Posted on 01/24/2009 6:43:23 PM PST by Daffynition
For seventy years, a prayer book moldered in the closet of a family in France, passed down from one generation to the next. Its mildewed parchment pages were stiff and contorted, tarnished by burn marks and waxy smudges. Behind the text of the prayers, faint Greek letters marched in lines up the page, with an occasional diagram disappearing into the spine.
The owners wondered if the strange book might have some value, so they took it to Christie's Auction House of London. And in 1998, Christie's auctioned it off—for two million dollars.
For this was not just a prayer book. The faint Greek inscriptions and accompanying diagrams were, in fact, the only surviving copies of several works by the great Greek mathematician Archimedes.
An intensive research effort over the last nine years has led to the decoding of much of the almost-obliterated Greek text. The results were more revolutionary than anyone had expected. The researchers have discovered that Archimedes was working out principles that, centuries later, would form the heart of calculus and that he had a more sophisticated understanding of the concept of infinity than anyone had realized. [snip]
(Excerpt) Read more at sciencenews.org ...
Well, from the Principle of Inclusion and Exclusion, I can immediately write down an expression for the number of ways to distribute N items in M bins such that at least one bin remains empty. Since there are M^N ways to distribute them without conditions, and presuming these to be equally probable, this gives the probability of at least one bin remaining empty.
For (M,N) = (12,20) I get a probability of ~.968 which matches the fraction obtained from a trial of 1,000,000 , so I’m satisfied the expression is correct.
Is this enough to convince you that it wouldn’t be “unusual” to find birthdayless months among a class of 20 students? I would have thought that 10 successes in 10 (model) trials would have done that, so who knows.
I just checked using matlab.
>>0/0
NaN
>>1/0
Inf
>>-1/0
=Inf
>> 0*Inf
NaN
A department manager where I work had a Chinese surnamed kid in whose resume claimed a 4.0 average at (iirc) Notre Dame in EE and an advanced degree from someplace like UCLA. First question he asked was to describe an FFT in his own words. When the kid couldn’t begin to answer, he canceled the rest of his interviews that day (I was on the schedule) and sent him home. True story. I wonder if he asked, “Does this mean I don’t get the job?”
Exactly - no point in wasting everybody’s time.
For instance, there is a perfectly good and interesting function, the sinc function, whose value is sinc(x) = sin(pi*x)/(pi*x), everywhere except at zero, when it equals 1.0, by definition. This might seem like a fudge, but differential calculus gives us a method, L’Hopital’s Rule, for determining rigorously such limiting cases. [lim x->0 f(x)/x = f’(0)].
Archimedes was known to have used limits to solve problems. For instance, he was the first to devise a method of calculating pi to arbitrary precision by expressing it as bounded by perimeter of inscribed and inscribing polygons of a unit circle. Computing the perimeters of such polygons requires continued fractions and repeated root extraction which is highly tedious and time consuming even today. Until the invention of calculus, pi was only known to a few decimal places, but Archimedes at least provided a rigorous method of bounding it.
Lim dx —> 0 := 0
(a-b)=(1-1)=0
Only Chuck Norris can divide by zero... :-)
;)
The efficient representation is what was invented, not zero.
IMHO, numbers exist regardless of ways to represent them.
Apologies, I missed your reply.
The detrministic joke was that is you average the results of your ten tries, there are no empty months.
a=b=1
aa=ab
aa-bb=ab-bb
(a+b)(a-b)=b(a-b)
a+b=b
2=1
Tee Hee, I haven’t seen that for a while. Always good at parties and with liberal friends.
I remember reading a theory about just this subject years ago. Supposedly, he approached integration (sorry, unintentional derivative there) when he computed the formula for the volume of a sphere.
A Daedalus, Newton and an Einstein, all rolled into one.
In my darkest hours, I fear the Muslim is about to perform the only useful function he has ever performed: Help to guard the achievements of the West while the West resurrects itself.
Sigh - I need to get out more...
Let’s hope it won’t be necessary. I don’t want to pray with my butt to the sky and my nose towards Mecca.
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