HaloO,
Mark J. Reed wrote:

In what the heck mathematical world is the square root of two an
infinite value? Irrationality and infinitude are not the same thing;
in particular, there are an (uncountably) infinite number of
irrational numbers...

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I don't know what you accept as an infinity but there is e.g.
an infinite set of complex numbers
subset InfSqrt2 of Complex where { $_ ** sqrt(2) == 1 }
whereas
subset Inf141 of Complex where { $_ ** 1409/997 == 1 }
# 1409/997 approx 1.41
only has 1404773 entries. That is one way to transform the infinity
of an irrational number into an infinite set. Rationals only render
infinities of the kind a circle has, where you can run in one direction
without reaching an end.
Or take the tan function that can be used to wrap the real number line
around a circle at Zero. Then you get a point Inf on "the other side" of
Zero that just behaves like Zero, i.e. on one side you have all the
positive reals and on the other all the negative ones. So you can make
the transition from +Inf to -Inf just as easily as from +0 to -0. IOW,
there are properties that are usually ascribed to Zero that Inf can
claim as well. For that very reason I gave here, IEEE-754 distinguishes
-0 and +0 just like it distinguishes -Inf and +Inf. And there are
contexts where -Inf === +Inf makes sense just as -0 === +0 makes sense
and some where it doesn't. Now, how are these contexts distinguished?
Regards, TSa.
--
"The unavoidable price of reliability is simplicity" -- C.A.R. Hoare
"Simplicity does not precede complexity, but follows it." -- A.J. Perlis
1 + 2 + 3 + 4 + ... = -1/12 -- Srinivasa Ramanujan