What operation? You're thinking of the steps used in long division, as I said. This is how it's done, you think. There is only one way to do it, you think.MD said:What is "immediate" about .999...? You have a REPEATING DECIMAL, so the operation is continuously repeating and never complete, rate of repeat being irrelevant!
What operation? You're thinking of the steps used in long division, as I said. This is how it's done, you think. There is only one way to do it, you think.
Why do you think that?
They are equally well built on axioms as are all valid inferences.Thanks for your comments. You are much better versed in formal mathematics than I (a physicist) ever was but I do have a question:
If rational decimal, RD = 0.ab.... where a&b are integers, perhaps equal as in 0.3333... or not as in RD =0.121212... is it less well established from axioms that 10RD =a.babab.... than that terminating decimal, TD =0.ab00000... has 10TD = a.b0000...? Or are they equally well built on the axioms?
To the formalist there is no "fundamental" without axioms. To the formalist the excursion towards "fundamental" is all timey-wimey twaddle until you establish a ground floor by stating your axioms. I agree with you that the necessary equivalence between 10×3.1415... and 31.415... is easy to conceptualize as a necessary consequence of our base-ten number system, but proving their equivalence requires talking about infinite, something that mere arithmetic of rationals is ill-suited to do.I only said I "multiplied" to more clearly explain my general procedure but as note earlier post, my moving of the decimal points I think is well defined and FUNDAMENTAL in the meaning of the notational system used.
Of course I agree, as this follows from finite addition of terms and is provable specifically or generally for all finite numbers of terms via finite induction. Arithmetic of rational numbers alone, however, does not give us an axiom system strong enough to address the case when the number of terms is larger than any finite number.I.e. I believe if 0.abc = 0.def then a.bc = d.ef and both =10x0.abc or 10x 0.def just from what the base 10 notational system means. (First place to left of decimal point tells how many units of one, second to the left tells how many units of ten, etc. and etc. for the right side of the decimal point)
Do you agree?
That's Peano multiplication. In the real numbers the definition of multiplication has to be defined in terms that make sense with whatever is adopted in place of the axiom of completion. If it is Cauchy completion, then $$\sqrt{2} \, \cdot \, \pi $$ is defined as the limit of the sequence you get by multiplying pairwise corresponding terms of any sequences of rational numbers that converge to $$\sqrt{2}$$ and $$\pi$$, respectively. Thus the theorem that $$\sqrt{2} \, \cdot \, \pi = \pi \, \cdot \, \sqrt{2} $$ in the reals is a theorem proven from the commutative law of the rationals (which itself can be proven from the commutative law of natural numbers which may be axiomatic or rest on the Peano definition of multiplication of natural numbers).Also do you agree that multiplication is only defined for multiplication by integers as multiplying M by n is defined as adding M to itself n times. The multiplying M by some non-integer is defined from an algorithm known to be valid for multiplying by an integer or IFF at least one of M & n is and integer with help of the commutative law (I think that is the name) which for ordinary math states: AxB = BxA.*
This procedure has to be handled for negative numbers (how do you add -2 to itself -5 times?), for rational numbers and the reals, so the concept is understood. But the actual details seem to differ from how you are thinking of them because we abandon the Peano axioms as soon as we allow negative numbers. Thus we bootstrap ourselves natural numbers -> integers -> rational numbers -> real numbers -> complex numbers and algebraic completeness.E.g. 3x7.5 is not computed directly from the definitions of Mxn but makes use of this commutative law first to get 7.5 x 3 as you can add 7.5 up 3 times but not add 3 up with itself 7.5 times but 3.4 x7.5 can not be computed directly from the definition of multiplying. We can only assume (or define) the result via some algorithm known to be valid by it producing the same results as the products the definition can be applied to.
The decimal is a repeating pattern of digits, the pattern is otherwise known as the period. In the case of 1/3, the period is 1.Motor Daddy said:What do you think the term "repeating" means? Why does it repeat? How does something repeat instantaneously?
Wow, a simple glitch/loop in the decimal representation of thirds generates 18 pages of discussion!
No wonder I always found philosophy to be boring.
$$(\frac{1}{\2}=.5)$$ and $$(.5*2=1.0)$$
$$(\frac{1}{\4}=.25)$$ and $$(.25*4=1.0)$$
$$(\frac{1}{\5}=.2)$$ and $$(.2*5=1.0)$$
So it appears to me, Tach, that according to your concept of the above, zero times infinity should equal one according to you.
$$(\frac{1}{\infty}=0)$$ and $$(0*\infty=1)$$
Is that what you are saying, Tach, that since $$\frac{1}{\infty}=0$$ then it stands to reason that $$(0*\infty=1)$$???
Dude, get your meds adjusted, you're not thinking straight!!
Good, you pass 3-rd grade arithmetic.
Nope, basic calculus says that $$0* \infty=undetermined$$
Yep
Nope, you fail. Again.
