Wikipedia:Reference desk/Archives/Mathematics/2008 June 7

= June 7 =

How small can small get?
Is there an infinity for the small? Or is there a certain point where you simply can't divide anymore? I've heard of Planck lengths, but I've always had a suspicion that these are just man's mental limits because the math backs me up (1/2, 1/4, 1/8, etc.)Of course this begs the question of how big something can be also. It seems there's no limit there either.--Sam Science (talk) 16:44, 7 June 2008 (UTC)
 * In maths, there is no limit to how big or small numbers can get. If there was a smallest positive number, you could just half it and get a smaller one, so there clearly isn't a smallest. Likewise, you could double the biggest number and get a bigger one, so there can't be a biggest. The equivalent of infinity for the very small is called "infinitesimal", however for regular numbers (real numbers), the only infinitesimal is actually zero. You keep halving again and again and again and you get closer and closer to zero, you'll never actually get there after a finite of steps, but the limit as the number of steps approaches infinity is zero. If you want the technical version, see Archimedean property.
 * However, in physics, things are a little different. Mathematical numbers are just an abstract concept, they don't always apply to the real world. While you can keep halving numbers, you can't keep halving physical objects. In quantum mechanics, things are "quantised", which means they can only come in integer multiples of a particular value, so the smallest it can get (without being zero) is fixed at one times that value. That's where the Planck length comes in, roughly speaking (the Planck length isn't quite "the smallest length possible", but it's kind of related - read the article for more information). Zero-point energy is a better example of a smallest possible value - it's the lowest energy level a given physical system can have, and it's not zero. --Tango (talk) 17:37, 7 June 2008 (UTC)

Thank you for answering my question. While were on the subject of size extremes I think I will ask if there is indeed a limit to "big". A small limit I can kinda understand because at some point small can become meaningless. But, big- that seems to be a different story. You can always add matter or space, correct? I hear the universe is continually expanding, but into what? Doesn't that mean it gets "bigger"? Is there a limit to how big big can get? Or at some point does matter's own weight collapse upon itself and cease to be?--Sam Science (talk) 21:39, 7 June 2008 (UTC)
 * I don't believe the laws of physics have any upper limits in the way they have lower limits. I'm an idiot, see below. Yes, the universe is expanding, which does indeed means it is getting bigger. It's not really meaningful to ask what it's expanding into, it's more space itself stretching. There isn't an "edge" that's moving further and further from the "centre", each bit of space is just getting bigger and bigger. It might help to imagine a balloon being blown up. Ignore the 3D space that the balloon occupies and just think about the 2D rubber surface as being the universe. There is no centre and no edge to the rubber (ignoring where you're blowing the air into, at least!), it expands by each bit of rubber stretching more and more. There is no limit to how big the universe can get (it may stop expanding eventually, but not because it's reached any kind of size limit - current observations suggest it will just keep going, anyway). I think the closest you get to an upper limit of anything is black holes. Once density increases past a certain point in a region (that point depends on the size of the region, it's smaller for larger regions), it will collapse into a black hole. That kind of puts a limit on densities. --Tango (talk) 21:54, 7 June 2008 (UTC)
 * Well, universal expansion doesn't necessarily mean the universe is getting bigger (it may already be infinite); what it means is that things are moving farther apart. As for the observable universe, it may be getting smaller. --Trovatore (talk) 22:15, 7 June 2008 (UTC)
 * True, I guess doubling something infinite doesn't really count as making it bigger. --Tango (talk) 22:49, 7 June 2008 (UTC)


 * It isn't possible for the size of the observable universe to decrease as such, because past light cones from later times always lie outside past light cones from earlier times, so have a larger cross section in every era. I fixed the article. -- BenRG (talk) 12:07, 9 June 2008 (UTC)
 * ... aaand while we're on the subject, may as well cover the middle. What is the ultimate middlest middle? The "planck middle", if you will? The absolute centerest center of all existence? It can't be my consciousness, because the universe was here long before my chaotically firing neurons showed up :)   --Sam Science (talk) 00:00, 8 June 2008 (UTC)
 * Well, how about zero? That's the middle of numbers. As for a middle in physics, there isn't one. The universe is generally assumed to be (roughly) homogeneous, which means every point is pretty much the same as every other point. You'll always have an origin for your coordinates, which is basically the centre, but you can move that origin to a different place and, on the cosmic scale, it makes no difference to the laws of physics (obviously, on a smaller scale, it matters - things are far different in deep space than they are on the surface of the Earth, for example, but there isn't much difference between the local cluster of galaxies and some cluster billions of light years away). --Tango (talk) 01:04, 8 June 2008 (UTC)
 * I would think that, assuming there was a big bang of sorts, that the location of that would be a middle of sorts. I’m thinking along the lines of suppose one could easily travel anywhere in the universe, if they travel out far enough from this place they should be able to see it taking place. But travel any farther and they would be able to see nothing that resulted from it. e.g. there is a sphere from that center (that constantly gets larger at the speed of light) outside of which the light-borne information from the big bang cannot be observed. GromXXVII (talk) 11:02, 8 June 2008 (UTC)
 * The observable universe has a centre - the point where the observer is. The universe as a whole doesn't. The big bang happened "everywhere", since the whole universe was a single point at the beginning (assuming it's finite, which it could well not be, but the basic concept is the same for an infinite universe, just more difficult to visualise). See a couple of paragraphs up where I explained the balloon analogy, note that the balloon has no centre, it's basically a sphere and no point on it is special. --Tango (talk) 11:19, 8 June 2008 (UTC)
 * [ec]This is a common misconception regarding the big bang. Think about the balloon again (assuming it is perfectly round with no actual air filling hole). When there is no air in the balloon, its surface is a point and its density is infinite. When you start filling it with air, it expands and the distances start increasing. But there is no point on the surface in which the expansion "started". In other words, it's not like there was an infinite space and all the mass was concentrated in a point in this space. The entire space was a point. -- Meni Rosenfeld (talk) 11:21, 8 June 2008 (UTC)
 * Hmm. I guess whether or not it was valid I was thinking of the universe as being nested within some larger space where it therein expanded. GromXXVII (talk) 16:28, 8 June 2008 (UTC)
 * Yes, it's very tempting to do that, but it's best not to if you can help it, it can lead to significant confusion. --Tango (talk) 20:27, 8 June 2008 (UTC)
 * The free physics textbook Motion Mountain describes how all the results of general relativity can be derived starting from the assumption that there is a maximum power (physics) in the universe (or equivalently maximum force, or equivalently maximum rate of mass flow). These maximum amounts are all equal to 1/4G times the appropriate power of the speed of light, e.g. the maximum power is c5/4G. (G is the gravitational constant.) —Keenan Pepper 03:51, 8 June 2008 (UTC)
 * I'm an idiot... of course the laws of physics have upper limits, that's the whole premise of relativity... thanks! (See speed of light for the most obvious one.) --Tango (talk) 11:19, 8 June 2008 (UTC)