Side Topic: 'It's Not a Toom-ah'

[ewmayer]

Originally Posted by retina

Hmm. I have no idea what you mean here. I though it was simply one thing is numerically the same as the other.

Well, for starters, on the left-hand sides of your alleged equalities you have a rational, whereas on the RHS you have notation indicating a decimal expansion. As I did in Proof B it is easy to show that for any finite-term truncation of the RHS expansion, != holds. Thus one must bring in the the concept of the limit of such an expansion (if it exists), which involves the notion of closure of the infinite set {0.9,0.99,0.999,...}. This is the stuff about "equivalence classes of Cauchy sequences of rational numbers" mentioned in the Wikipage I linked above. We nearly always take this stuff for granted, but in fact it is highly nontrivial in its details.

There are many examples where people blithely write '=' but the meaning of that needs to be made precise. Another example is from the theory of convergence of approximations in Hilbert spaces. Well-known specific example of that is approximation of a nonsmooth function - say a step function - via a summation of smooth ones. When we say that an infinite Fourier-series summation of sine and/or cosine functions 'equals' a step function, what do we mean? In fact due to the Gibbs phenomenon, no matter how many terms we take in the expansion we end up with 'wiggles' in the summed expansion which never settle down - they get narrower as we take more terms, but their amplitude does not decrease, in fact quite the opposite. Thus when we write '=' what we really mean in the pointwise-sense is equality 'almost everywhere', which means 'except on a set of measure zero' (measure theory), and in the sense of the smooth-functional approximation equality only holds in the sense that the L^2 norm of the difference of the 2 functions - the step function and its Fourier-series-approximation - vanishes in the limit of an infinite-term expansion.

[ewmayer]

Originally Posted by owftheevil

Proof B seems to depend on the mistaken assumption that an uncountable set cannot be written as a countable union of uncountable sets.

That's one way of looking at it - I was trying to highlight the 'what do we mean mean when we transition from the finite to the infinite?' aspects of the rational approximations. Any finite-term approximation is 'infinitely far away' from the limit in the sense of number of intervening reals, yet we make sense of the infinite-term limit existing. (This is basically Zeno's famous paradox in disguise.)

As for the TDH problem, if you think about it for a minute, you will realize that if T,D, and H represent times to finish the job, then T + D =2, T + H =3, and D + H = 4 make no sense. In Toom's defense, i think this was what he wanted the student to realize.

The 'how can we assume the work-times are simply additive' aspect occurred to me while I was writing that, but I didn't think that was what Toom was driving at. Fail on my part.

[retina]

Originally Posted by ewmayer

Well, for starters, on the left-hand sides of your alleged equalities you have a rational, whereas on the RHS you have notation indicating a decimal expansion. As I did in Proof B it is easy to show that for any finite-term truncation of the RHS expansion, != holds. Thus one must bring in the the concept of the limit of such an expansion (if it exists), which involves the notion of closure of the infinite set {0.9,0.99,0.999,...}. This is the stuff about "equivalence classes of Cauchy sequences of rational numbers" mentioned in the Wikipage I linked above. We nearly always take this stuff for granted, but in fact it is highly nontrivial in its details.

There are many examples where people blithely write '=' but the meaning of that needs to be made precise. Another example is from the theory of convergence of approximations in Hilbert spaces. Well-known specific example of that is approximation of a nonsmooth function - say a step function - via a summation of smooth ones. When we say that an infinite Fourier-series summation of sine and/or cosine functions 'equals' a step function, what do we mean? In fact due to the Gibbs phenomenon, no matter how many terms we take in the expansion we end up with 'wiggles' in the summed expansion which never settle down - they get narrower as we take more terms, but their amplitude does not decrease, in fact quite the opposite. Thus when we write '=' what we really mean in the pointwise-sense is equality 'almost everywhere', which means 'except on a set of measure zero' (measure theory), and in the sense of the smooth-functional approximation equality only holds in the sense that the L^2 norm of the difference of the 2 functions - the step function and its Fourier-series-approximation - vanishes in the limit of an infinite-term expansion.

Oh yeah, of course. Clear as mud to me.

I think you just made me realise this online course too complicated. There is no way I could write a two paragraph explanation like the above just for my usage of '=' when trying to prove something. "Cauchy sequences", "Hilbert spaces", "nonsmooth function", "Fourier-series", "Gibbs phenomenon", "pointwise-sense", "measure theory", "L^2 norm", "vanishes in the limit": I didn't expect the Spanish inquisition.

