Re: Replacement of Cardinality

Am 05.08.2024 um 08:44 schrieb Chris M. Thomasson:

On 8/4/2024 5:24 PM, Moebius wrote:

Am 05.08.2024 um 01:07 schrieb Chris M. Thomasson:

[...] there is no smallest [unit fraction]...

>
Yeah.
>
Proof: If s is a unit fraction then 1/(1/s + 1) is a unit fraction which is smaller than s (for each and every s).

 Yup.
 always_smaller(i) = (1/(i + 1))
 So, starting at 1/1:
 always_smaller(1) = 1/2
always_smaller(2) = 1/3
always_smaller(3) = 1/4
always_smaller(4) = 1/5
...

Let's try
    next_unit_fraction(u) = (1/(1/u + 1))
So, starting with 1/1:
    next_unit_fraction(1/1) = 1/2
    next_unit_fraction(1/2) = 1/3
    next_unit_fraction(1/3) = 1/4
    next_unit_fraction(1/4) = 1/5
    :
:-P
Though in Mückenland we will have
    next_unit_fraction(1/n) = NaN
for some natural number n, I guess.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Btw. we might implement this function (in Python) the followimg way:
def next_unit_fraction(n, m):
     return (1, m//n + 1)
Testcode:
(n, m) = (1, 1); print((n, m))
(n, m) = next_unit_fraction(n, m); print((n, m))
(n, m) = next_unit_fraction(n, m); print((n, m))
(n, m) = next_unit_fraction(n, m); print((n, m))
(n, m) = next_unit_fraction(n, m); print((n, m))
Output:
(1, 1)
(1, 2)
(1, 3)
(1, 4)
(1, 5)
Works! :-P