Wikipedia:Reference desk/Archives/Mathematics/2009 April 21

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April 21

A*B*C = B cubed – B, where A, B and C are successive integers?

Hi - my brilliant 15-y-o cousin Tom Schwerkolt-Browne came up with the above rule or observation. Is it original to him? Is it significant or non-trivial or whatever the term is?

Thanks - Adambrowne666 (talk) 12:23, 21 April 2009 (UTC)

It's pretty straightforward if you start with A*B*C = (B-1)*B*(B+1), and multiply it out. -- Coneslayer (talk) 12:28, 21 April 2009 (UTC)

(After edit conflict) Yes, it's a fairly obvious identity. If your cousin is generalising from a few numerical examples, he might want to see if he can prove that it is always true for any three conecutive integers (positive or negative). Difficult way - call your integers x, x+1 and x+2, expand x(x+1)(x+2) and compare it to (x+1)³-x. Simpler way (as per Coneslayer) - call your integers x-1, x and x+1 - then there is less algebra. And notice that there is nothing in the proof that assumes x is an integer ...

Related problem - show that n³-n is always a multiple of 6, for any integer n. Gandalf61 (talk) 12:36, 21 April 2009 (UTC)

The point to note after having read the comments here is that mathematics is broader than what your 15 year old cousin and you think. A reasonably bright 12 year old would be able to verify this identity, and usually an indicator of whether your "discovery" is trivial or non-trivial is to check the length of the proof of your claim. Algebraic identities are almost always trivial (with respect to high-school algebra). The other thing that you should learn is that it is very rare for someone as young as your cousin to discover something new in mathematics, unless they have had training in mathematics at undergraduate level or so. There is nothing to discover within the bounds of high-school mathematics as far as I know the difficulty of high-scool mathematics to be. Lastly, please do not be ignorant. I do not mean to sound rude, but it is very ignorant to think that a mathematical discovery can be achieved within the bounds of a few minutes thinking; not to mention the fact that it is insulting to reasearchers in mathematics. I do not mean to discourage you from asking questions here, but it is important that you understand the intended meaning in my comments. --PST 13:39, 21 April 2009 (UTC)

I am certain that Adam's pride in having a bright and inquisitive younger cousin was not intended as a savage attack on the practitioners of your field. With further encouragement, I am sure that his cousin will make meaningful contributions to mathematics or whatever field he chooses to go into, even if that day is some years off. -- Coneslayer (talk) 14:01, 21 April 2009 (UTC)

Certainly not. My comment was not that his question was unacceptable, but rather that he should understand that mathematics is broader than he thinks. --PST 14:14, 21 April 2009 (UTC)

We are here to answer reasonable questions in the same spirit, not to guess the quality of anyone's thoughts.Cuddlyable3 (talk) 14:34, 21 April 2009 (UTC)

PST: my reasonably bright 12 year old grandson was stumped. -hydnjo (talk) 21:49, 21 April 2009 (UTC)

Thanks, everyone, for the answers, and thanks for the defence, Coneslayer and others; but maybe Point-set is right - I was being ignorant, or at least romantic - had a naive hope that Tom had chanced on something interesting. Adambrowne666 (talk) 05:51, 22 April 2009 (UTC)

In my nursery school they taught me the multiplication table of 5 (1x5=5, 2x5=10,...,10x5=50). Some days later, I realized that one can go further (11x5=55,.....) and I was deeply impressed by this fact... Grown-up people didn't show much interest, but for me it was an important observation and I was quite proud of it, and some of my playmates warmly congratulated. --pma (talk) 10:22, 22 April 2009 (UTC)

Indeed. I felt the same way at 10 or 11 when I discovered how to use forward differences to extrapolate sequences of integer powers. I imagine most amateur and professional mathematicians have had some similar experience in childhood or adolescence that motivated their continuing interest in mathematics. A teenager who is actively exploring the world of mathematics should be encouraged by all means possible, not put down because they have not yet reached the edge of the map. Adambrowne666 - if your cousin is interested in widening their mathematical horizons, I recommend they read Courant and Robbins' classic What Is Mathematics?. Gandalf61 (talk) 11:38, 22 April 2009 (UTC)

Absolutely. Just because a discovery isn't original doesn't mean it isn't impressive or significant that you discovered it. --Tango (talk) 12:22, 22 April 2009 (UTC)

Certainly. I never discouraged his desire to discover more identities in algebra. Nor did I criticize his eagerness when he thought that he made a discovery. All that I said (or what I intended to say), which he will carry on in the future (and make better judgements as to what is a discovery and what is not), is that it is unlikely for something like that to be a new discovery - and that although it is nice that he is interested, he should understand this. --PST 13:31, 22 April 2009 (UTC)

In the context of undergraduate research in mathematics (much less research at the high-school level), a common metric for novelty is that the results are new and interesting to the student who discovers them, not that they are new from the point of view of the research literature. Of course not all results that meet this metric will be publishable, but publication is not usually the goal. There are many interesting problems in elementary combinatorics and number theory that, while they may be easily solved by an expert using high-powered techniques, still provide a great experience for a novice. The same can be said for assembling a car in your garage from a kit; you know ahead of time that the factory could do it faster, but that isn't the point.