[owftheevil]

Originally Posted by ewmayer

That's one way of looking at it - I was trying to highlight the 'what do we mean mean when we transition from the finite to the infinite?' aspects of the rational approximations. Any finite-term approximation is 'infinitely far away' from the limit in the sense of number of intervening reals, yet we make sense of the infinite-term limit existing. (This is basically Zeno's famous paradox in disguise.)

First thing I thought when reading the proof was "This is just a version of Zeno's paradox." I agree that having to deal with infinity makes that problem mush more subtle than the others. I also think that Retina's proof is fine. Most middle school students would understand it without having to know exactly what the = means.

[retina]

Originally Posted by owftheevil

As for the TDH problem, if you think about it for a minute, you will realize that if T,D, and H represent times to finish the job, then T + D =2, T + H =3, and D + H = 4 make no sense.

Erm, T=0.5, D=1.5, H=2.5?

Sorry that I didn't explain my usage of '=' there. I really have no idea how to explain what I mean above. But to me it makes sense for what I know about '='. I intend it to mean that one side is the same as the other; and there are no wiggles, or sequences and any of other stuff that the '=' might imply but which I am unaware of at this time. To anyone else, please feel free to properly explain what I mean

[ewmayer]

Originally Posted by retina

Oh yeah, of course. Clear as mud to me.

I think you just made me realise this online course too complicated. There is no way I could write a two paragraph explanation like the above just for my usage of '=' when trying to prove something. "Cauchy sequences", "Hilbert spaces", "nonsmooth function", "Fourier-series", "Gibbs phenomenon", "pointwise-sense", "measure theory", "L^2 norm", "vanishes in the limit": I didn't expect the Spanish inquisition.

Well, I miserably flunked the Tom/Dick/Harry problem, so after overseeing a lengthy torture session the Inquisitor just tripped over his shoelaces and broke his face, or something.

Anyway, back to number theory, preferably with nice round integers and the only twist on '=' being the 'congruent to' sense. :)

[owftheevil]

Originally Posted by retina

Erm, T=0.5, D=1.5, H=2.5?

Sorry that I didn't explain my usage of '=' there. I really have no idea how to explain what I mean above. But to me it makes sense for what I know about '='. I intend it to mean that one side is the same as the other; and there are no wiggles, or sequences and any of other stuff that the '=' might imply but which I am unaware of at this time. To anyone else, please feel free to properly explain what I mean

So each one can finish the job faster alone than when working with someone else?

[ewmayer]

Originally Posted by retina

Erm, T=0.5, D=1.5, H=2.5?

I had the same nosebleed from tripping over my shoelaces. :facepalm:

We must recast things in terms of the *rates* at which T,D,H do work (and yes, we do assume these are additive in order to obtain a solution)

T and D together: 1/2 units per hour
T and H together: 1/3 units per hour
D and H together: 1/4 units per hour

Then set up the linear system same as before but the right-hand-sides are the pairwise rates-of-work, solve for the individual rates-of-work, sum all three and convert that to a total-time-for-the-trio to do the job.

Originally Posted by owftheevil

So each one can finish the job faster alone than when working with someone else?

Well, in the real world this is in fact often how things work out - the old 'too many cooks' kind of deal. Maybe Toom is being wickeder than we thought, and the 'correct' answer is 'When all three work together the job takes forever, because politics and labor/management disputes.'

[owftheevil]

Originally Posted by ewmayer

Well, in the real world this is in fact often how things work out - the old 'too many cooks' kind of deal. Maybe Toom is being wickeder than we thought, and the 'correct' answer is 'When all three work together the job takes forever, because politics and labor/management disputes.'

This made me laugh.

[retina]

Originally Posted by ewmayer

Well, in the real world this is in fact often how things work out - the old 'too many cooks' kind of deal. Maybe Toom is being wickeder than we thought, and the 'correct' answer is 'When all three work together the job takes forever, because politics and labor/management disputes.'

One woman takes nine months to make a baby. How many months would nine women take to make one baby?

Strangely, no mention of men in there!

[Batalov]

Originally Posted by retina

One woman takes nine months to make a baby. How many months would nine women take to make one baby?

Strangely, no mention of men in there!

Nine spherical women in vacuum in one month will make nine times 1/9th parts of spherical babies in vacuum. This much is clear.

Which sums incidentally to 9 * 0.111111111111... = 0.999999999999... which we will not be able to compute (at least we couldn't over the last dozen posts). It may or may not be equal to 1 spherical baby in vacuum. I have no problems with the first paragraph. But this one is much harder and may never be resolved.