Banknotes featuring scientists and mathematicians. Including the two in-print US bills that we're all least likely to see: the $100 (Franklin) and the $2 (Jefferson). For the non-US readers: the $100 is the largest bill in general circulation. For some reason the $2 bill has fallen out of favor, and although it's legal it's very rare, to the point that some people don't know about them and urban legends circulate about the $2 being suspected as counterfeit)
There seem to be more "scientists" than "mathematicians" on the list, but this may just reflect the fact that there are more scientists than mathematicians in general. In fact, "scientist" is a broad enough category that I don't think too many people would describe themselves as "scientists" when asked "what do you do?", rather responding with something like "physicist" or "biologist"; but I think a lot of mathematicians would answer "I'm a mathematician" to this question. (This seems to correspond roughly with the way departments are organized in most universities; there's usually a "department of mathematics" but very rarely a "department of science".)
(via a comment at Gil Kalai's blog)
Edit, 6:20 pm: the linguists seem to be compiling their own list of linguists-on-money, over at Language Log.
Showing posts with label money. Show all posts
Showing posts with label money. Show all posts
18 June 2009
04 March 2009
A fool and his money are soon parted
Did you know that there are people who think that by reducing their income from over $250,000 to under $250,000, they can take home more money? For those of you who aren't aware of this, President Obama is planning to increase taxes on families earning more than $250,000 per year.
Of course, the way the US tax code is set up, the amount of tax you pay as a function of your taxable income is continuous, monotone increasing, and Lipschitz with parameter 1. That is, say that T(x) is the tax due if your taxable income is x. Let y > x. Then T(y) > T(x), and T(y) - T(x) < y - x. As you may note, you can derive from the second of these that
y - T(y) > x - T(x)
which tells us that if you make more money, you get to keep more of your money.
Note that T is actually not differentiable, because it's piecewise linear. Your "tax bracket" is in fact the amount of tax you pay on the last dollar of your income; that is, it's T'(x) where x is your income.
I'm not saying that there are no situations where this sort of thing might make sense. (The tax code is complicated.) But it's certainly not as common as these people would have you believe.
Original article from ABC News; I followed a link from The New York Times via The New Republic.
Of course, the way the US tax code is set up, the amount of tax you pay as a function of your taxable income is continuous, monotone increasing, and Lipschitz with parameter 1. That is, say that T(x) is the tax due if your taxable income is x. Let y > x. Then T(y) > T(x), and T(y) - T(x) < y - x. As you may note, you can derive from the second of these that
y - T(y) > x - T(x)
which tells us that if you make more money, you get to keep more of your money.
Note that T is actually not differentiable, because it's piecewise linear. Your "tax bracket" is in fact the amount of tax you pay on the last dollar of your income; that is, it's T'(x) where x is your income.
I'm not saying that there are no situations where this sort of thing might make sense. (The tax code is complicated.) But it's certainly not as common as these people would have you believe.
Original article from ABC News; I followed a link from The New York Times via The New Republic.
25 October 2007
Math for America plays with numbers
From the November 2007 Notices, an ad for Math for America (p. 1305):
"Do you know someone who loves π as much as pie? Would they also love a full-tuition scholarship for a master's degree in mathematics education, a New York State Teaching Certificate, and a $90,000 stipend in addition to a competitive salary as a New York City secondary school math teacher? Math for America... [etc.]"
I don't know about you, but when I see a five-figure number followed by the word "stipend" I automatically assume it's an annual stipend. It seems somehow disingenuous to put that number there; it turns out it's a five-year stipend. This is in addition to the usual salary one gets for teaching, though; the idea appears to be that this program is attracting teachers who actually know math by making up at least some of the difference between what they would make teaching and what they could make elsewhere. Also, the people in this program receive a full-tuition scholarship for a master's in math ed.
