This is a special case of Euler's Formula, and is considered by many to be the most amazing combination of the most fundamental elements of mathematics:
|e, the base of the natural logarithms|
|i, the fundamental imaginary number, equal to|
|, a very important irrational number|
|0 and 1, real numbers (and the identities for addition and multiplication!)|
that is as obvious as that twice two makes four is to you!"
Sounds like a class I'd think about dropping! (Though, as Nahin goes on to show, it's not really that hard to show. I still don't fit the description, though!)
This is something I dreamed up in college, when I had nothing better to do:
Here's the answer in Microsoft Word format. (500k)
Here it is zipped. (150k)
Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 5 miles per hour, instantaneously changing direction as it meets each train. It does this until the trains collide and crush the fly (not to mention the really awful mess they make in the process!). What is the total distance the fly has flown?
The fly actually hits each train an infinite number of times before it gets crushed, so one could solve the problem the hard way with pencil and paper by summing that infinite series of distances. (Or at least by using the formula for the sum of an infinite Geometric Series.) The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 5 miles per hour, the fly must have flown 10 miles. That's all there is to it.
When this problem was posed to John von Neumann, he immediately replied, "10 miles." "That's very strange," said the poser, "nearly everyone tries to sum the infinite series." "What do you mean, strange?" asked Von Neumann. "That's how I did it!"
A little mathematical limerick for you:
Say what?? Maybe I should translate it:
|Integral z squared dz,|
|From 1 to the cube root of 3,|
|Times the cosine|
|of 3 pi over 9|
|is log of the cube root of e.|
Work it out -- it's true! Sorry about the 3/9 part not being simplified - poetic license.
I'm going to give you an assignment now. Say someone wants to add up all the numbers from some number to some other number, like, oh, 55 to 275. I'm going to do it like this:
1) Take 275 and add one to it: 276
2) Take 55 and subtract one: 54
3) Take the 275 times 276 (getting 75900), and take 54 times 55 (for 2970). Subtract these two products to get 72930.
4) Take half of that, and that's your answer: 36,465.
Try it again with an easy one to verify: Add the numbers from 10 to 15. It's pretty easy to add them and see that the sum is 75. Does my algorithm work? Sure, because [(15x16)-(10x9)]÷2 = 75.
Why does this work? Why is it always true that
for integer i and j (with i<j)?
And notice it's true for integers as well, not just naturals!
If you're studying sequences and series, the proof is trivial. But, you can prove it with just simple Algebra (nothing higher than the FOIL method) and some reasoning if you're feeling lucky!
And, of course, the infamous
Remember? log(cabin)? +C?
A parting question for you:
And finally, an amazing proof, with apocolyptic ramifications for all humanity!
© 1998-2002 Dan McGlaun