The Contractor's Problem
A masonry contractor notes that for 1000 bricks, he needs 8 bags of mortar.
For another 2000 bricks, he needs 16 bags of mortar. For yet another 3000 bricks, he needs 24 bags of mortar. He writes the fraction, where in the denominator 1 = 1 0 0 0
1 + 2 + 3 + 4 + 5 + 6 + ⋯ 8 + 1 6 + 2 4 + 3 2 + 4 0 + 4 8 + ⋯ = 8
and so he reasons that he needs 8 bags of mortar for every 1000 bricks used. Is he right? Remember, he is also a mathematician.
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1 solution
Regardless of the occupation of the contractor, the sums don't diverge! :D
Oh is this your "philosophically I have problems with this issue." problem?
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You're getting it now. This makes perfect sense to a mathematician, but explain that to a regular brick contractor.
The best analogy I have as to why we have these kinds of conventions is similar to why computer programmers follow certain guidelines---to keep things from blowing up. However, that does not necessarily mean there aren't valid alternative ways of dealing with this. As I said in that other problem, there's just too much to go into with this. It's like arguing about Cantor's infinite sets---and that one is not even settled.
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This is a trick question: this brick contractor committed a fallacy and is not a true mathematician.
See No true Scotsman .
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@Pi Han Goh – See Is Infinity / Infinity = 1? How can anybody argue with that?
Are you suggesting one set of rules for mathematicians and another for bricklayers?
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@Michael Mendrin – Haha, my "trick question response" is complete sarcasm. I thought you would pick up on that.
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@Pi Han Goh – Yeah but I just wanted to get in the comment about having different set of rules for different "occupations".
Really, though, what would your suggestion be for any alternative way of dealing with this? To a bricklayer, it's abundantly clear that it's 8 bags of mortar per 1 0 0 0 bricks, but he is being told he is wrong. Tell him don't do fractions in that way?
I also invite Michael Huang to come in with some alternatives as well.
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but he is being told he is wrong. Tell him don't do fractions in that way?
Tell him about order of operations . You need to add up all the positive integers first before doing the division.
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@Pi Han Goh – The simplest explanation would be to tell the contractor to get rid off the ellipsis (...). Problem solved! Suddenly he is perfectly correct in saying that it's 8 bags per 1000 bricks. And that gives one a clue of an alternative way of dealing with this.
A major problem with infinite series is the ambiguity inherent in trying to add (or multiply) an infinite number of terms. Just by rearranging the terms we could come up with different results. What if infinity itself is just a limiting value, and not a real thing? And then we can specify exactly how the terms are to be summed, such as "the first n terms"---and should that parameter n drop out in the evaluation, then the resulting value should be valid. This is actually not so different from the problem of finding limits to certain functions---it can be path dependent.
The way I see it, mathematics is rife with difficulties stemming from trying to grapple with infinities and we are forever creating new "flavors" of infinities, with ever more relations between them. I just wonder if there's a better alternative.
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@Michael Mendrin – Good point you made here!
Another question: What if suppose we add zeroes in both numerator and denominator aka 0 + 0 + 0 + ⋯ 0 + 0 + 0 + ⋯
I was intentionally being funny with the mathematics and how the contractor adds numbers in the numerator and denominator like that in the problem.
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@Michael Huang – Oh I thought you were being sincerely funny!
Here, even if we had a finite number of terms, so that we're not dealing with any possible divergence, the fraction is still indeterminate. So, we're still stuck with that.
Mathematics right now has a kind of a toolkit of rules about dealing with these kinds of things, but I heartily do not believe that such a toolkit is uniquely the only useful one. I'm suggesting other possible toolkits, but that is probably beyond the scope of this problem or note here.
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@Michael Mendrin – Wish I were either Ramanujan or Abel. The thing is: it's hard to find the unique topic to create and prove. I wish I can create my own work, discovering the possible summations.
Other than adding and subtracting terms in the numerator and denominator, I believe there isn't much to see for the existence of summation fraction.
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@Michael Huang – Let me illustrate. Compare the following two expressions
k = 1 ∑ ∞ k k = 1 ∑ ∞ 8 k
n → ∞ L im k = 1 ∑ n k k = 1 ∑ n 8 k
The first one is indeterminate. The second one has the value 8 . Now, here's where the ambiguity comes in. When that brick contractor wrote that fraction, which one was he thinking of?
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@Michael Mendrin – He was actually thinking the second expression.
slap the contractor
How dare you fool us with that kind of equation?! You are the contractor, not some sort of mathematician! Argh. Seems like you don't have university degree in Mathematics! ^.^
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Michael, do you have any ideas of how else this can be handled, other than to tell the bricklayer, "don't do fractions like that!"?
Is it necessarily invalid to evaluate fractions in this way, the way the bricklayer is trying to do it?
If the masonry contractor is a mathematician, he knows better to try to divide an infinity by an infinity. The fraction is indeterminate
See Is Infinity / Infinity = 1?