They don't report it as "$18,000 per year for five years" is because it's not; it's paid as $28,000 in the first year (which is mostly spent being trained as a teacher, and which doesn't carry a salary) and $11,000, $14,000, $17,000, and $20,000 in the second through fifth years (these are in addition to the usual salary a New York City public school teacher would receive). Still, why not say "$90,000 over five years"? I feel like they're trying to fool the people reading the ad -- but the people reading the ad are the people in the world who are least likely to be fooled by tricks played with numbers.
I'm not saying that the financial package isn't valuable. I'm just saying that it feels like the ad is hiding something because they don't give the time period.
It reminds me of a letter I got from a graduate school which claimed that my support package was something like $50,000 per year, which was actually $20,000 in annual stipend and $30,000 in tuition. (This was a school with a comparatively high tuition, obviously; I suspect they did this because the $50K number was larger than schools with lower tuitions but the same stipend would have reported.) But anybody who's actually comparing financial offers would think of them as "full tuition plus [dollar amount]" and not even care what the dollar amount of the full tuition was.
"Do you know someone who loves π as much as pie? Would they also love a full-tuition scholarship for a master's degree in mathematics education, a New York State Teaching Certificate, and a $90,000 stipend in addition to a competitive salary as a New York City secondary school math teacher? Math for America... [etc.]"
I don't know about you, but when I see a five-figure number followed by the word "stipend" I automatically assume it's an annual stipend. It seems somehow disingenuous to put that number there; it turns out it's a five-year stipend. This is in addition to the usual salary one gets for teaching, though; the idea appears to be that this program is attracting teachers who actually know math by making up at least some of the difference between what they would make teaching and what they could make elsewhere. Also, the people in this program receive a full-tuition scholarship for a master's in math ed.
They don't report it as "$18,000 per year for five years" is because it's not; it's paid as $28,000 in the first year (which is mostly spent being trained as a teacher, and which doesn't carry a salary) and $11,000, $14,000, $17,000, and $20,000 in the second through fifth years (these are in addition to the usual salary a New York City public school teacher would receive). Still, why not say "$90,000 over five years"? I feel like they're trying to fool the people reading the ad -- but the people reading the ad are the people in the world who are least likely to be fooled by tricks played with numbers.
I'm not saying that the financial package isn't valuable. I'm just saying that it feels like the ad is hiding something because they don't give the time period.
It reminds me of a letter I got from a graduate school which claimed that my support package was something like $50,000 per year, which was actually $20,000 in annual stipend and $30,000 in tuition. (This was a school with a comparatively high tuition, obviously; I suspect they did this because the $50K number was larger than schools with lower tuitions but the same stipend would have reported.) But anybody who's actually comparing financial offers would think of them as "full tuition plus [dollar amount]" and not even care what the dollar amount of the full tuition was.
28 August 2007
the lighter side of financial mathematics
Engraved Portraits of Gauss for sale, just 40 dollars!
These are in fact 10 Deutsche Mark notes. I have the feeling that the seller, Acme Klein Bottle, sold them at a lower price before 2002.
I'm kind of tempted to order one, but forty bucks is forty bucks, and I'm a grad student. It kind of seems like a nice conversation piece, though. Besides, for two dollars more I could get a Klein bottle. Or I could get a Klein bottle hat. I knew someone who tried to knit one of these once; I don't remember if she succeeded. Acme also sells Mobius band scarves, which I suspect would be quite annoying because I like my scarves to have ends. These would have novelty value and keep me warm.
(You might also consider these portraits of Euler, or any of the portraits from this gallery. Rather strangely, all the portraits have numbers on them.)
But let's say, hypothetically, I bought the portrait of Gauss. I could hang it on my wall and people would wonder why I hung money on my wall instead of spending it. It would kind of be like a Knuth reward check -- Knuth pays a bounty of $2.56 for each error people find in his books.
