[RESOLVED] .6 with a line over it

[Spoo]

Was watching Jeopardy! the other night...

.. there was a .6 with a line over it.

The correct response was .. "What is 2/3?"

I'm not familiar with such a "notation" or "designation".

I believe the line is called a vinculum,
and that it means that the "6" is repeated
forever, such as 2/3 = .6666666666666....

What is this "notation" called?

Spoo

[szlamany]

It is a vinculum - but it has several meanings

http://en.wikipedia.org/wiki/Vinculum_(symbol)

Of course this is a STRAIGHT line - if it was a tilde it would indicate approximation...

[Spoo]

SZ

Yes, thanks for the Wiki ref .. but been there, done that.

What I was more interested in here was what the "notation" is called.
For example:

• Exponent .. a 2 with a superscript 2 (ie, 2^2)
• Root .. a "squiggly" line (aka radicand) with superscript (ie 2 -\/-- 2 = square root of 2)
• Repeater .. is that what "this" would be called?

Vinculum is the line, in a similar sense to superscript
Repeater is the term, in a similar sense to exponent (or "raise to the power of")

Is repeater the "term"? If so, sounds kinda weak.
I was hoping for a more elegant term.

Spoo

[szlamany]

Might not be exactly what you are looking for but this is a good read on the vinculum

http://www.pballew.net/arithme6.html

recurrent - recurring decimals - repetends

Circulating - circulating decimals - circulants

1/3 or .3333 is a pure circulate

.12333333 is a mixed circulate

I even saw reptile in one text - but that's really hard to google for because of it's obvious fauna-references...

[Spoo]

SZ

I support recurrent

Thanks

Spoo

[jemidiah]

0.6 with a line over the 6 would be read "point six repeating", so I suppose "repeating" is the verb form you're after. It's not a terribly important operation, so it doesn't get much attention outside of some basic school arithmetic. It's almost invariably more convenient to express the decimal number 0.(digits) as a fraction like (digits) / (the digit 9 repeated the length of 'digits' times): eg. 0.(142857) = 142857 / 999999 = 1/7. The proof of this equation relies on the geometric series formula,

a*(1 + x + x^2 + x^3 + ...) = a / (1-x)

0.(142857)
= 142857 / 1000000 + 142857 / 1000000^2 + 142857 / 1000000^3 + ...
= (142857 / 1000000) * (1 + 1/1000000 + 1/1000000^2 + ...)
= (142857 / 1000000) / (1 - 1/1000000)
= (142857 / 1000000) / (999999 / 1000000)
= 142857 / 999999

The general case is exactly analogous.


It's mildly interesting to consider how many digits are used before 1/n repeats. Note that the answer very much depends upon the base; for instance, 1/5 uses 0 repeating digits in base 10, while it uses 4 in base 2 (since 1/5 = 0.(0011)_2). There is actually an interesting pattern in these two examples alone: 1/7 used 6 = 7-1 digits (in base 10) and 1/5 used 4 = 5-1 (in base 2). 1/7 in binary uses 3 repeating digits, which at least evenly divides 6. Since I have some experience with these things, I'm lead to hypothesize that the number of digits 1/n uses to repeat in base b evenly divides n-1 when n is prime and when n does not divide b. Beforehand, we don't even know that the expansion will eventually repeat, though this property would ensure repetition in the relevant cases.

As noted above, we wish to find some integer N and some denominator of length M where...
1/n = N/(b^M - 1)
or
N*n = b^M - 1
or
b^M = 1 (mod n)

Fermat's Little Theorem ensures that there is a solution M to this last equation. Indeed, it says that the solution M divides n-1, just as I said initially. (I could have written this much more generally, but that would require Euler's generalization of Fermat's Little Theorem and more elaborate reasoning/definitions, so I decided against it.)

The smallest value of M which solves the above equation is called the "order" of b (mod n). So, to figure out the number of digits repeated in the base b expansion of 1/n, one need only compute the order of b (mod n). Interestingly, you can do this without even computing N--that is, you can answer the question of "how long until 1/n repeats in base b?" without actually computing 1/n in base b via long division.

Incidentally, this is a case where the convention 0^0 = 1 makes things work out nicely without having to add in special cases. Often 0^0 is left undefined, though there are a number of good arguments for using the above convention, apparently including this example.

[Spoo]

Jem

A bit more that I was anticipating, but very
nicely done. Thanks.

As far as ..

It's not a terribly important operation, so it doesn't get much attention outside of some basic school arithmetic.

.. it was, after all, a Jeopardy! question. ..

While I made it thru basic calculus in college
(albeit barely), the vinculum notation was,
while "basic", still a new one for me.

