Thinking about real numbers in the 01_ programming language, the fractional part can be represented as big endian base 1/2. (Or is it little endian? Bits to the left represent larger numbers, but smaller powers of 1/2.) Infinite lists of bits can represent numbers that are ≥ 0 and ≤ 1.

Addition of fractional numbers can be defined as

+/fractional 0a 0b = +/fractional/carry a b 0_ 1_ +/fractional a b.
+/fractional 1a 0b = +/fractional/carry a b 1_ 0_ +/fractional a b.
+/fractional 0a 1b = +/fractional/carry a b 1_ 0_ +/fractional a b.
+/fractional 1a 1b = +/fractional/carry a b 0_ 1_ +/fractional a b.

where evaluating the carry of fractional addition is

+/fractional/carry 0a 0b carry-zero carry-one = carry-zero.
+/fractional/carry 1a 0b carry-zero carry-one = +/fractional/carry a b carry-zero carry-one.
+/fractional/carry 0a 1b carry-zero carry-one = +/fractional/carry a b carry-zero carry-one.
+/fractional/carry 1a 1b carry-zero carry-one = carry-one.

And the subtraction of fractional numbers is

-/fractional 0a 0b = -/fractional/borrow a b 0_ 1_ -/fractional a b.
-/fractional 1a 0b = -/fractional/borrow a b 1_ 0_ -/fractional a b.
-/fractional 0a 1b = -/fractional/borrow a b 1_ 0_ -/fractional a b.
-/fractional 1a 1b = -/fractional/borrow a b 0_ 1_ -/fractional a b.

where evaluating the borrow of fractional subtraction is

-/fractional/borrow 0a 0b borrow-zero borrow-one = -/fractional/borrow a b borrow-zero borrow-one.
-/fractional/borrow 1a 0b borrow-zero borrow-one = borrow-zero.
-/fractional/borrow 0a 1b borrow-zero borrow-one = borrow-one.
-/fractional/borrow 1a 1b borrow-zero borrow-one = -/fractional/borrow a b borrow-zero borrow-one.

Unlike the addition and subtraction of integers, these operations, in general, require infinite time and memory to calculate a finite number of bits, due to the carry and borrow.