In 2002, Knuth said in this article in the notices of the AMS, when asked what would happen if all his reward checks were cashed:
On the contrary, Erdos once said that he would not be able to pay out all the rewards he had put on various problems, and compared this to that the strongest bank would not be able to survive if all its customers simultaneously wanted their money -- but believed the bank run to be more likely. I'm kind of curious if there's a list of Erdos problems out there. This article indicates that people usually did cash Erdos checks, perhaps because the amounts of money involved are greater -- and back then, you got a cancelled check back from the bank anyway. Ronald Graham estimates that the total outstanding bounties on Erdos problems are about $25,000, although it seems he's not sure because as of that writing there was no list of them. I was able to track down a list of a couple dozen or so.
And while I'm talking about money: I previously wondered about the density of money, and I concluded that the density of U.S. coinage -- if we make certain reasonable assumptions about how change is given -- is $28.58 per kilogram. This was inspired by the fact that I had some change I needed to cash in.
I cashed in my change recently; I had 95 quarters, 126 dimes, 76 nickels, and 339 pennies, for a total of $43.44. This is also 2,052 grams, for a money density of $21.17 per kilogram. As I said in my earlier post, I tend to use quarters for laundry. One expects dimes, nickels, and pennies to occur in a 2:1:5 ratio in randomly occuring change; my actual experience is not too far from that. In randomly occuring change, though, one expects three-fourths as many quarters as pennies; I didn't have nearly that many quarters.
These are in fact 10 Deutsche Mark notes. I have the feeling that the seller, Acme Klein Bottle, sold them at a lower price before 2002.
I'm kind of tempted to order one, but forty bucks is forty bucks, and I'm a grad student. It kind of seems like a nice conversation piece, though. Besides, for two dollars more I could get a Klein bottle. Or I could get a Klein bottle hat. I knew someone who tried to knit one of these once; I don't remember if she succeeded. Acme also sells Mobius band scarves, which I suspect would be quite annoying because I like my scarves to have ends. These would have novelty value and keep me warm.
(You might also consider these portraits of Euler, or any of the portraits from this gallery. Rather strangely, all the portraits have numbers on them.)
But let's say, hypothetically, I bought the portrait of Gauss. I could hang it on my wall and people would wonder why I hung money on my wall instead of spending it. It would kind of be like a Knuth reward check -- Knuth pays a bounty of $2.56 for each error people find in his books.
In 2002, Knuth said in this article in the notices of the AMS, when asked what would happen if all his reward checks were cashed:
There's one man who lives near Frankfurt who would probably have more than $1,000 if he cashed all the checks I've sent him. There's a man in Los Gatos, California, who I've never met, whom I've never met, who cashes a check for $2.56 about once a month, and that's been going on for some years now. Altogether I've written more than 2,000 checks over the years, and the average amount exceeds $8.00. Even if everybody cashed their checks, it would still be more than worth it to me to know that my books are getting better.Knuth didn't answer the question directly, but I assume he'd be okay if they all were cashed -- I have a feeling he's got some money saved up.
On the contrary, Erdos once said that he would not be able to pay out all the rewards he had put on various problems, and compared this to that the strongest bank would not be able to survive if all its customers simultaneously wanted their money -- but believed the bank run to be more likely. I'm kind of curious if there's a list of Erdos problems out there. This article indicates that people usually did cash Erdos checks, perhaps because the amounts of money involved are greater -- and back then, you got a cancelled check back from the bank anyway. Ronald Graham estimates that the total outstanding bounties on Erdos problems are about $25,000, although it seems he's not sure because as of that writing there was no list of them. I was able to track down a list of a couple dozen or so.
And while I'm talking about money: I previously wondered about the density of money, and I concluded that the density of U.S. coinage -- if we make certain reasonable assumptions about how change is given -- is $28.58 per kilogram. This was inspired by the fact that I had some change I needed to cash in.
