I know what I mean. Perhaps you need to read better. Mea culpa, double négatives are bound to confuse some. Read “It’s possible to infer a well formed meaning of 0.000…0001”.
Hangon, I like where you are going with this because it least it drills down.
You deliberately use double negatives aye?
You are not very clear and forthright then.
Answer my questions else.
There’s no requirement (for many valid reasons) that a human should be able to list the decimal digits of a real number or computation involving real numbers. Nevertheless, I am quite confident that sin(\pi) = 0 and that the unique solution of the equation y’(x) = y(x) with y(0) = 1 has y(1) = e. Nor, for that matter, is there any requirement that I should have to “specify” each real number as, say, the solution to a particular equation or in some other similar fashion. The real numbers are— again, by definition— the completion of Q under the usual metric. That’s it. They’re not required to satisfy any other property, and in particular not computability.
A single number doesn’t “converge” to anything
In physics the concept of something approaching the speed of light is analogous to mathematical convergence…and in terms of the magnitude of velocity it would be expressed as a percentage as .999… converging to 1. THAT is why the Lorentz equation is ruined by the substitution.
The x/2 is a nice question.
But we can break other things. Like multiplicative inverses.
What is M (the multiplicative inverse of x)? It can’t be one of our humdrum every day real numbers. They have multiplicative inverses that are greater than 0 but smaller than every other real number.
For example, 1/1000000 is still larger than 1/1000001. Just add 1 to M and 1/(M+1) will be smaller than 1/M.
So, whatever M is has to be larger than every regular real number.
We can call that “BIGNUM”. But “BIGNUM” is a funny beast. It has to be larger than every other regular real number. What other properties must it have?
Actually, at this point let’s go back to x/2. To avoid contradiction, we need x/2 < x. So, suddenly, we don’t have a single infinitesimal. We have infinitely many infinitesimals (keep dividing by 2) with ordering so that we can definitely say some are smaller than others. And since there are infinitely many, there’s no “smallest” one.
Now look at their multiplicative inverses. We don’t just have one “BIGNUM”, we have infinitely many "BIGNUM"s now with ordering so that some are larger than others while they are all bigger than normal real numbers.
And that leads us to hyperreals, nonstandard analysis, and a bunch of other stuff.
But back to the mundane world the standard reals. By introducing any sort of infinitesimal, you need to extend the standard real number system in some very strange ways. That’s not really a problem but it also means that infinitesimals absolutely can’t exist in the standard reals.
Itself, why did you cite the spigot algorithm?
Why.
You stated:
“pi is exceedingly well-understood”
I disagree.
Pi is mostly not understood because its decimal representation as far as we know is infinitely random…and infinite randomness is the precise definition of that which is unknown mathematically
Nowhere was decimal representation a necessary, you said
I took one sentence to follow on from the other (that is normal convention, let me introduce you, “normal convention” meet “Cognitive Tide”).
Why pick on pi, what’s wrong with the square root of two?
Do irrationals perhaps scare you? Or is it just the idea that we cannot represent them completely as a decimal number? But so what? Doesn’t mean that they don’t exist, nor does it mean that we don’t know what the limit of their decimal expansions converge to.
1.41421356… etc converges to √2
3.14159265… etc converges to pi
It’s not a mystery, and it’s not circular, it’s definitional and has no bearing on the limit of 0.999…
ε*ε = ε
(just as 1 x 1 = 1)
Er, no. It really isn’t. Numbers are not particles; a number is not the same as a sequence of particles; massive particles can’t reach the speed of light; convergence in some metric space doesn’t require any potential; Lorentz invariance, which leads to special relativity, has no analogue in the real numbers; and so on.
The next one is harder.
Infinitesimals are something between zero and finite.
So I would be inclined to see where defining an axiom as ε/anything = ε leads to just
as 0 over anything = 0, that one would need to be evaluated to see how the system behaves.
Piffle.
The outcome of the roll of a single fair die is random. Roll it infinitely many times. We can’t predict the outcomes.
That doesn’t mean the results are totally unknowable. There’s an entirely branch of mathematics that deals with what we can know about it.
But that’s a digression and a distraction.
In the case of pi, the decimal representation is NOT random. It’s deterministic. I can’t tell you the 1895th digit of pi off hand, but I can guarantee that it will be the same each time anybody computes it.
Likewise, I can’t tell you the 1895th digit of the decimal expansion of 581738/458373, but it will be the same each time anybody computes it.
There’s a differences between something that looks random and something that truly is random.
And none of this is truly relevant to the fact that for the standard real number system, 0.999… truly is 1. That’s the way the standard reals are built up in the first place.
Then you have provided your contradiction.
Let M be the multiplicative inverse of ε, so that ε*M = 1. (If ε > 0 is a real number, it has a multiplicative inverse)
εε = ε
εε * (MM) = ε (MM)
εMεM = (ε*M)M
1 * 1 = 1M
1 = M
So, 1 would be the multiplicative inverse of ε, or ε*1 = 1, or ε = 1. Since the original supposition was that ε < 1, this poses a bit of a problem. It means ε doesn’t exist, is 1, or is 0.
ε/2 might have to be undefined since epsilon is the least element
“Infinitely random” is a nonsensical term. Presumably you’re talking about normal numbers. It’s unknown whether \pi is normal, and that has absolutely nothing to do with whether it’s a real number, whether it’s possible to do computations with it, whether computations involving it are well-defined, or whether 0.999… = 1. And ‘random’ is not the same as ‘unknown’; there’s a quite well-developed field involving probability spaces, stochastic differential equations, etc.
LOL
These guys cannot step down off of their high horses into the real world.
NO, NO, and NO
No such thing as a perfect cube in the real world.
System Halt.
All real word dies would converge due to the imperfections in the cube.
Can someone who isn’t Cognitive Tide tell me if I should laugh or cry at this post?
ok let’s assume ε*ε = ε, is
ε < 1 ?
ε = 1 ?
ε > 1 ?
like wise
ε < 0 ?
ε > 0 ?
ε = 0 ?
Jeesh!
What on earth could this possibly mean?
It’s like a four-year old who’s learned a new word.
Laugh. There’s a difference between simply not knowing anything about a subject and willful, deliberate ignorance. Combine that with arrogant rants about how everyone else is arrogant and refuses to accept your obvious brilliance, and it’s comedy gold.
ε < 1
The second part, it has the BOTH The properties of zero and finite. (ie it is somewhere in between)
So it might be greater than OR equal to zero
ε ≥ 0