Which infinity? Because some infinities are larger than others... For example continuum (the number of real numbers) is larger than aleph-zero (the smallest infinity, the number of integers). I love calculus class...
Do they really teach that in Calculus? Because I've never taken that, so I might be totally wrong.
The very nature of infinity is that it never begins and never ends; sure, there are an infinite amount of numbers between 0 and 1, so it would seem like there are more numbers than integers, but infinity can't be measured like that... I don't know how to explain it well enough. I'll let Biggles tackle it. :p
[QUOTE][i]Originally posted by Random Chaos [/i]
[B]Well...I might just point out to A% and MartianDust that the levels are completely random so if you think you hit a limit...in the future you might have a "good" random level that you can go real far in :) [/B][/QUOTE]
Well I reckon that was an easy random level, cos I never got that far since lol! Will try again in abit, not tried today. Will put it down to tiredness last night! ;)
Hey the fish game was funny! Liked that too. Oh and the penguin one, did that few days ago think I got 327, can't get higher than that yet! Never knew these types of games existed online.
[QUOTE][i]Originally posted by the_exile [/i]
[B]Do they really teach that in Calculus? Because I've never taken that, so I might be totally wrong.[/B][/QUOTE]
They do in the Calculus class I'm taking right now. Sometimes the teacher gets bored and starts going off on tangents. Like the day he told us the buttered-bread-and-cat theory (bread always lands buttered side down, cats land feet down. tie bread face up to back of cat. Drop. Observe ;)).
His hobby is researching how mathematicians die...Did you know that almost every mathematician that studied the theory of the nature of infinity, went insane? You do now :)
As I said, I love Calculus...tis the best flavour of math.
Φ was absolutely correct; some infinities are larger than others. The proof is somewhat easy to see if you understand binary.
First, I'll list a couple integers in binary:
[quote]0
1[/quote]
Now, I'll add some of the real numbers that come between those two:
[quote]0.0011
0.0101
0.0111
0.1011[/quote]
Note that I listed them such that the numbers beyond the decimal point form a square; that's so that I can demonstrate that, by flipping digits along the diagonal of that square marked with x's below, I can come up with a number that's not in that list:
[quote]0.x011
0.0x01
0.01x1
0.101x[/quote]
Taking just the digits from those x's, we get:
[quote]0.0111[/quote]
Flipping the digits, we get:
[quote]0.1000[/quote]
...which is a new number, not seen in the list. A number generated in this manner will [i]always[/i] be a new number, because it must by nature differ from every other number in at least one digit.
Still with me?
Really?
Wow.
Moving on:
How do we decide if one value of infinity is equal to another? Let's use set theory.
Two sets are said to be equal in size if there is a function [i]f[/i] from set [i]A[/i] to set [i]B[/i] that is both one-to-one and onto. That is, if it's possible to match every element in set [i]A[/i] to a unique element in set [i]B[/i] and vice-versa, then the two sets are equal in size.
As an example, let's show that the set of strictly positive integers (integers greater than zero) and the set of strictly negative integers (integers less than zero) are equal in size.
We can pair them using the function [b]f(i) = -i[/b]. Visually:
We can clearly see that every positive integer matches up with exactly one negative integer, and that none are left over on either side. Thus, even though the two sets are infinite, they can still be said to be the same size
Have I lost you yet?
No?
You really are a glutton for punishment, you know. Anyway:
Try to match up integers and reals however you like. Use whatever function you want, or just mix and match randomly. Just make sure you don't leave any out. You'll find that you always have reals left over. Here's a demonstration of that, presented in such a way that I'm clearly using all the integers. Reals will just be assigned randomly, for now.
Clearly, all the integers are used here. I can prove, however, that not all the reals are used. Remember when I created a new number by flipping digits along the diagonal? Well, you can do that here, too, and the new real number is guaranteed to be different from every other real number in that list. Since all the integers are already paired with their own real numbers, this real number has no more integers to be paired with. Thus, there are more real numbers than integers.
Are you beginning to understand why mathematicians who study this stuff extensively go insane?