I cashed in my change recently; I had 95 quarters, 126 dimes, 76 nickels, and 339 pennies, for a total of $43.44. This is also 2,052 grams, for a money density of $21.17 per kilogram. As I said in my earlier post, I tend to use quarters for laundry. One expects dimes, nickels, and pennies to occur in a 2:1:5 ratio in randomly occuring change; my actual experience is not too far from that. In randomly occuring change, though, one expects three-fourths as many quarters as pennies; I didn't have nearly that many quarters.
10 August 2007
links for 10 August
Why Stuff Is Hard, at the Everything Seminar. I asked this in physics class in high school and never got a satisfactory answer. I suspect I'm not alone here. I was told it was some sort of electromagnetic repulsion between the outermost electrons, but apparently the Pauli exclusion principle is really doing most of the heavy lifting.
Mark Chu-Carroll at Good Math, Bad Math comments on the way math is taught at certain religious schools. If you don't want to bother reading the post, check out the course descriptions at one such school. They all start out "Students will examine the nature of God as they progress in their understanding of mathematics". The descriptions of the non-mathematics courses begin similarly. Today I was reading parts of Laplace's A Philosophical Essay on Probabilities (available in Hawking's anthology God Created the Integers: The Mathematical Breakthroughs That Changed History
); among other things, Laplace mocks the idea of Pascal's wager. I couldn't help but thinking of what Laplace is said to have said to Napoleon when asked why he didn't mention God in his work on celestial mechanics: "I had no need of that hypothesis."
Compound interest isn't intuitive, from Adventures of BruteForce; if you invest a little money now that's like investing a lot of money later. People just aren't set up to understand exponential growth, which isn't surprising; unrestrained exponential growth isn't common in the situations for which we evolved. A population can't keep doubling every ten years without pretty quickly running out of space; a sum of money can. There's a persistent rumor that Ashkenazi Jews are actually better equipped for understanding this particular sort of abstraction than other classes of people, because of certain unique historical circumstances -- for quite some time they lived among Christians, who were forbidden to lend money for religious reasons, but these same Christians wanted to borrow money, and therefore turned to the Jews, who were not subject to those same religious laws. The Jews who were better at understanding this fact ended up with more money themselves, their kids didn't starve, and supposedly this explains why about a quarter of Nobel laureates are Jewish. I can't find exact numbers overall. One thing I can find is in this Wikipedia article which says that "Of American Nobel Prize winners, 37% have been Jewish Americans (19 times the percentage of Jews in the population) [...]" But I'm not sure who they count as "American". (Wikipedia used to have a list of Jewish Nobel laureates, but it's been deleted.)
It would be interesting if this were true, because it seems to imply that evolution can work ono the scale of a few hundred years. As humanity heads more and more towards working with its brains instead of its hands, will we get smarter? (On the other hand, at least at the present time in the United States, intelligence and number of children seem to be inversely correlated; it seems difficult for Darwinian evolution to work in a population when almost everyone survives long enough to have children.)
Mark Chu-Carroll at Good Math, Bad Math comments on the way math is taught at certain religious schools. If you don't want to bother reading the post, check out the course descriptions at one such school. They all start out "Students will examine the nature of God as they progress in their understanding of mathematics". The descriptions of the non-mathematics courses begin similarly. Today I was reading parts of Laplace's A Philosophical Essay on Probabilities (available in Hawking's anthology God Created the Integers: The Mathematical Breakthroughs That Changed History
); among other things, Laplace mocks the idea of Pascal's wager. I couldn't help but thinking of what Laplace is said to have said to Napoleon when asked why he didn't mention God in his work on celestial mechanics: "I had no need of that hypothesis."Compound interest isn't intuitive, from Adventures of BruteForce; if you invest a little money now that's like investing a lot of money later. People just aren't set up to understand exponential growth, which isn't surprising; unrestrained exponential growth isn't common in the situations for which we evolved. A population can't keep doubling every ten years without pretty quickly running out of space; a sum of money can. There's a persistent rumor that Ashkenazi Jews are actually better equipped for understanding this particular sort of abstraction than other classes of people, because of certain unique historical circumstances -- for quite some time they lived among Christians, who were forbidden to lend money for religious reasons, but these same Christians wanted to borrow money, and therefore turned to the Jews, who were not subject to those same religious laws. The Jews who were better at understanding this fact ended up with more money themselves, their kids didn't starve, and supposedly this explains why about a quarter of Nobel laureates are Jewish. I can't find exact numbers overall. One thing I can find is in this Wikipedia article which says that "Of American Nobel Prize winners, 37% have been Jewish Americans (19 times the percentage of Jews in the population) [...]" But I'm not sure who they count as "American". (Wikipedia used to have a list of Jewish Nobel laureates, but it's been deleted.)
