# Deep Thinkers Answered

This topic will be a place to post all those deep questions you sometimes ask, and might get answers. Here's one I've been thinking about: (which no longer needs answering) Space. It's black, right? Why? If there is nothing for color to reflect off of, shouldn't space be void of any color?

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## Comments

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"does the circumference of C equal infinity, or (infinity x Pi)?"

Yes. Sorry, I know that's an aggravating answer but it does equal infinity and (infinity*pi) because they are the same thing. That's infinity for you. Also, remember that the perimeter of an infinitely large circle is actually a straight line :D

In answer to your previous question, I briefly confused my maths teacher by asking "if you cut a 30cm ruler into an infinite number of infinitely small pieces is it still the same length?". To her eternal credit, rather than fob the 12-year-old me off with some rubbish like "Yes" or "No" she said "Undefined. Go and look up calculus" :)

No, and sure. The infinite set of all positive integers is pretty big. But, the infinite set of all positive and negative integers is exactly twice as large, as there is a corresponding negative and positive in that set for each positive in the first infinite set. :)

Ah, something I know something about!

No, not all infinite sets are the same size, but the proof is bizarre and quite involved, and there is a lot of counterintuitive stuff.

The set of positive integers is actually the same size as the set of positive and negative integers. If you have a function that
• doubles a positive integer
• makes a negative integer positive, doubles it and adds one

you have a one-to-one mapping from all integers to just positive integers. 1 maps to 2, -1 maps to 3, 2 maps to 4, -2 maps to 5, and so on. Ergo there are the same number of integers as just positive integers. (See the second problem of the Infinity Hotel, housing an infinite number of new guests. If anyone is not familiar with the Infinity Hotel I suggest you look it up, it's fascinating)

Here's a weird one. There are an infinite number of fractions between 0 and 1- given any two fractions you can find one between them ad infinitum, so there must be an infinite number. So, the number of fractions between 0 and infinity must be infinity squared, mustn't it?

Nope. You can map any fraction to two integers (the top and bottom of the fraction), and you can map any two integers to one integer using the uniqueness of prime factorisation (another strange involved process, but trust me that it's possible) so you then have a one-to-one mapping from all fractions to positive integers. Ergo there are the same number of fractions as whole numbers. This one really confused me for a while because while I can accept that infinity + infinity = infinity, this proof demonstrates that infinity * infinity = infinity.

The sets I've mentioned are called countably infinite because you can map them onto the positive integers so they can be "counted" like the rooms in the infinity hotel. The first set that is a "bigger sort of infinity" than these is the set of irrational numbers, ie non-terminating and no-recurring decimals, like pi or the square root of two. This set is an "uncountable infinity", because you cannot map them onto the integers. The proof for this is delightfully common-sense but too long for me to explain here without diagrams so I will point anyone who survived the previous blurb with their brain and curiosity intact to Cantor's Diagonal Argument. It was stuff like this that kept me sane through the first year of my degree.

Goodhart, in answer to your question the set of real numbers is qualitatively larger than the set of integers. Not sure if you wanted the answer or were just provoking discussion, but hey.

Lets pursue this then: If .99999 repeating equals one, then we can assume that .8888888 repeating equals .999999 repeating. If this is so, .88888 repeating also equals one. We can do this for each repeating number down to and including .11111 repeating. Logically, and mathematically then, it can not. But by reason statistically, it is known as "close enough".

No one is asserting that sequential numbers are equal to each other! :) Please, go back and read ALL the refutations on that first page I linked to. As Weissensterinburgenheimerwinkydink (get a shorter name, willya!) has almost pointed out, there actually an infinite number of real between .8999repeating and .999repeating. There's .91 .92 .93 .90000000001 .98888989898989 etc. Really, the problem is a limitation of the decimal notation. It's just like saying 2+2 is another way of saying 1+3 or 4! As in many things in math, some problems can only be solved by changing the terms they are expressed in. Or, in other words, by changing one's perspective, OR in other words, using a different notation. Can you imagine if the only definition we had of 1/3 was .33333repeating how hard it would be to grasp fraction operations? Try to think of it like this . . . 1/3 is .3 repeating because when you divide it out longhand, you get 1.0 divided by 3 which gives you .3 with a remainder of 1. So you keep dividing longhand. and get .03 for a total of .33 with another remainder of 1. Well, it's obvious that you could keep this up for eternity, so we simplify the notation and say .333repeating. So in actual fact .333repeating and not .33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 is equal to 1/3. Now do the same with 2/3 and you get .666repeating. Add those two infinite decimals together and you get .9999 repeating!

