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Sounds similar to how Haskell does all number computations by default. When you do 1/3 for example it leaves the types generic/ambiguous using typeclasses until it finds something to refine them. 1/3 could be a floating-point number, or a fixed-point number, or a rational (datatype holding the numerator and denominator), or any other user-defined datatype. You could put these computations in a library, keeping their types generic with typeclasses, and a user of your library can use their own number datatypes. Your library would automatically support that datatype in your number computations.

No really, computable real store the AST of the computation to lazily compute digits as they are needed (which is an interesting idea but memory hungry at scale) while Haskell approach is more akind to a form of general type deduction.

Note that I first meet computable numbers in Haskell : https://wiki.haskell.org/Applications_and_libraries/Mathemat...

http://hackage.haskell.org/package/exact-real-0.12.2

I’ve used this package to get exact real computations in Haskell which get evaluated at n bits when needed for printing or equality checking etc.

I don't understand how this is supposed to work. For one thing iterative code wouldn't even do the same number of iterations when you change the desired precision, so merely plugging in higher precision numbers wouldn't give the higher precision approximation you'd be looking for. You'd have to actually re-execute the entire code with higher precision numbers so that intermediate control flow decisions use those numbers instead too, right?

If you made decisions based off of precision at some point in time, then you’re right. You’ve “baked” into the number some aspect about a precision decision you’ve previously made. That doesn’t invalidate the number, as it reflects precisely the number you asked for, it just means you’re expressing your computation in a dubious manner.

Sometimes it’s unavoidable though, such as when checking equality where you have to call it quits at some point at some precision.

Generally, precision is pulled on-demand, as it is needed. If you compare two values as x < y, x and y will be evaluated to higher and higher precision until the answer is unambiguous. This can be an issue if the values are equal, since they often cannot be proven to be equal, so the evaluation may just outright fail to terminate.

That doesn't address the situation I outlined?

You store the expression. Because that expression is a computable real, you can calculate it to arbitrary precision.

This is quite inefficient for things like summing over a list. Or really, any kind of accumulation.

Interesting caveat to the computable reals. Equality is hard. I need to look into this but I'm pretty sure it's undecidable.

Edit: looking into it was just reading futher on wikipedia. Even ordering is undecidable.

Undecidable in principle but I wonder how much it comes up in practice.

In practice, you get guarantees “up to precision”. This is totally reasonable. If you’re doubtful, you can either prove theorems, or bump up the precision.

In general this works well.

Good point. It's not worse than floating point numbers.

That's interesting because that's how humans actually do algebra. Who cares about the decimal expansion of some irrational if in the end we divide it by itself for example.

Unfortunately, most infinite-precision real-number implementations (including the one here, I bet, but I haven't checked) will not recognize that you're dividing something by itself. So you'll get a super-inefficient implementation of the number 1 :-).

[EDITED to add:] I checked; the implementation here indeed won't recognize that you're dividing something by itself. For the avoidance of doubt, nor do I think trying to recognize such things would be an improvement.

(Also, if you ask for e.g. the digit immediately preceding the decimal point, you will wait for ever because no amount of precision will enable the implementation to tell whether your number is just less than 1 or 1 or just over. So don't ask it for that, ask it for a very close approximation and draw your own conclusions.)

The notion of "by itself" is dubious in this context at best. You can only check up to finite precision that two numbers are equal. For something more, you need computer algebra, which is a tall order.

Even if you have computer algebra, math says it's not possible to check that arbitrary expressions are equal. It's an undecidable problem. So with that, I don't fault a computable reals package not being able to detect that x == y in an expression x/y.

You can imagine an infinite-precision-real package that checks whether numerator and denominator are the exact same object and optimizes that case away.

(And I wonder idly from time to time about making one that does know a bit about some particularly nice classes of number -- e.g., algebraic numbers -- and does all the things you'd like it to as long as you stay within such a class. Obviously as soon as you ask it for sqrt(2)+pi all that would go out the window.)

Also note that dividing something by itself is not necessarily 1.

I dunno, I think 0/0 is 1, but I cannot be sure.

https://www.wolframalpha.com/input/?i=lim+x%2Fx+as+x-%3E0

You can say 0/0 is 1, but it's also 2 and 3 and every other number, since x*0=0 for any x. That's why it's undefined.

Could we define it as a one to many function that maps from 0/0 to the set of all Complex numbers?

If 0/0 = Q, then

0 = 0 × Q. So, we get to define this operation. If we treat 0 as an integer, then we could treat Q (a set) like a 1 x inf. matrix and we have scalar multiplication.

But

0 × Q = [0]' × Q

So 0 can be an infinite set of 0. This is cyclical, I know. This cyclical nature leads to an interesting effect if we also define integer division. Try this around 0 in Q.

We could also define the × as an inner (dot) product, or as a cross product.

This is fun!

Is there a number set larger than a complex number? If so you can’t map to any concrete thing as it is really anything.

And I guess it is easy to define one more dimension of number.