*Cape Cod Times* reports:

[Headline:] Truro zoning decision hinges on single vote

Voters narrowly approved one of four zoning amendments late Tuesday night at the annual town meeting. But town officials were still looking at the exact vote count on that article yesterday.

In a vote of 136 to 70, voters passed a new time limit on how quickly a cottage colony, cabin colony, motel or hotel can be converted to condominiums....

The exact count of the vote — 136 to 70 — had town officials hitting their calculators yesterday. The zoning measure needed a two-thirds vote to pass. A calculation by town accountant Trudy Brazil indicated that 136 votes are two-thirds of 206 total votes, said Town Clerk Cynthia Slade.

Brazil said she used the calculation of .66 multiplied by 206 to obtain the number.

But using .6666 — a more accurate version of two-thirds — the affirmative vote needed to be 137 instead of 136, according to an anonymous caller to town hall and to the Times.

Slade said that she called several of her colleagues to see how they calculate a two-thirds vote, and the answer varied widely. In Provincetown, Town Clerk Doug Johnstone uses .66. But Johnstone said he'd never had a close vote where it might matter.

A spokesman from the Secretary of State's office was not available to comment yesterday.

Slade said she will let the state Attorney General's office decide on the correct count, as part of their normal review of town meeting decisions....

Now let's be careful here, since there are so many problems here that it's easy to miss some. (I assume, by the way, that the article is correct in saying that a two thirds vote is required to pass the amendment, which the Truro Charter seems to confirm.)

1. Voters didn't "narrowly approve[]" the amendment; they narrowly disapproved it. It doesn't matter what the town official says; 136 is not 2/3 of 206. [UPDATE: The reporter e-mails me, in response to a query, "the article is considered to have passed by town meeting. but it will be reviewed (as all general and zoning bylaw articles are) by mass. attorney general before it is enacted." But whatever the town authorities might have said, the *voters* didn't narrowly approve the amendment — their vote fell short of the legally required 2/3.]

2. The affirmative vote needed isn't 137, since 2/3 of 206 is 137.333..., which means 137 is less than 2/3 of 206. The affirmative vote needed is 138.

3. Therefore, the zoning decision didn't "hinge[] on [a] single vote" — even if the one vote had changed, so the total would be 137 to 69, still not 2/3. To pass with 206 people voting, two voters would have had to switch sides.

4. I realize this item is a bit pedantic, but .6666 isn't a more accurate version of two-thirds. There is only one 2/3, not multiple versions. .6666 is a more accurate *approximation* to 2/3 than is .66.

5. If you need to calculate 2/3 on a calculator, there's no need to choose an approximation of two-thirds — you can just multiply by 2 and divide by 3.

6. But beyond this, who needs "an anonymous caller to town hall and to the Times," or even a calculator here? If you need 2/3 ayes (and assuming, as the article does, that this is 2/3 of those who cast a vote, not 2/3 of those present), then the only question is whether there are at least twice as many ayes as nays. 136 is less than two times 70, so 136 is less than 2/3 of the total.

7. It's not just one town, but two! And the answers in different towns "varied widely," so who knows what else is going on there?

8. They need the state Attorney General's office to multiply 2/3 in Massachusetts.

9. Finally, the best part — this was supposedly done by *the town's accountant*. Hey, a lawyer I could understand, but shouldn't the accountant be the one person who knows better?

Thanks to Adam Bonin for the pointer.

In order to repair the fan clutch, one might have to calculate the appropriate thickness of a shim. So I directed the mechanic to take a particular measurement, divide it by half, and then look the result up on a table to find the right shim.

The Sergeant who was in charge of testing the manuals said, "If you want the mechanics to divide by 2 you have to include a calculator in the required tool kit."

We decided to double the table ranges instead.

As an aside, I like point 6. I'm good at math, but every so often someone comes up with an incredibly simple, and glaringly obvious factoring that makes me feel a bit more humble.

8. They need the state Attorney General's office to multiply 2/3 in Massachusetts.You

rascal!IOW, you directed him to multiply by two.

Yes, yes it does. I've already started.

I find myself doing this sort of thing a lot these days. It may be because my mind is somewhat mathematical, but not mathematical enough to do anything interesting in my head.

Of course, the multiplying by two and dividing by three gives the answer that two people would have had to change their votes, which my calculation really didn't do for me.

Anyone who went to school after that had no hope until Texas Instruments introduced the portable calculator some 10 years later. Tom Lehrer wrote a song about it, and Losantiville described the sad results.

So thank you to my teachers for grades 3 through 5, Mrs. Ott, Miss Hurwitz and Mrs. Monson (I hope I spelled that one right, it's been a while), and to my parents for not letting me skip my homework.

