11th January 2006, 8:48 PM
Well, let's see!
First test, there are infinite whole numbers. Are there as many even whole numbers (2,4,6,8) as there are both even and odd whole numbers?
Both are infinite, so you'd have to conclude "yes". After all, name any whole number you can think of and I can give you an even number to match it "set for set". I'll just double your number!
You: 1
Me: 2
U 4!
M 8!
U 8
M 16
U 10000000
M 20000000
U -5
M -10
U Infinity
M Infinityx2
I'll match every unique natural number you come up with with an even number, EVERY one! It won't stop!
But, that just doesn't SEEM right does it? By all rights, the number of even numbers should be "half of infinite" or something, shouldn't it? Or, are there greater and lesser infinite sets? Seems more likely, but how defined? How about definining them by their limitations? Yes, infinite sets can have limits. The infinite set of even numbers is limited to even numbers for example. Odd numbers are absent, and yet it is still infinite.
Next riddle: is .9999~ < 1?
Same issue, that number gets closer and closer to 1 forever. Any single number you can ever list, that number will beat it. There is no "in between" number you can ever name. The only time this stops is when you name "1", so that's it's limit.
So... .999~ = 1
So then, why did I bring this up? Well, for those of you that already knew this, and I know anyone who's taken a decent math class does, never mind this. This is for a couple people who in the past didn't seem to get that infinity can have limits.
First test, there are infinite whole numbers. Are there as many even whole numbers (2,4,6,8) as there are both even and odd whole numbers?
Both are infinite, so you'd have to conclude "yes". After all, name any whole number you can think of and I can give you an even number to match it "set for set". I'll just double your number!
You: 1
Me: 2
U 4!
M 8!
U 8
M 16
U 10000000
M 20000000
U -5
M -10
U Infinity
M Infinityx2
I'll match every unique natural number you come up with with an even number, EVERY one! It won't stop!
But, that just doesn't SEEM right does it? By all rights, the number of even numbers should be "half of infinite" or something, shouldn't it? Or, are there greater and lesser infinite sets? Seems more likely, but how defined? How about definining them by their limitations? Yes, infinite sets can have limits. The infinite set of even numbers is limited to even numbers for example. Odd numbers are absent, and yet it is still infinite.
Next riddle: is .9999~ < 1?
Same issue, that number gets closer and closer to 1 forever. Any single number you can ever list, that number will beat it. There is no "in between" number you can ever name. The only time this stops is when you name "1", so that's it's limit.
So... .999~ = 1
So then, why did I bring this up? Well, for those of you that already knew this, and I know anyone who's taken a decent math class does, never mind this. This is for a couple people who in the past didn't seem to get that infinity can have limits.
"On two occasions, I have been asked [by members of Parliament], 'Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?' I am not able to rightly apprehend the kind of confusion of ideas that could provoke such a question." ~ Charles Babbage (1791-1871)