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

[axn]

Originally Posted by ewmayer

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.

Or, you could just short cut all that and note that (T+D) + (T+H) + (D+H) = 2(T+D+H) which is 2x what is asked for, without solving for individual rates.

[Dubslow]

Originally Posted by axn

Or, you could just short cut all that and note that (T+D) + (T+H) + (D+H) = 2(T+D+H) which is 2x what is asked for, without solving for individual rates.

That's a nifty way to do the elimination.

The problem boils down to simple linear algebra ("system of equations" for those who don't know linear algebra), but that is indeed a very elegant way to solve it

What Toom was complaining about, of course, was the inability of most people to set up the correct system of equations.

In particular, going along as Ernst suggested, let T represent the work done by Tom in one hour, and likewise D and H. Then, since Tom and Dick can complete "the job" in 2 hours, we say that 2 Tom-hours + 2 Dick-hours = J, the total amount of work to complete the job. In other words, 2(T+D) = J. Likewise, 3(T+H) = J and 4(D+H) = J. Applying axn's inspired way to solve the system (other more "brute force" ways are possible, I myself just let Wolfram Alpha do the work ), we rewrite this as (T+D)=J/2, (T+H)=J/3, (D+H)=J/4, and then add them all together to get that 2(T+D+H)=J(1/2+1/3+1/4), or J = (T+D+H)*2/(1/2+1/3+1/4), where the latter number is 2/(6/12+4/12+3/12) = 2/(13/12) = 24/13 hours, or a bit under 2, as the final answer. (Harry is a slacker! His help only cuts 9 minutes from the 2 hours that Tom and Dick can do it in!)

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For those who haven't followed Ernst on the 0.9 problem, perhaps this will help:

The "standard" way shown is schools is this:

x = 0.999999....
=> (1)
10x = 9.99999... ("well multiplying by ten just moves the decimal point over by one place!")
10x - x = 9.99999... - 0.999999....
=> (2)
9x = 9 ("well we can do a place by place subtraction, clearly each 9 cancels out")
=>
x = 1

The problems Ernst mentions arise from trying to rigorously justify the steps labelled (1) and (2); the quotes are the school-intuitive justifications that aren't quite suitably rigorous for a mathematician.

This is of course exactly the same way one sums a geometric series ("series" just means a bunch of things added together, and "geometric" means those things are all a multiple of each other):

S = a + ar + ar^2 + ar^3 + ...
r*S = ar + ar^2 + ar^3 + ar^4 + ...
rS - S = (ar + ar^2 + ar^3 + ar^4 + ...) - (a + ar + ar^2 + ar^3 + ...) = -a
S(r-1) = -a
S = -a/(r-1) = a/(1-r)

We recover the specific problem asked by letting a = 0.9 and r = 0.1; then 0.9 + 0.09 + 0.009 + ... = S = 0.9/(1-0.1) = 0.9/0.9 = 1. Again, for a mathematician to accept this, we need to have rigorous ideas about how and when an infinite sum actually exists, and how to manipulate such series (such as subtraction). These topics are covered in some detail in a calculus course, though in less detail than Ernst is diving to.

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Can a mod please split this discussion into a separate thread as previously suggested?

[MooMoo2]

Originally Posted by wblipp

This article includes four problems that Toom used as an indicator of mathematical preparedness for college math - and that most students in his business calculus classes could not solve. Already one person here has bravely admitted that he could not solve three of these. ... I believe these problem involve a handful of simple concepts that anyone here can master, but I also believe that the American educational system has failed to impart these simple concepts to many graduates.

It took me almost an hour, but I solved all four problems on my own.

My problem for the forum:
Problem: The text on page 122 states that the author "came to a huge state university" to teach business calculus. What state was this university located in?
Extra credit: Obtain the solution (before reading it below!) without using Google or any other search engine.

Solution: Texas
The url of the text provided is: [url]http://www.de.ufpe.br/~toom/my-articles/engeduc/ARUSSIAN.PDF[/url]
Try to go to Toom's home page by deleting the last part of the url to obtain: [url]http://www.de.ufpe.br/~toom/[/url]
This redirects you to: [url]http://toomandre.com/[/url]
From there, click his biography to obtain his CV: [url]http://toomandre.com/myself/CV.pdf[/url]
Page 122 states that he taught business calculus the next (school) year after teaching at Boston University. If you read through his CV, you'll find that the "huge state university" that he taught business calculus at was the University of Texas at Austin.

I wonder whether the average person has a greater chance of getting the extra credit for this problem or of figuring out one of Toom's problems.

[Dubslow]

Originally Posted by MooMoo2

It took me almost an hour, but I solved all four problems on my own.

