Thread: Interesting, surprising facts

Originally Posted by Karloky

why would this first one be surprising..... look at first five positive whole numbers 1, 2, 3, 4, 5 and look at decimal numbers in that range..... how much decimal number's are there between just 1 and 2 ? lol 1.000001...1.000002.........1.00021545454 lol it's infinite lol I guess....now how much are there between 1 and 5 whole numbers decimal....so for every whole number there is even a million more decimal so I guess that's the reason there are more decimal numbers and idk and if between every whole number was a million of decimals then for 100 whole numbers there would be like 100 million decimals and for 1000 whole nubers there would be 1000 million decimals so I guess it's the reason there are more decimals or at least th

Neither is bigger actually, as they're both infinite. But this is something I have to discuss with my math teacher at university, he's a cool dude and he will like it. I'll report back.

Originally Posted by Athylus

Neither is bigger actually, as they're both infinite. But this is something I have to discuss with my math teacher at university, he's a cool dude and he will like it. I'll report back.

There are different levels of infinity. In this case, you have infinity times infinity decimals.

Originally Posted by Athylus

Neither is bigger actually, as they're both infinite. But this is something I have to discuss with my math teacher at university, he's a cool dude and he will like it. I'll report back.

I actually agree with you, even though I know the reasoning. Your math teacher will no doubt tell you that there are less decimal numbers than whole numbers. And this is true in a way. In most cases, if you can map one set one-to-one with items from another set and use up all the items from the first set, but not even come close to using up all the items from the second set, then there must be more items in the second set than the first set. For example, you could map numbers from positive integers to positive decimals like so:
1 = 1.1
2 = 1.12
3 = 1.123
4 = 1.1234
5 = 1.12345
....
14 = 1.1234567891011121314
15 = 1.123456789101112131415
...

You could go on and on like that until you get:
∞ = 1.1234567891011121314151618.....100101102103...... .

You've then mapped every single positive integer to a unique decimal number, and you still have so many decimal numbers left over (nothing that isn't in the 1.1-1.3 range has been used). There are still an infinite number of decimal numbers left after the mapping.... so there must be infinitely more decimal numbers than positive integers.

That's the reasoning anyway. And in a way this makes sense, perhaps it's useful in some situations. But I think, on a more common-sense level, one infinity can't be greater than another. Infinity is a concept, not an actual number, though it can be treated like a number in some circumstances. And the concept of infinity refers to some abstract quantity which is by definition greater than any amount we can possibly name. To say that ∞1 > ∞2 contradicts the definition of infinity. Because by definition nothing can be greater than ∞2.

Originally Posted by dutchraptor

That's ridiculous, you can use whatever the hell you want. Since when must you use the letter h to do represent a variable height.

Well if you can name them whatever you want then there are a lot more chances for it to spell a name. It doesn't seem like such an amazing coincidence. I mean you could say there's a box with length B, width O and height X..... What's the volume? BOX! !!!!!

The Capgras Delusion is a psychological disorder where you think everybody you know has been replaced with an impostor.

A cool hypothesis to explain this (and one which has some evidence, I think) is that the neural pathways from the part of the brain which recognises people to the part of the brain which causes an emotional response has been in some way severed. So you see your loved ones and yet don't feel any kind of emotions towards them. Your brain rationalises this by insisting that they're somebody different.

The Fregoli delusion is the opposite. You think that different people are in fact one person. It's pretty funny, really.

Originally Posted by Marvo

There are different levels of infinity. In this case, you have infinity times infinity decimals.

Ah well, technically the infinity of the whole numbers (let's call it N0) multiplied by itself should be the number of pairs of whole numbers (in the same way that multiplying 2 by 3 means count how many points there are in a 2 by 3 grid), in other words the number of whole points in the first quadrant of the plane, but this turns out just to be equal to the original infinity; i.e. N0*N0 = N0. That's surprising fact #2 about infinity.

The infinity of decimals, call it N1, turns out that it should equal, if anything, 2 to the power of N0. But this turn out to be impossible to prove from the axioms of number theory or set theory, and in fact you can consistently take it to be either true or false. Extremely surprising fact about infinity #3.

Originally Posted by Dianeva

There are still an infinite number of decimal numbers left after the mapping.... so there must be infinitely more decimal numbers than positive integers.

Technically this reason doesn't work. For example, you could double every whole number. This mapping sends every whole number to every even number inside the whole numbers. There are an infinity quantity of whole numbers still left after the mapping, but you can't then conclude that there are infinitely more whole numbers than whole numbers.

In the same way you could map the infinite quantity of decimal numbers into the space between, say, -1 and 1.

The problem is that you're trying to show that there are some decimals left untouched no matter what mapping you use, not that any specific map fails (it's always possible to come up with maps which fail). In other words, given a map from the whole numbers to the decimals, you need to find a methodical way of producing a decimal which can't be touched anywhere on the list. There's an ingenious way of doing this.

That's the reasoning anyway. And in a way this makes sense, perhaps it's useful in some situations. But I think, on a more common-sense level, one infinity can't be greater than another. Infinity is a concept, not an actual number, though it can be treated like a number in some circumstances. And the concept of infinity refers to some abstract quantity which is by definition greater than any amount we can possibly name. To say that ∞1 > ∞2 contradicts the definition of infinity. Because by definition nothing can be greater than ∞2.

Perhaps I've become numb through overexposure, but I don't feel there's anything wrong with it. The maths of different infinities does behave in a nice consistent way, anyway. I'd turn what you said on its head and interpret it in the opposite way - these results simply tell us that our intuitions about infinity are inconsistent.

After all... the infinity of the decimals is bigger than the infinity of the whole numbers. But surely we don't want to say the infinity of the whole numbers is not infinite? There's always a bigger whole number than any whole number you can think of. They go on forever and ever. We've gotta call them infinite.

Originally Posted by Dianeva

Well if you can name them whatever you want then there are a lot more chances for it to spell a name. It doesn't seem like such an amazing coincidence. I mean you could say there's a box with length B, width O and height X..... What's the volume? BOX! !!!!!

You aren't helping your argument very much in all fairness, take a look at your own picture of a pie. The figure they used for pi, is not the actual value of pi, the decimal point is disregarded and the only reason the whole thing works is because of a non standard font. Try mirror an image of 3.14 in standard font and I'm willing to bet it bears very little resemblance to the word pie. It doesn't really seem like an amazing coincidence at all.