Biggles<font color=#AAFFAA>The Man Without a Face</font>
Logical proof, but next time: less tabs, more spaces. :)
Unfortunately, I won't get to take Calculus, at least not before college. I missed 30 out of 90 days (1 to 2 days per schoolweek, and that's not counting the times I got one midway through the day, either) of school last semester thanks to migraines, and let's just say that it's hard to make up tests when you a) haven't learned the material and b) you're already waiting to make up 5 previous tests. Failed Algebra II Pre-Calc last semester, which meant I couldn't take Chemistry this semester, and instead had to take a [i]ridiculously[/i] easy Algebra II class.
Comments
-Φ
The very nature of infinity is that it never begins and never ends; sure, there are an infinite amount of numbers between 0 and 1, so it would seem like there are more numbers than integers, but infinity can't be measured like that... I don't know how to explain it well enough. I'll let Biggles tackle it. :p
[B]Well...I might just point out to A% and MartianDust that the levels are completely random so if you think you hit a limit...in the future you might have a "good" random level that you can go real far in :) [/B][/QUOTE]
Well I reckon that was an easy random level, cos I never got that far since lol! Will try again in abit, not tried today. Will put it down to tiredness last night! ;)
Hey the fish game was funny! Liked that too. Oh and the penguin one, did that few days ago think I got 327, can't get higher than that yet! Never knew these types of games existed online.
:)
[B]Do they really teach that in Calculus? Because I've never taken that, so I might be totally wrong.[/B][/QUOTE]
They do in the Calculus class I'm taking right now. Sometimes the teacher gets bored and starts going off on tangents. Like the day he told us the buttered-bread-and-cat theory (bread always lands buttered side down, cats land feet down. tie bread face up to back of cat. Drop. Observe ;)).
His hobby is researching how mathematicians die...Did you know that almost every mathematician that studied the theory of the nature of infinity, went insane? You do now :)
As I said, I love Calculus...tis the best flavour of math.
-Φ
First, I'll list a couple integers in binary:
[quote]0
1[/quote]
Now, I'll add some of the real numbers that come between those two:
[quote]0.0011
0.0101
0.0111
0.1011[/quote]
Note that I listed them such that the numbers beyond the decimal point form a square; that's so that I can demonstrate that, by flipping digits along the diagonal of that square marked with x's below, I can come up with a number that's not in that list:
[quote]0.x011
0.0x01
0.01x1
0.101x[/quote]
Taking just the digits from those x's, we get:
[quote]0.0111[/quote]
Flipping the digits, we get:
[quote]0.1000[/quote]
...which is a new number, not seen in the list. A number generated in this manner will [i]always[/i] be a new number, because it must by nature differ from every other number in at least one digit.
Still with me?
Really?
Wow.
Moving on:
How do we decide if one value of infinity is equal to another? Let's use set theory.
Two sets are said to be equal in size if there is a function [i]f[/i] from set [i]A[/i] to set [i]B[/i] that is both one-to-one and onto. That is, if it's possible to match every element in set [i]A[/i] to a unique element in set [i]B[/i] and vice-versa, then the two sets are equal in size.
As an example, let's show that the set of strictly positive integers (integers greater than zero) and the set of strictly negative integers (integers less than zero) are equal in size.
We can pair them using the function [b]f(i) = -i[/b]. Visually:
[quote]1    -1
2    -2
3    -3
.    .
.    .
.    .
i    -i
.    .
.    .
.    .[/quote]
We can clearly see that every positive integer matches up with exactly one negative integer, and that none are left over on either side. Thus, even though the two sets are infinite, they can still be said to be the same size
Have I lost you yet?
No?
You really are a glutton for punishment, you know. Anyway:
Try to match up integers and reals however you like. Use whatever function you want, or just mix and match randomly. Just make sure you don't leave any out. You'll find that you always have reals left over. Here's a demonstration of that, presented in such a way that I'm clearly using all the integers. Reals will just be assigned randomly, for now.
[quote]0    0.000110...
-1    0.100011...
1    0.010010...
-2    0.011000...
2    0.100110...
-3    0.1010101...
.    .
.    .
.    .[/quote]
Clearly, all the integers are used here. I can prove, however, that not all the reals are used. Remember when I created a new number by flipping digits along the diagonal? Well, you can do that here, too, and the new real number is guaranteed to be different from every other real number in that list. Since all the integers are already paired with their own real numbers, this real number has no more integers to be paired with. Thus, there are more real numbers than integers.
Are you beginning to understand why mathematicians who study this stuff extensively go insane?
Edit: I don't suppose you can edit that with good html that will make things line up properly?
On the other hand I did realize what I love doing after that.