Your basic error is that $$\frac{1}{\infty} =0$$ does not imply $$0 * \infty=1$$. Math is tough.
So then calculus must think $$undetermined* \infty=0$$, right???
Nope $$undetermined* \infty=undetermined$$. Math is really tough. The fact that you'll never learn it should start sinking in.
The decimal is a repeating pattern of digits, the pattern is otherwise known as the period. In the case of 1/3, the period is 1.
But that does not mean the division operation repeats, and 1/3 certainly implies a single (i.e. not repeating) operation. The numbers are distinct from any operation on them.
Since we've been discussing the properties of the number 1, how many times can you multiply or divide 1 by itself? Is 1 x 1 = 1 or 1 x 1 x 1 = 1 "immediate", or do you have to do in steps?
Operations are defined on sets, like the integers.Undefined said:Using such trivial and self-selecting 'devices' is tantamount to using a NON-action, rather than a valid OPERATION.
No idea what you mean by this.The operation is essentially an ACTION, not a hypothetical one-off conclusion/assumption as to result
I have been observing the above exchange, and it again highlights clearly the incompleteness and insufficiency of the relevant axioms when the excuse of 'undetermined/undefined' must be invoked because the axioms cannot treat the consequent logic-flow 'results' in a complete and consistent manner.
It is the symptom of the gap between the maths Axioms and the reality Physics, which is at the heart of the problem for the mathematics,
which produces these 'undetermined/undefined' outcomes (like the one pointed out by MD above) whenever the maths/axioms are pushed to the limit of their competence/reliability.
That is why we need the overhaul of the maths axioms to reflect the reasonableness of Physical reality FIRST, and the Maths exercise SECOND.
Good luck trying to make sense from the current INCOMPLETE and (as MD has highlighted in the above exchange) patently inadequate axiomatic formulation, guys!
There is no "incompleteness". There is no "insufficiency".
The issue being discussed has nothing to do with physics, it is an elementary math problem. This has been explained to you countless times.
I am thrilled to see that you and MD are in agreement while sharing fringe misconceptions. Mainstream scientists, not so much. The math as we know it doesn't suffer from any issues and it is fully reliable, contrary to your persistent fringe dronings.
We need no such things, your theories belong in "Alternate Theories". Current axioms are firmly established.
Most of us can make perfect sense of the math, why can't you?
what this tells us is that when a mathematician reckons that 1/infinity = 0 then he is talking in approximations.strictly speaking, 1 / infinity is infinitesimal (very small), and 1 / undefined is zero. that is the language when talking about Pure Mathematics. However, in applied sciences, where approximations are often prevalent, 1 / infinity is considered as zero, and the word "undefined" is seldom used.
How long would you say mathematics has refused to acknowledge this paradox? Since a few centuries B.C,, since the 19th century, or just since this thread started?Quantum Quavk said:I think the bottom line with this thread is that mathematics refuses to acknowledge that a paradox exists when considering infinities.
They don't communicate instantaneously, there is no FTL communication.Now when talking about quantum entanglement and how two 1/2 particles can communicate instantaneously over vast distances one can see potential for a solid explanation for F.T.L. communications.
They don't communicate instantaneously, there is no FTL communication.
strictly speaking, 1 / infinity is infinitesimal (very small), and 1 / undefined is zero. that is the language when talking about Pure Mathematics. However, in applied sciences, where approximations are often prevalent, 1 / infinity is considered as zero, and the word "undefined" is seldom used.
actually a fair jab..How long would you say mathematics has refused to acknowledge this paradox? Since a few centuries B.C,, since the 19th century, or just since this thread started?
Well, I think I'll stick with the idea that infinity is defined in a non context-free way, That is to say, it depends on the context.Quantum Quack said:eh?
I suppose you failed to read or agree with what the Phd Guy said.. that in pure mathematics 1/infinity =/=0
by all means... as you wish...Well, I think I'll stick with the idea that infinity is defined in a non context-free way, That is to say, it depends on the context.
If zero in the physical world is defined as being a point that is as described by the reduction of a ball to the diameter of 1/infinity [ which is not zero ] it means that the zero point exists in a three dimensional sense even though it is zero dimensional. [zee paradox!]But what do you think a mathematical definition of infinity has to do with communication? What's the context?
Your 'explanation' to MD was "undetermined".
And now you say "Current axioms are firmly established".
So, those current axioms are established so well that an 'answer' of "undetermined" is OK with you as a "result" of those axioms?
Not all of us are so 'accepting' of such inadequacy/insufficiency in our maths axioms.
The only way to remedy those axioms is to make them based on more real rather than abstract foundations.
You, of course, are welcome to stay put in your perfectly abstract but insufficient axiomatic 'world' where "undetermined" is an OK 'result' from said current axioms.
Those who claim that 0.9999... is not 1.0 or true are (1) illogical
or (3) have assumed different axioms than common ones
or (3) poorly understand the common base-10 notational system.
or are (4) just internet trolls.