Anyway, here's the part I'm stuck on:

138 is

alsoless than twice-70, yet it *is* greater than two-thirds of 206 total voters -- but is not more than twice the number of votes against. So there's a part of my brain that wants 140 to be the correct answer, but that requires there to be more voters.206 * 2 / 3

Gives you the answer right off.

figure out the percentage of the ayes (136/207=.6570)--.

LOL. How's your addition? Want a calculator for that?

Adam B.wrote:If there are 138 ayes and 70 nays, there are 208 votes total, not 206. 138/208 ~= 66.35%.

Remember that the Indiana state legislature once tried to set pi equal to a rational number.

.

Said another way, just weight each NAY vote double, and do a direct comparison. Now, if "greater than less than" poses a problem, we're in politician territory.

This is pretty lame on Truro's part. As a Town Meeting member in my town (MA towns can either be "open" (any registered voter can debate and vote) or "representative" (any registered voter can debate, but only elected Town Meeting members (there are between 50 and 300 of them, depending on the town) can vote)) the Moderator and Clerk always use the 2:1 rule when declaring whether a 2/3rds vote passes or fails. And our town always lists such an item as "defeated" when it fails to get 2/3rds, even if it gets a simple majority.

The scientists and engineers where I work were having an argument over whether 0.999999.... (infinitely repeating 9s) is approximately 1.0 or is 1.0. (Hint, if 9x = 9, what does x equal? 10x - x = 9x. No disappearing gnome tricks, it's real.)

But sheesh, hadn't anyone at the town meeting ever seen a puzzle involving bags of coins and balance scales?

The .66 practice sounds like just a local custom, and I know of no legal rule that would allow city officials to implement a city charter's express reference to "two-thirds vote" by using .66 instead.

Some of us used chalk and slate.

The two thirds majority was actually pretty relevant when the town I grew up in wanted to buid a new high school. The bond needed to pass with 2/3's of the vote and repeatedly came up just shy of that in the 65 - 66.6 % range and it took a couple of elections to get to two thirds + 1. A few years after the bond passed California voters passed proposition 39 to reduce the 2/3rds majority to a 55% majority when voting for local school bonds.

However, once the number of significant digits goes to three, then .667 becomes more appropriate four digits is 0.6667 and so on...

They are one and the same, and the thing they are the same as is... one.

It's hardly a proof (although a proof isn't that hard), but consider that 2/3 = 6/9 = .666....

7/9 = .777....

8/9 = .888....

9/9 = ?

or, 1/3 = .3333... * 3 = .999... = 1.

10x-x=9x i.e.

10x -> 9.9999999.......

x -> .9999999.......

as can be readily seen, there is exactly one "9" on the lower line for each "9" on the upper line, so when you subtract, you get

10x-x -> 9.00000000.......

or

9x = 9

so x = 1

the same method works for any infinitely repeating fraction, so, e.g.

x -> .707707..... where the 707 repeats forever

1000x -> 707.707707.....

so 999x = 707

so x = 707/999 a fraction in (I think) reduced form...

Anbody remember when Walt Kelly had a series of cartoons in Pogo dealing with this very issue? Try googling "SNORBERT ZANGOX over in Waycross" or go to http://www.langston.com/Fun_People/1994/1994AXG.html

If the whole article is correct it would be as silly as the most insulting comments claim - it's not like 2/3 votes in Massachusetts town meetings are rare; any bonds, zoning bylaw changes, transfers from Stabilization (a form of municipal savings), tax limit (prop 2.5) overrides, as well as a number of less common votes are all 2/3 vote to pass, not a simple majority.

I will test the local town clerk to see if she can get it right (I suspect she will, but it should be fun). The town I live in never needed an accountant to add votes up and do simple math; we do use someone from the town's Finance Committee to double check and certify the math of the town clerk.

Oddly enough, you can do the same with pencil and paper, as many of us learned to do up through the early 1960s. Then the schools replaced arithmetic with the new math. Anyone who went to school after that had no hopeOddly enough, I went to elementary school in the early 70s, didn't have any access to electronic calculators until maybe the 8th grade, and would have had no problem getting the correct answer by the 6th grade at the latest.

Nitpick, but is a town accountant a "politician?"In MA, they are a hired employee, at least in every town I know of.

To those who proffered an argument for why 1 = .999..., you do know that your argument requires that you first prove that addition/subtraction are well-defined for the class of infinite series to which .999... belongs?

notinvolve asking the nearest talking computer. Since it involves making marks on paper according to a rigid set of rules with a stick of graphite encased in wood, he calls it "graphitics."It's weird that anyone would think a plausible decimal approximation to 2/3 is 0.66. Why not 0.6? Or just 1? They're all equally (in)valid, because they're all constructed the same way, by simply dropping digits after the last one you feel like keeping.