Thanks again.
Spoo

[jemidiah]

While I quite enjoy Jeopardy!, whenever they have a category I'm particularly knowledgeable about I'm always struck by how superficial the questions are. For instance, there was recently a "Yay, Physics!" category that required only a few bits of basic physics vocabulary: energy, torque, mechanics, normal force, and electromagnetism. Knowing the relationships between any of these concepts and knowing some mathematical models involving them are not at all necessary to answer those questions, even though they're absolutely fundamental to actually understanding physics.

And then a question on modern art comes up and I'm clueless.

[Spoo]

Jem

Haha!!
I couldn't agree more.

One of course has to take into consideration that
the show needs to be somewhat "dumbed-down" to
attract and audience, and hence advertisers.

Nonetheless, after watching a recent PBS NewsHour
piece about vocational training integration with
manufacturing in small German companies, only to
be followed by a Jeopardy! show wherein the Final Jeopardy
topic was Comic Books, I was sufficiently perturbed that I
resolved not to watch the Final outcome. Three minutes later,
I relented and low and behold, I got the answer!

The clue had to do with a 1920's comic about a boy.
It seemed to have a French connection, so the only thing
I could think of was Tin Tin. If I recall correctly, that may
have been the same show with the Yay, Physics category.

Movies and country music usually leave me clueless.

Spoo

[jemidiah]

It was the same show. The page I linked lists the final question:

An inspiration for this character introduced in 1929 was 15-year-old Palle Huld's 1928 44-day voyage around the world

I didn't know the answer. My only exposure to Tintin is Roger Ebert's review of the recent movie (which I didn't end up seeing). The review briefly discussed the character's popularity in France and Europe. It also said something about the character traveling the world. Had I been really on my game that day I might have made the connection, even with this extremely superficial knowledge. I imagine if I had seen the movie I would have gotten it.

The one contestant who got it had been outplayed by the other two the rest of the game. I found myself wishing one of the others had won since they seemed to deserve it more--a question that boils down to "do you know Tintin?" seems to be a strange way to decide a game. I might be biased against the final round because I so often miss the question, though.

[Spoo]

Jem

My bad .. I had not bothered to check your link
prior to my previous post.

I have since done so ... who knew??
And a fantastic link it is.

FWIW, I actually found myself rooting for Monica.
While she only "won" 2 of the 5 Yay Physics topics,
she did get torque (as a DD) and electromagnetism.
She also seemed to me to be the most "studious" ..
how's that for a nebulous evaluation.

EDIT

BTW, I could have sworn she was going either to
Harvard or MIT. Turns out it's North Texas. Well,
I guess we can't hold that against her ..

Spoo

[jemidiah]

Hehe, yeah, J! Archive is amazing. That level of fandom is impressive, especially for just a TV show. A similar but larger example of the same type of fandom is Pokemon and Bulbapedia, which has something like 20,000 articles, though that's a much larger franchise.

I think I preferred the other two because she was so far below them for most of the game (there's a graph at the bottom of the page I linked). She was also a distant third in Coryat score (that is, the score the players would have had if daily double wagers used only the value of the question and if the final round is ignored), and she answered the fewest questions correctly, also by a ways (19 to 26 to 11).

She was more even with them through the second round than I remembered, though, and she did do well with the daily double she had. Truth be told, I also thought one of the guys was cute and wanted to see him in a later match since it was a tournament, but that's not exactly fair to her .

[B61Nuke]

While I quite enjoy Jeopardy!, whenever they have a category I'm particularly knowledgeable about I'm always struck by how superficial the questions are. For instance, there was recently a "Yay, Physics!" category that required only a few bits of basic physics vocabulary: energy, torque, mechanics, normal force, and electromagnetism. Knowing the relationships between any of these concepts and knowing some mathematical models involving them are not at all necessary to answer those questions, even though they're absolutely fundamental to actually understanding physics.

And then a question on modern art comes up and I'm clueless.

I E-mailed them a couple years ago and told them they need to enhance the show with a tournament on specific categories such as physics or diesel mechanics and get into more detail on questions. I think I mentioned maybe spawning a new improved jeopardy. I'd love a chemistry tournament.

[jemidiah]

tournament on specific categories such as physics or diesel mechanics and get into more detail on questions

I really like that idea. It might alienate viewers, though--I know I start zoning out on eg. art history categories I know nothing about. A "science tournament" might work, though, where every category is science-themed (maybe science-and-math-themed). A history/geography tournament might work too....

[B61Nuke]

Yeah, that would be cool. Most people know some detail in one of the sciences. You are right though, alot of the questions are very general. School teachers do not do well on there most of the time. I hope that is not a graph of our education system in America.