It would be interesting if this were true, because it seems to imply that evolution can work ono the scale of a few hundred years. As humanity heads more and more towards working with its brains instead of its hands, will we get smarter? (On the other hand, at least at the present time in the United States, intelligence and number of children seem to be inversely correlated; it seems difficult for Darwinian evolution to work in a population when almost everyone survives long enough to have children.)
27 July 2007
checks for nothing, and why English is useful
Karl Fogel attempts to pay a bill for $0, because of course you have to pay bills for zero, because otherwise the companies that issue them keep sending them.
In this case, the bill was for the purchase of a book from the Mathematical Association of America, so he wrote a check for eiπ+1 dollars. They didn't deposit it, because "check needs to be wrote out in U. S. dollars", as they put it.
I can see a more legitimate reason for rejecting the check. Usually, when writing a check, one puts, say, "3.14" in the little box on the right, and "Three and 14/100" on the line. The reason for writing out the value of the check in both figured and words is for redundancy. (Although then why don't we write "three dollars and fourteen cents" on the line? I suppose redundancy doesn't matter quite as much when we're talking about sub-dollar amounts.)
So you might say he should have written "e to the i π plus one dollars" on the line. But even that seems a bit suspect, because e, i, and π are themselves bits of mathematical notation. It seems that he really should have written something like
"The base of the exponential function, raised to the product of the imaginary unit and the ratio of a circle's circumference to its diameter, plus one"
for the number of dollars he wanted. Of course, this is the sort of thing that makes it obvious why having a compact mathematical notation is a good idea. I am not enough of a mathematical historian to have looked at the way things used to be written, but from what I understand this is the sort of thing they would have written five centuries ago, and I can't imagine working like that.
Unfortunately, the fact that we have such a good mathematical notation creates another problem -- people think that they can just put a bunch of symbols on a page and not explain what they mean by them, and that's "mathematics". Terry Tao, at his blog, has lots of writing advice; of particular interest in this discussion is his advice to take advantage of the English language. Here he gives a couple dozen ways to say that two statements are true, which are logically equivalent but have a wide variety of connotations. To take two examples of his examples at random, "P(x) is true. Unfortunately, Q(y) is also true." and "P is satisfied by x. Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct. I think that mathematicians as a whole are not sensitive enough to the connotations of their words; this is useful when doing formal mathematics but not so useful when trying to express the results of it. Perhaps we kneel too much at the altar of Bourbaki.
In this case, the bill was for the purchase of a book from the Mathematical Association of America, so he wrote a check for eiπ+1 dollars. They didn't deposit it, because "check needs to be wrote out in U. S. dollars", as they put it.
I can see a more legitimate reason for rejecting the check. Usually, when writing a check, one puts, say, "3.14" in the little box on the right, and "Three and 14/100" on the line. The reason for writing out the value of the check in both figured and words is for redundancy. (Although then why don't we write "three dollars and fourteen cents" on the line? I suppose redundancy doesn't matter quite as much when we're talking about sub-dollar amounts.)
