Of course infinity cannot be treated as an ordinary number but it still
'exists' and you can compute larger and larger numbers as long as you
like.
Infinity is not a number. Because of this, when you get infinity as an answer, final or intermediate, you need to look closer. That said...
There is more than one size of infinity. Consider for example the positive integers and the odd positive integers. Obviously the set that is missing all the even integers has to be smaller. But it ain't so. You can pair each member of one set
with a member of the other set, and you'll never run out.
1..2
2..4
3..6
etc.
Ok, so much for that. What about fractions? There have to be more fractions than integers, because the fractions include the integers (6/3 is just another way of spelling 2). Let's arrange a grid however:
(note - at this point you'll probably want a fixed spacing font to see the grid)
* 1 2 3 4
1 1/1 1/2 1/3 1/4 ...
2 2/1 2/2 2/3 2/4 ...
3 3/1 3/2 3/3 3/4 ...
4 4/1 4/2 4/3 4/4 ...
....
This chart has to have all the fractions in it. And all the integers are listed in the column to the left, going down. Each row has an infinite number of fractions in it, so there just =have=to= be more fractions than integers. Well, no. (and it
has nothing to do with the fact that 2/1 is the same as 4/2)
Consider a path that starts with 1/1, and goes diagonally up right as far as it can (which isn't far at all!) then goes to the next diagonal down, and the next diagnoal up, zigzagging along, until it reaches the end (which is, of course, never). It
would cover (and I've paired them with integers starting with 1, below)
1/1 1/2 2/1 3/1 2/2 1/3 1/4 2/3 3/2 4/1 ...
1 2 3 4 5 6 7 8 9 10 ...
I'll never run out, and I'll never miss a beat. There must be the =same= number of fractions as there are integers.
Ok, we get three strikes in baseball, I get three ups here. Lets look at all the real numbers beween 0 and 1 and try to list them. They are listed as infinite decimals,
though some of them may end in lots of zeros - i.e. .5 is the same as .500000.... (and also .4999999..., which I won't get into here)
Here's my list. Itegers on the left, decimals on the right (in no particular order):
1 .348791037984....
2 .500000000000....
3 .000023416898....
4 .142857142857....
5 .141592653589....
6 .414213562373....
.... ...
No matter what I do, I can't list all the decimals on the right, and not because I can't afford the paper. However the list is created (and I have not put them in an order for several reasons), there is always at least one number that's not on the
list. Create it thusly: Write a decimal point and then take the first decimal place of the first number on the list, and write it down. Take the second decimal digit of the second number and write it down... take the nth decimal digit of the nth
number on the list.. and write it down... (see below)
..300893...
Now, just below it write a number whose digits differ in every place.
..411904
That number IS NOT ON THE LIST! ("Sure it is... it's the 52342th one, you must have missed it." "nope, the 52342nd digit is different." "oh yeah... oh wait, here it is, it's the 230498103984th one on the list." "nope... "
So, there are more real numbers between zero and one than there are integers! There are at least two sizes of infinity... the "original size" and the "giant size".
(actually, there are an infinite number of sizes of infinities, but that step is less mind boggling than the first one)
So, what does this have to do with aviation? Well, it will give you something to ponder on those long cross countries, it will explain why your fuel calculations were a bit off in the headwind (Oh, it must have been a big infinity in the
calculations instead of a little one), and it will give you something to impress the girls with when a pilot certificate doesn't do the trick.
Ok, maybe not. But it's still interesting.
Jose
--
Freedom. It seemed like a good idea at the time.
for Email, make the obvious change in the address.