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If I asked you for the sum of the integers from 1 to 100 inclusive, you could solve the problem by an easy formula [ref.: formula for the sum of an arithmetic series ]. But if I asked you for the product of all of the integers from 1 to 100 (or 100!, as this is written), if you didn't have a shortcut you'd come back to me 3 days and several gallons of Red Bull [I hate the stuff] later -- and odds are, if you did the calculation manually, you've introduced at least one error, which makes your result a study in 1 ) discipline and devotion; and 2 ) high unreliability. Happily, over the years, mathematicians have pondered this problem and found a formula to approximate the result. Note that the factorial formula is increasingly accurate as the amount of numbers multiplied grows, i.e., the formula would be rotten for calculating 10!, poor for 100!, passable for 1,000! and very darn close and useful for 10,000! Now "here", as the ignorami [a Lingovation of the highest...

Image via Wikipedia In our last posting, we discussed the notion of asymptotes , and I posed two problems for your consideration. One involved the eventual (but unreachable) sum of a convergent series of numbers, and the other involving a ever-more troubling fraction. You can quickly refresh your memory by clicking on http://braintenance.blogspot.com/2011/09/asymptotes-closer-but-never.html , and by then hitting your browser's "BACK" button. The answers are unsatisfying, but they were promised: 1) In adding the sum of the series 1 + 1/2 + 1/4 + 1/8....and so forth, the sum will eventually approach, but never quite reach a limit of 2. 2) In dividing (n-1)/n, as n increases, the value of the expression approaches, but never reaches 1. There are examples of this type of complex conundrum in nature, in such things as trying to solve 22/7 (which is a never-ending decimal), and in determining the halflives of certain radioactive materials (isotopes), where one half of the ma...

The number Pi is a mathematical constant . It represents the ratio of the circumference of a circle to its diameter . If you divide the circumference of any circle (or perfectly-formed, hand-tossed pizza) by its diameter (the measure of a straight line cutting the pizza in half), you will get Pi, which is equal to 22/7, or approximately 3.14. Now let's review Fabian's quandary, from a couple of days ago... Background : Fabian Focaccia (not his real name, which is Sal Monella)  is a struggling 'artist' who is working at a neighborhood pizzeria in Brooklyn, New York (not his real location, as he is in the Federal Witness Protection Program , but which is still the best geographical location to make a pizza purchase if you/ youse should ever get around to it) to pay his bills until he can sell one of his paintings. He is faced with a decision and needs your help. He has cardboard boxes for 'take out' pizza (this pizza parlor does a big take-out busines...

Your mind is a muscle. You must use it to strengthen it. If you don't, it will atrophy and hasten your descent toward dementia (which is not the name of a town in the Midwestern USA ). Following are a pair of practical problems involving Pizza, Pi and Packaging . Try 'em. Background : Fabian Focaccia (not his real name, which is Sal Monella)  is a struggling 'artist' who is working at a neighborhood pizzeria in Brooklyn, New York (not his real location, as he is in the Federal Witness Protection Program , but which is still the best geographical location to make a pizza purchase if you/ youse should ever get around to it) to pay his bills until he can sell one of his paintings. He is faced with a decision and needs your help. He has cardboard boxes for 'take out' pizza (this pizza parlor does a big take-out business -- even for Arizona...oops!) which are each three inches deep (irrelevant for solving this problem) and measure exactly 20 inches by 20 inche...