So you might say he should have written "e to the i π plus one dollars" on the line. But even that seems a bit suspect, because e, i, and π are themselves bits of mathematical notation. It seems that he really should have written something like
"The base of the exponential function, raised to the product of the imaginary unit and the ratio of a circle's circumference to its diameter, plus one"
for the number of dollars he wanted. Of course, this is the sort of thing that makes it obvious why having a compact mathematical notation is a good idea. I am not enough of a mathematical historian to have looked at the way things used to be written, but from what I understand this is the sort of thing they would have written five centuries ago, and I can't imagine working like that.
Unfortunately, the fact that we have such a good mathematical notation creates another problem -- people think that they can just put a bunch of symbols on a page and not explain what they mean by them, and that's "mathematics". Terry Tao, at his blog, has lots of writing advice; of particular interest in this discussion is his advice to take advantage of the English language. Here he gives a couple dozen ways to say that two statements are true, which are logically equivalent but have a wide variety of connotations. To take two examples of his examples at random, "P(x) is true. Unfortunately, Q(y) is also true." and "P is satisfied by x. Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct. I think that mathematicians as a whole are not sensitive enough to the connotations of their words; this is useful when doing formal mathematics but not so useful when trying to express the results of it. Perhaps we kneel too much at the altar of Bourbaki.
13 July 2007
ten thousand pennies
Philadelphia Fish & Company is running a promotion in which, once the Phillies lose their ten-thousandth game, anybody who brings in ten thousand pennies can get a dinner for ten.
This is from the Don Polec's World segment on WPVI's 6 PM newscast yesterday. (To see the actual segment, look at the menu under the "Action News on demand".) The guy who came up with the promotion, Kevin Meeker, says it comes from an ancient Greek tradition -- when the ancient Greeks had lost ten thousand men in battle, they had a feast of fish and it was believed to bring them good luck.
(To answer the obvious question -- you actually have to bring ten thousand pennies. You can't just show up with a $100 bill. And they have to be rolled. He wants people to suffer.)
The Phillies are currently at 9999 losses. The probability of the Phillies' ten-thousandth loss tonight, which I originally looked at here and re-examined here and here? Just under 40%. The full distribution -- which is now nothing more than the distribution of the time until the team's next loss is as follows:
And how much do ten thousand pennies weigh? Twenty-five kilograms, or about fifty-five pounds.
This is from the Don Polec's World segment on WPVI's 6 PM newscast yesterday. (To see the actual segment, look at the menu under the "Action News on demand".) The guy who came up with the promotion, Kevin Meeker, says it comes from an ancient Greek tradition -- when the ancient Greeks had lost ten thousand men in battle, they had a feast of fish and it was believed to bring them good luck.
(To answer the obvious question -- you actually have to bring ten thousand pennies. You can't just show up with a $100 bill. And they have to be rolled. He wants people to suffer.)
The Phillies are currently at 9999 losses. The probability of the Phillies' ten-thousandth loss tonight, which I originally looked at here and re-examined here and here? Just under 40%. The full distribution -- which is now nothing more than the distribution of the time until the team's next loss is as follows:
| Jul 13 | v. Cardinals | 0.394329 |
| Jul 14 | v. Cardinals | 0.238834 |
| Jul 15 | v. Cardinals | 0.144655 |
| Jul 16 | @ Dodgers | 0.130046 |
| Jul 17 | @ Dodgers | 0.053929 |
| Jul 18 | @ Dodgers | 0.022364 |
| Jul 19 | @ Padres | 0.009184 |
| Jul 20 | @ Padres | 0.003861 |
| Jul 21 | @ Padres | 0.001623 |
| Jul 22 | @ Padres | 0.000682 |
| Jul 24 | v. Nationals | 0.000174 |
| Jul 25 | v. Nationals | 0.000113 |
| Jul 26 | v. Nationals | 0.000073 |
| Jul 27 | v. Pirates | 0.000048 |
| Jul 28 | v. Pirates | 0.000031 |
| Jul 29 | v. Pirates | 0.000020 |
| Jul 30 | @ Cubs | 0.000018 |
| Jul 31 | @ Cubs | 0.000009 |
| Aug 01 | @ Cubs | 0.000004 |
| Aug 02 | @ Cubs | 0.000002 |
| Aug 03 | @ Brewers | 0.000001 |
| Aug 04 | @ Brewers | 0.000001 |
06 July 2007
the density of money
I have a jar of assorted coins. I sort out the quarters separately (because I need them for laundry) but otherwise every so often I throw my change in here. It's getting kind of heavy; one of these days I'll cash it in for real money.
