.999...=1 and other irrational implications of logic & math

[Revoltingest]

but if the diameter is 1......your point
then the circumference cannot be 1

The circumference is different for different diameters.
But sometimes it's exactly 1.

 

[Thief]

The circumference is different for different diameters.
But sometimes it's exactly 1.

I see confusion....
now divide 1 by (pi)

 

[Revoltingest]

I see confusion....
now divide 1 by (pi)

I did.
I made it the diameter of a circle.

 

[Poisonshady313]

I am not only very familiar with how certain aspects of logic, formal reasoning, and mathematics can be unintuitive or even appear to be illogical, but have actually used examples of such cases when teaching. However, an off-topic discussion in a recent thread made me realize that, at least for some, notions that I have found most students grasp quickly can be challenging. Also, there exists a number of books Counterexamples in [X-topic] (e.g., Counterexamples in Probability and Statistics, Counterexamples in Probability and Real Analysis, Counterexamples in Topology) filled with subject-specific examples of proofs/derivations of unexpected results. This is because when it comes to mathematics (including logic), truly understanding the implications of various axioms, theorems, propositions, proofs, etc., means knowing how and when they fail to hold, don’t apply, or yield what appear to be (or actually are!) paradoxes.


So I was hoping that others would be willing to provide examples of logical puzzles, paradoxes, seemingly paradoxical aspects of any area of mathematics, amusing implications of formal logic to informal life, or even just a line or two of something they were told is true of mathematics they had trouble grasping (“And remember, this is for posterity so be honest” –Count Rugen from The Princess Bride).


To be fair (and to give some idea of the kind of examples/things I’m looking for), I’ll go first (see below)

It's mathematically impossible to perfectly tune a piano.

The first 1:40 of this video is some background on tuning instruments and harmonics... the problem presents itself as of 1:41.

 

[freethinker44]

the exact circumference of a circle cannot never be measured to the last decimal.
one of the factors has an infinite number of digits behind the decimal point.

yet the circle is finite

Only if you're a slave to base 10. Try it in base pi.

 

[LegionOnomaMoi]

What has more points?

The surface of a sphere, or the whole sphere (including its surface)?

Ciao

- viole

And can you take a single sphere, chop it into pieces, and use the pieces to create 2 spheres equal in size/volume to the original sphere? If so, why can't you do the same with a circle?

 

[Acim]

1. 0.999…= 1

Why didn't you just number that first one as .999...? Ha!

I still have faith that you have faith in proposition (that .999... = 1). And still believe it is not verifiable.

Was thinking earlier today that if say 99.999... = 100, then would this mean that 999... = 100...?

 

[LegionOnomaMoi]

Why didn't you just number that first one as .999...? Ha!

I have to admit, that's pretty clever!

Was thinking earlier today that if say 99.999... = 100, then would this mean that 999... = 100...?

No. Recall that decimals represent a "fractional" part of a number (or are just simply fraction). So, for example, 99.99 is closer to 100 than 99.9, 99.9999999 is closer to 100 than 99.99, 99.999999999999999999 is closer to 100 than 99.9, 99.9999999, etc. We get closer and closer to a single number (i.e., the sequence of decimals converge). 999... gets closer and closer to infinity. Any number with infinitely many digits before the decimal place goes off to infinity.

 

[viole]

And can you take a single sphere, chop it into pieces, and use the pieces to create 2 spheres equal in size/volume to the original sphere? If so, why can't you do the same with a circle?

This is the sort of things that happen when you can partition a measurable set in a finite set of parts that are not measurable, or for which no meaningful definition of measure can be given. Once you have not measurable parts, you can assemble things that have different measure from the original whole. Those parts seem to have lost all information about the original measure of their whole, so to speak. The fact that measurable things can be split into non measurable ones (and vice-versa) is also pretty weird.

And funny that in lower dimension you can always define a measure for those parts. Which entails you cannot do the same for circles.

Pity that all this relies on the axiom of choice, about which I am not so sure.

Ciao

- viole

 

[viole]

Number 6 (The Monty Hall Problem) got me when I first saw it.
Now it's obvious why the contestant should switch doors.
But oddly, when I conducted an experiment to show someone else the
advantages of switching, probability was working against me that day.
Stupid randomness!

Next time, try with 1000 doors and let the host open 998 empty ones. Especially, if you are betting money.

Ciao

- viole

 

[LegionOnomaMoi]

This is the sort of things that happen when you can partition a measurable set in a finite set of parts that are not measurable, or for which no meaningful definition of measure can be given.

That much I didn't find very surprising by the time I learned of the Banach-Tarski paradox. I was, however, surprised by the fact that such paradoxes can't exists for R^n | n< 3.

Pity that all this relies on the axiom of choice, about which I am not so sure.

Neither am I. But then, I don't deal much with axiomatic set theory and the fact that there exists non-measurable sets because of the AC doesn't really concern mean as such sets are basically constructed to be non-measurable and don't, as far as I can see, have any physical relevance. That said, I would still much rather that the standard set theory drop the final "c" from ZFC without any of the difficulties that would result. The AC is so clearly intuitive an plausible for finite sets and even (IMO) for many a case with infinite sets, but there exist those examples which are sufficiently distasteful as to question the axioms inclusion.[/QUOTE]

 

[Thief]

I did.
I made it the diameter of a circle.

not sure if we are yet on the same page.

This is not quite correct.
Example.
If the diameter is 1/(pi) then the circumference is exactly 1.
Of course, then one can't measure the diameter to the last decimal point.
Nor the radius.

but I see what you did....you gave the quality of measure to the diameter.
that much is finite.....point to point.
and the circle is finite......round about.
but the calculation for having used pi......will not stop

I've seen a program about savants ...one fellow inp articular.....

he can recite the value of pi at great length.
with several people around the table doing check off.....
he will pronounce the value for six hours......as the computer generated list is watched.