My problem for the forum:
Problem: The text on page 122 states that the author "came to a huge state university" to teach business calculus. What state was this university located in?
Extra credit: Obtain the solution (before reading it below!) without using Google or any other search engine.

Solution: Texas
The url of the text provided is: [url]http://www.de.ufpe.br/~toom/my-articles/engeduc/ARUSSIAN.PDF[/url]
Try to go to Toom's home page by deleting the last part of the url to obtain: [url]http://www.de.ufpe.br/~toom/[/url]
This redirects you to: [url]http://toomandre.com/[/url]
From there, click his biography to obtain his CV: [url]http://toomandre.com/myself/CV.pdf[/url]
Page 122 states that he taught business calculus the next (school) year after teaching at Boston University. If you read through his CV, you'll find that the "huge state university" that he taught business calculus at was the University of Texas at Austin.

I wonder whether the average person has a greater chance of getting the extra credit for this problem or of figuring out one of Toom's problems.

I merely did Wikipedia -> homepage -> CV, which nearly amounts to what you did. Also, mayhaps delete some of your links, they show through the spoilers

[Batalov]

My bonus question for you is how do you pronounce Andre's last name.
Because it is actually of Estonian origin, 'oo' is actually simply long 'o'. New Yorker journal spells 'cöoperation', and should have spelled his name 'Töom' if they ever wrote an article about him. If you ask me, they should.

[ewmayer]

Originally Posted by Batalov

My bonus question for you is how do you pronounce Andre's last name.
Because it is actually of Estonian origin, 'oo' is actually simply long 'o'.

Long 'o' like in 'tOmography', you mean?

[blip]

...

[0PolarBearsHere]

Originally Posted by Dubslow

We recover the specific problem asked by letting a = 0.9 and r = 0.1; then 0.9 + 0.09 + 0.009 + ... = S = 0.9/(1-0.1) = 0.9/0.9 = 1. Again, for a mathematician to accept this, we need to have rigorous ideas about how and when an infinite sum actually exists, and how to manipulate such series (such as subtraction). These topics are covered in some detail in a calculus course, though in less detail than Ernst is diving to.

Or if you don't believe it you can start from first principles, though that will take you waaaaaay down the mathematical rabbit hole.
http://us.metamath.org/mpegif/0.999....html

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How do you do the huck finn one? Does it require some knowledge of rafts that I don't know about?

[Nick]

Let x=1-0.9999...
Any reasonable definition of what we mean by 0.9999... (with the nines going on forever) will give us the folowing:
0.9999... is greater than 0.9 so
0.9999... is greater than 0.99 so
0.9999... is greater than 0.999 so
etc.
If x>0 then we also have a number with the following properties:
as , we have
as , we have
as , we have
etc.
So if 0.9999... is less than 1 then we have a real number greater than any integer.
It is possible to construct number systems in which such a "number" exists, but they do not have the properties we need to do analysis and calculus. (In technical terms, a complete ordered field has the Archimedean property, so an ordered field with such an element is not complete.)

Anyone interested in looking at this seriously can consult free notes from prof. Tom Körner's lectures at Cambridge: https://www.dpmms.cam.ac.uk/~twk/ExAn1.pdf
(or if you are really serious I recommend his book http://www.ams.org/bookstore?fn=20&a...ew&item=GSM-62 )

The Wikipedia article referred to above by Ernst fails to mention the uniqueness of the isomorphism, which gives us another exercise:

Let f be a function such that, for any real number x, f(x) is also a real number.
Suppose f respects addition, multiplication and the ordering, i.e. for any real numbers x, y and z:
if x+y=z then f(x)+f(y)=f(z)
if xy=z then f(x)f(y)=f(z)
and if x>y then f(x)>f(y).
Show that f is simply the identity function, i.e. that f(x)=x for all real numbers x.

[retina]

Originally Posted by 0PolarBearsHere

Does it require some knowledge of rafts that I don't know about?

One presumes the raft will just float along with the currant current.

[ewmayer]

Originally Posted by 0PolarBearsHere

How do you do the huck finn one? Does it require some knowledge of rafts that I don't know about?

Beyond 'raft moves at same speed as river, which flows at constant uniform speed', no. Ah, we do also need to assume that the boat moves at constant uniform speed relative to river and makes no stops along the way, there are no wind or Coriolis effects, no relativistic effects, etc. :P

Those allow you to eliminate the unknown boat speed and solve for the raft (i.e. river) speed, hence the raft travel time needed.

Call boat and river speeds v and r, both constant. Then for the downstream trip we have v+r=1/5 and for the upstream we have v-r=1/7. Solve for r, eliminating the unspecified v, and invert the result.