We can consider all numeric operations to be defined by recursions. For more details, read about the Peano axioms that define the essential features of numbers at: http://en.wikipedia.org/wiki/Peano_axioms.
From these axioms, we can see that addition is defined recursively using more primitive notions of the next number, or successor of a number, n,
.
To simplify the presentation, we'll assume that we can define a predecessor function,
, such that
, as long as
. This formalizes the idea that a number is the successor of the number's predecessor.
Addition between two natural numbers could be defined recursively as follows:
If we use the more common
and
instead of
and
, we can see that
.
This translates neatly into Python, as shown in the following command snippet:
def add(a: int, b: int) -> int:
if a == 0:
return b
else:
return add(a-1, b+1)We've rearranged common mathematical notation into Python.
There's no good reason to provide our...