so, my coffee and i will attempt to merely wax philosophical
it is true that pi and √2 (etc.)
will appear in their exact form
(naturally slash ...automatically)
when happened upon accidentally
such as using the length of a square's diagonal for anything further
(regardless of anyone's current understanding of irrationality, etc.)
but if an irrational number is used
without being brought to naturally,
we rationally suspect foreknowledge,
wandering unarmoured into a gauntlet,
and shielded only by our calculators, but:
without consulting my notes, i can recount (like a TNG epidose)
that the poor bloke who discovered that √2 is indeed irrational
...did not end well (though history is ...divided, as to ...specifics)
the ancient greeks were convinced that, since:
there are infinite fractions between 1 and 0,
there should be no such thing as irrationals;
the fractions should account for all infinity
...ah, but there are many flavors of infinity
for instance, in our modern infinite appreciation, we now know that
if you count to infinity using only 1's, it will take you twice as long
vs counting to infinity using 2's, which gets there twice as fast
but there is no "there" to get to, it just keeps going forever...
(so, infinity is kinda only as far as your own personal horizon)
sorta like how dividing by zero, etc. produces DNE (does not exist) errors
but as a denominator approaches zero, the limit/result approaches infinity
i used to like to say "anything divided by nothing equals infinity"
(thinking myself simultaneously clever, and clever, and clever...)
but our now modern math is very careful about such dealio thingies
cuz so many things go to infinity
many roads lead in that direction
(so, the original name was sposedta be Roam... y’know, like, to “roam”?
n we were thinkin maybe: all roads could lead, like, away from there...)
as i say: coffee + vacation = qed
a curve known as Gabriel's Horn
has infinite area in 2 dimensions
(as x approaches infinity, y approaches 0)
but in 3 dimensions, the volume is pi (aha, hehe)
- probably... infinite thingies... like that... one imagines?
...anyways, using humble 22/7 (etc) in place of proud pi (etc)
easily seems like known ancient historic mere muggle attempts
...as per their limited understanding...
and our limited understanding of that
"no one expects the spanish inquisition"
Satire is a little like proof by contradiction
which first assumes a false estimate is true
and plugs that into the equation in question...
which then naturally produces erroneous results
...such as saying wanton silly things like 2 + 2 = 5
so, now we know the presumption is indeed false
and may more easily track down the actual truth
so, just because you have someone using 22/7 in place of pi
doesn't necessarily mean that they (obviously) didn't know
that proud pi wasn't equal to humble 22/7 exactly
...it may have just been more ...convenient... (hmm)
(which may have been the impetus for inventing math)
...for example; pi squared is kinda approximately 9.8
...which is also coincidentally approximately "little g"
or Earth's gravitational effect at or near the surface
9.8 meters per second squared
(or 32 feet per second squared)
my favorite math professor likes to joke
that one can be substituted for the other
which, of course, is true if you're in a hurry
and exactitude is unnecessary, such as his tests/exams
where proof of understanding is the whole/only point
(ironically, higher math sheds a need for calculators)
(...the scientific mode of the windows 10 calculator
goes out to like 20 digits past decimal or something)
in this fashion, political satire works by giving your opponent the benefit of the doubt
(let's say the world is flat, now let's see what else would have to be true... aha, hehe)
and often points out blatant hypocrisy or willful ignorance
due to the inherent hilarity of the erroneousness produced
thus, often very effective at exposing disingenuousness, etc.
(or at least saving poor saps from wandering off alleged edges)
"on second thought, let's not go to camelot, it is a silly place"
so, how does one prove anything, mathematically,
by other than a meticulous process of elimination,
using proof by contradiction, all out into ...infinity?
aha - proof by induction - hehe
(we will induce the truth from it)
to prove that some equation containing n is true for all n
- basically, you start with n=1, then plug in n+1... for fun...
if still true, we then substitute n+1 for n, giving us (n+1)+1
and usually end up with some version of the original premise
n both sides being equal, we pat our and each other's shoulders
seemingly relaxed now that we have evidence of our own cleverness
...perhaps oblivious to our own obliviousness
regarding ancient awareness of irrationality
so, if we want to prove that the ancient builders/architects...
had knowledge of the inherent irrationality of some numbers,
we just hafta catch em in the act
of actually being that irrational
everyone watch a vid
on Pascal's Triangle