I am a purest I suppose. I can see what you are saying, but I also hold that my point of view can be just as valid if one is able to understand what I am saying. Take this point, you know what a mobius strip is, right? Now imagine 2 dimensions turned that way. If you can do this, imagine 3. This gives us the model of reality as it stands in space. It can't be drawn, but it can be imagined. Such are the limitations of our mathematics.

I also hold that my point of view can be just as valid if one is able to understand what I am saying. HAH!

I can see what you are saying ;-)

Now imagine 2 dimensions turned that way. --Isn't that what a mobius strip sort of is? Not sure what you are saying here! So your point must be valid! :)

Well, not exactly. A mobius strip only represents a twist in a single dimension. If you twisted in both dimensions it would have a twist going in two directions. A third twist in the third direction or dimension is the final product. I only use this as an illustration of how we can sometimes "represent" reality in an completely non-realistic way. The reality then is that 1/3 is represented with .3333 repeating, but it is no more accurate then our representation of Pi for instance. Being inaccurate, one can not do accurate math with that representation and expect to end up with a precise answer.

"The reality then is that 1/3 is represented with .3333 repeating, but it is no more accurate then our representation of Pi for instance."

Weeeeeelllll, .3333 repeating is an accurate representation of 1/3 but it's also unwieldy. It's not very practical for performing operations on. And I have no idea how they calculate PI, but the accuracy of that IS limited by time, whereas because of the repetitive nature of .333 repeating we can easily predict what future digits of 1 divided by 3 will be, hence the .333 repeating notation. . . .

And I still have no idea what the heck you mean with the mobius strip thing. Care to post a picture? :)

And I still have no idea what the heck you mean with the mobius strip thing. Care to post a picture?

If I could illustrate it in drawing I would. You have a strip of paper. Although it is 3-D, if it represents only one dimension (say the length is right/left in front of you), when you twist it, you are only twisting the one direction, left to right. The narrow section of the strip (up / down) follows the twist of the right to left direction, but it itself it not "twisted" like in the right to left direction. It is easier to imagine than to explain. It is nearly impossible to draw.

A three dimensional mobius, where all three dimensions / directions are given a twist is impossible to illustrate. Still, it is a form of what some would call hyperspace. The representation of what exists, what is in front of us, yet we have no way of "modeling" it. The mathematics are way beyond me in the field (2-dimensional mobius strip mathematics). What it entails then is that space curves back on itself in all directions eventually, even though you would never "change" directions (traveling in a straight line) you would eventually be able to return to your start position.

This link is probably the closest we'll get to 3-D mobius

I'm not sure. Does the number 0.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051... have Pi in it? I'd say not because the two numbers are both equally infinite and yet different. But it is PROBABLY a transformation of Pi.

I wonder why ...
1 / ((10n) - x)
gives all the powers of x to n decimal places in order???
How can one division make squares, cubes, and all higher powers?
(You need a calculator with plenty of digits for this though.)
1/(1000-2)=0.001 002 004 008 016 032 064 128 256 512 :)

And how do some autistic savants recognize large prime numbers?
(I think I know this one.)

Hypothetical Autistic Prime-o-vision: 1.Really good imagination (visualization power or daydreams) 2.Able to instantly tell if the closest multiple of 6 is within 1 count 3.Failure to imagine a number as a rectangle if you are really good at doing that. example: 12 is a multiple of 6. It can be imagined as two rows of 6, or 4 rows of 3. (Two different rectangles). 11 and 13 are prime and are either one short or greater than rectangular. Numbers divisible by 6 are "even", AND also if you add up all their digits the sum is divisible by 3. I think this part works Recursively for really large numbers. 41 and 43 are prime Because they are one count away from a rectangle divisible by 6, AND SO they either have a piece missing or an extra piece, AND they make no rectangles. (sorry,Attempt to illustrate with text came out messed up.) Try again for 12,11,13: oooo ooo oooo oooo oooo oooo ooooo oooo oooo

No, not all infinite sets have equal number of members.

First, let me give a counter-examples that does *not* work :D

You might think that the set of even integers (and the set of odd integers) contains only half as many elements as the set of all integers. However, it turns out you can come up with a simple one-to-one mapping between the set of all even integers, and the set of all integers (by dividing by two). Therefore, the sets of all odd integers, all even integers, and all integers actually have the same numner of elements. Go figure...

In fact, these kind of sets are called "countably infinite". By definition, a set is countably infinite if there is a one-to-one mapping to the set of natural numbers, i.e., non-negative integers (I'll leave it up to the reader to show that you can map from the integers to the natural numbers).

One example of a set which is *not* countably infinite is the set of all real numbers. There's no way to map the real numbers to the natural numbers, i.e. there is no way to enumerate all real numbers - there's just infinitely more reals than naturals.

For more mind-twisting, look up aleph-null on wikipedia. ;)