I vaguely recall the Mass tax rate being 5%, that is, you take your income and multiply by 0.05 to get your tax. What a shame I didn't think of "approximating" my tax by dropping the second decimal place and hence multiplying my income by 0.0 to get the tax.

Everyone casting one of those votes ought to be required to read the book "Innumeracy."

To those who proffered an argument for why 1 = .999..., you do know that your argument requires that you first prove that addition/subtraction are well-defined for the class of infinite series to which .999... belongs?They are well defined due to the uniform convergence of Cauchy sequences, or something like that (the finite partial sums of an infinte series form a Cauchy (and therefore uniformly convergent) sequence. You can prove all this from the definitions of the terms involved, and it all becomes quite tautological

While it is unlikely to significantly alter the outcome, doing this will still result in using an approximation of two-thirds. Calculators are usually working with approximations of the non-integer numbers because of the way that their math libraries work. In this case, the calculator would give a better approximation than the accountant. But still an approximation.

What was interesting to me is that I could tell that they were wrong without actually thinking about it. I found that I had doubled 70 to 140, and 136 is less than 140. It took longer to figure out what I had done than to actually do it.I bet you went to elementary school before year dot, whatever it was, like I did.

The .66 practice sounds like just a local customCalculated to avoid invoking the devil's number. Wrong result, but a greater evil avoided.

Ithink this is pathetic.I can't tell you how happy I am that I went to school before calculators came along.

Put it this way. Suppose the task is to compute 2/3 of the total vote in 10 precincts, not just 1. A town clerk or other innumerate could imagine doing it two ways:

(1) Add all the votes, multiply by 0.67, round to the nearest vote.

(2) Multiply each precinct's vote by 0.67, round to the nearest vote, add them up.

Hopefully it's obvious (2) gives the more inaccurate answer. So there

isan important distinction between approximating 2/3 and approximating the result of multiplying a certain number by 2/3.Nitpicking, both your examples show aproxamation of an intermediate calculation. The way to aproxamate only the final result is to add the votes up, multiply by 2, then divide by 3.

It's been some years, but I do believe that even allowing transfinite numbers (and I suspect you mean infinitesimals, since these aren't transfinite - they're both certainly less than 1.1) the equation follows. The limit of the series 0.9, 0.99, 0.999,... is 1.0

Courts give abstract laws and principles concrete frameworks all the time, and it's not uncommon that when later courts apply these decision frameworks they sometimes reach results that seem to be in conflict with the plan meeting of these abstract principle taken alone, without interpretive precendents.

If "gold and silver coin" can include fiat paper money, surely one can't complain that letting "2/3" include .66 can represent a problem. It seems to me that if anything .66 has a much closer logical relationship to 2/3 -- it specifies a number of decimal places to use when using a calculator, which is what most people use, and one might as well specify some number of decimal places -- than "gold and silver coin" does to contemporary "legal tender" federal reserve notes.

Why not 0.6? Or just 1? They're all equally (in)valid, because they're all constructed the same way, by simply dropping digits after the last one you feel like keeping.The second example of that construction works only for exceptionally small values of 1. Like 0.

Nick

I can't do that, but there's an old Oregon case that holds that horse meat scrap that tests at 49.5 percent protein satisfies a contract clause that requires "not less than 50 percent" protein. I don't have the cite handy, but I discuss it in an article at 68 Ohio State L.J. 2 (2007).

Left" and has become "Left Behind".Or was this one of those "I need an answer now and don't have time to think" situations that we've all experienced?

No, sadly, not the case when you decide to ask for help.

Democracy is 2 wolves and a sheep voting on what to have for dinner and not being sure whether the vote has achieved the required 2/3.Well if we let 2/3 = .67, then in your example the wolves would need at least 2.01 votes to eat the sheep. So unless the sheep either had a death wish or abstained from voting, a sheep is legally safe from being eaten by two wolves in Massachusetts.

The "two-thirds" in the law does not mean some decimal approximation. It means *2 and /3. (or, equally, /3 and *2)

Most cell phones have a calculator, so even the pencil-challenged could have done it on the spot.

The result of 206*(2/3) is 137.3, so 137 is less than 2/3 and the required vote is 138. Period.

All the endless discussions of decimal near-equivalents could be useful in some other topic, but not here.

The in-my-head "reality check" comments, however, deserve another bit of attention: The 70-140 balance is no help because it would apply correctly only with 210 total votes.

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