As I was lifting it this morning, I began to wonder -- if I knew how much it weighed, could I tell from that approximately how much money it contained?
Then I went to buy breakfast -- which cost me $5.25, and I paid with a $20 bill. I got $14.75 in change -- a ten, four ones, seven dimes and a nickel, because there were no quarters. The woman working the cash register said she was sorry they were out of quarters; I replied that dimes were okay because at least they're small.
So, here's the question: what's the density of money in, say, dollars per kilogram?
A U.S. dime weighs 2.268 grams; that's $44.09 per kilogram.
A U.S. quarter weighs 5.670 grams; that's also $44.09 per kilogram.
A U. S. nickel weighs 5.000 grams; that's $10.00 per kilogram. Wikipedia says that"nickels have always had a value of one cent per gram", which is interesting if true in part because nickels have been minted since 1866. Clearly the designers of the nickel were thinking in metric.
A U.S. penny weighs 2.5 grams; that's $4.00 per kilogram.
I presume the fact that the quarter is exactly two and one half times the weight of the dime has something to do with the fact that they're both made out of the same alloy, an 11:1 copper/nickel mixture; the nickel is three-fourths copper and one-fourth nickel; the penny is 97.5% zinc and 2.5% copper.
Right now, metalprices.com reports that copper sells for $7.895/kilogram; zinc, $3.436/kilogram; nickel, $36.20/kilogram. Thus dimes and quarters, if you melt them down, could sell for $10.25/kg; nickels, $14.97/kg; pennies, $3.54/kg. I'm surprised to learn that nickels are worth less than the metal underlying them, because you hear this more often about pennies, even though it's not actually true. It does, however, cost more to make a penny than that coin is worth, and the U. S. Mint has passed regulations about the melting down of pennies and nickels.
But what's the density of "money"? That's a bit trickier. Let's assume that on any given transaction, I pay with a whole number of dollars; furthermore assume that the "fractional part" of my change is equally likely to be 0, 1, 2, ..., 99 cents, and that it's given back to me with the smallest number of coins possible. The easiest way to do the computation is to assume that I make 100 transactions, in which I get 0, 1, 2, ..., 99 cents back. Now I have $49.50. How much does it weigh?
Well, twenty times I got 0 pennies; twenty times I got 1 penny; and so on up to 4 pennies. So I have 20*(0+1+2+3+4) = 200 pennies.
I'll never get more than one nickel. I get a nickel if the fractional part of my change is 5-9, 15-19, 30-34, 40-44, 55-59, 65-69, 80-84, or 90-94 cents; there are 40 numbers there. So I get 40 nickels.
The rest of what I get is dimes and quarters; since dimes and quarters have the same "money density" I won't distinguish between them. I get $45.50 worth of dimes and quarters. (In fact, I get 150 quarters and 80 dimes.)
Together, all these coins weigh 1732 grams; thus the density of money appears to be $28.58 per kilogram.
But in reality, it won't be nearly this much. I try to get rid of change when I'm carrying it, and a lot of businesses now set their prices so that they don't have to deal with nickels. (At one of my favorite coffee shops, all the prices are multiples of 25 cents. The problem with this is that people will bitch and moan when that inevitable day comes when they raise the price of a large coffee from $1.75 to $2; if they were willing to deal with nickels they'd only have to raise it to $1.80.) And I'm more likely to spend quarters than any other coin, because they work the laundry machine (They buy newspapers, too; the Inquirer costs 50 cents. The machines take nickels, dimes, or quarters, but usually I use two quarters.)
