals not closed" seem to be far over my head.

Are there any operations you can even perform on the computables in the general case? Take addition, it seems simple until you try to add two computable numbers:

      0.59999999999999999999999...
    + 0.00000000000000000000000...
    ------------------------------
      0.?

Until you see a non-nine in that first number, or a non-zero in the second, you can't even emit the first digit of the output. From outside the black box, you don't know if the nines and zeros will stop or continue forever.

I think you can make pathological cases for every arithmetic operation, so maybe (I'm not sure) none of the operations are computable. (Need to be careful with the definitions though, and I'm being pretty sloppy)

That’s just 0.6 and 0.0 written cleverly.

If the 9’s repeat forever then there is no possible number between that and 0.6, so it must be the same…

Put another way, any single digit “n” repeating is just n/9.

Your problem is equivalent to (4/9 + 5/9 - 0.4) + 0/9

(Equivalent to 0.6)