I can't weigh my jar of money to tell you what its actual density is -- I don't have a scale. But when I cash it in I'll let you know how many of each kind of coin the coin-counting machine says it had.
As I was lifting it this morning, I began to wonder -- if I knew how much it weighed, could I tell from that approximately how much money it contained?
Then I went to buy breakfast -- which cost me $5.25, and I paid with a $20 bill. I got $14.75 in change -- a ten, four ones, seven dimes and a nickel, because there were no quarters. The woman working the cash register said she was sorry they were out of quarters; I replied that dimes were okay because at least they're small.
So, here's the question: what's the density of money in, say, dollars per kilogram?
A U.S. dime weighs 2.268 grams; that's $44.09 per kilogram.
A U.S. quarter weighs 5.670 grams; that's also $44.09 per kilogram.
A U. S. nickel weighs 5.000 grams; that's $10.00 per kilogram. Wikipedia says that"nickels have always had a value of one cent per gram", which is interesting if true in part because nickels have been minted since 1866. Clearly the designers of the nickel were thinking in metric.
A U.S. penny weighs 2.5 grams; that's $4.00 per kilogram.
I presume the fact that the quarter is exactly two and one half times the weight of the dime has something to do with the fact that they're both made out of the same alloy, an 11:1 copper/nickel mixture; the nickel is three-fourths copper and one-fourth nickel; the penny is 97.5% zinc and 2.5% copper.
Right now, metalprices.com reports that copper sells for $7.895/kilogram; zinc, $3.436/kilogram; nickel, $36.20/kilogram. Thus dimes and quarters, if you melt them down, could sell for $10.25/kg; nickels, $14.97/kg; pennies, $3.54/kg. I'm surprised to learn that nickels are worth less than the metal underlying them, because you hear this more often about pennies, even though it's not actually true. It does, however, cost more to make a penny than that coin is worth, and the U. S. Mint has passed regulations about the melting down of pennies and nickels.
But what's the density of "money"? That's a bit trickier. Let's assume that on any given transaction, I pay with a whole number of dollars; furthermore assume that the "fractional part" of my change is equally likely to be 0, 1, 2, ..., 99 cents, and that it's given back to me with the smallest number of coins possible. The easiest way to do the computation is to assume that I make 100 transactions, in which I get 0, 1, 2, ..., 99 cents back. Now I have $49.50. How much does it weigh?
Well, twenty times I got 0 pennies; twenty times I got 1 penny; and so on up to 4 pennies. So I have 20*(0+1+2+3+4) = 200 pennies.
I'll never get more than one nickel. I get a nickel if the fractional part of my change is 5-9, 15-19, 30-34, 40-44, 55-59, 65-69, 80-84, or 90-94 cents; there are 40 numbers there. So I get 40 nickels.
The rest of what I get is dimes and quarters; since dimes and quarters have the same "money density" I won't distinguish between them. I get $45.50 worth of dimes and quarters. (In fact, I get 150 quarters and 80 dimes.)
Together, all these coins weigh 1732 grams; thus the density of money appears to be $28.58 per kilogram.
But in reality, it won't be nearly this much. I try to get rid of change when I'm carrying it, and a lot of businesses now set their prices so that they don't have to deal with nickels. (At one of my favorite coffee shops, all the prices are multiples of 25 cents. The problem with this is that people will bitch and moan when that inevitable day comes when they raise the price of a large coffee from $1.75 to $2; if they were willing to deal with nickels they'd only have to raise it to $1.80.) And I'm more likely to spend quarters than any other coin, because they work the laundry machine (They buy newspapers, too; the Inquirer costs 50 cents. The machines take nickels, dimes, or quarters, but usually I use two quarters.)
I can't weigh my jar of money to tell you what its actual density is -- I don't have a scale. But when I cash it in I'll let you know how many of each kind of coin the coin-counting machine says it had.
Subscribe to:
Posts (Atom)