However, PST is correct that, since number theory has been heavily studied for centuries, one is unlikely to discover any truly new results at the elementary level.

Another book that a bright high school student might want to pore over (and will take work) is A course in pure mathematics by Hardy. — Carl (CBM · talk) 14:21, 22 April 2009 (UTC)

OK. Let me correct myself - if you find a proof that every even number is the sum of two primes (or an equivalent), publish it. It is not trivial. Similarly, there are many other facts of this nature which are seemingly trivial but in actuality non-trivial. The point is however, algebraic identities tend to be usually trivial or not worth any interest to professional mathematicians. It is perfectly OK (in fact, encouraged) to prove such identities, but what I wish to say is that such identities are unlikely to be of any interest (or non-trivial) to professional mathematicians (but certainly maybe of interest to high-school students). In my view (others may not share the same view), high-school students should understand the purpose of mathematics (or what mathematics really is) before venturing out to make discoveries. Therefore, you should probably read the texts indicated above but in any case, I encourage your cousin to continue mathematics - whatever the nature. --PST 03:16, 23 April 2009 (UTC)

There are other areas in which an amateur mathematician can make a difference. Martin Gardner has many very long lists of original discoveries by modern amateurs, and progress continues. This is generally in areas that professional mathematicians don't bother with, such as board games and origami and so on, but it's still original research and can be a good introduction to the subject. Black Carrot (talk) 22:42, 23 April 2009 (UTC)

Quantum mechanics?

I was bored sitting in work and a colleauge said look up Schrödinger's cat.... While reading it...... it seemed as if the only way to explain Quantum mechanics is "the end product of any situation is allready predetermined"; applying this to the riddle, as far as i see it would explain the answer to the Schrödinger's cat experiment...the cat is either pre destined to be dead or alive this was pre determined right back to the big bang hence all future things are all ready predestined....(god knows what that makes to a time theory?) For example we know the Sun will burn out but it hasnt happened yet' its fule will be used up it will swell then cool ECT....this is all ready predetermined.So my question is if the universe surrounding us is all ready played out and the end states are known (but not by the human race) is there an equation that would fit this model? Ok i dont want you to look at this and think this guy is a nutter 'which i am sure you will' but it makes sense to me....we just need to be able to see the future is all :) Adrian O'Brien —Preceding unsigned comment added by 214.13.113.138 (talk) 12:29, 21 April 2009 (UTC)

I'm not sure but I think there is. Even if there is, the equation has to be really complicated. Also, I recommend you on posting this question on the science reference desk instead. Superwj5 (talk) 13:18, 21 April 2009 (UTC)

It is nice that you are interested in this - I suggest you read the article Quantum mechanics. Especially, it is recommended that you read the "overview" section and the beginning of the introduction. Just read what you understand and ignore what you don't. The other thing I wish to comment upon is your belief that mathematics is described by "equations". This is false. Group theory does indeed apply in quantam mechanics as well as probability theory, and the deep purpose of either subject has nothing to do with equations. Just let me stress that mathematics is not equations. If you are referring to the uncertainty principle, then:

$\Delta X\Delta P\geq {\hbar \over 2}.$

Also, I recommend that you read Introduction to quantum mechanics first as this is (apparently) more accessible. I do not wish to discourage you from asking questions but I just wish to stress the mathematics that applies to theoretical physics is much deeper than you think (for example the application of knot theory (a branch of mathematics) to string theory (a branch of physics)). --Point-set topologist (talk) 14:14, 21 April 2009 (UTC)

All equations including mistaken ones fit the OP's notion of predetermination. It made me post this. Cuddlyable3 (talk) 14:25, 21 April 2009 (UTC)

I was under the impression that it was classical mechanics that deals with a deterministic world, while quantum mechanics deals with a probabilistic one. What I mean to say is that if the position and momentum of every particle in the world were known at one instant, the state of the universe at any future instant is easily obtained by applying Hamilton's (or Lagrange's) equations. However in quantum mechanics it is intrinsically impossible to know the position and momentum of even one particle to arbitrary precision, let alone all of them. I would disagree with the OP's statement that the end result of any situation is predetermined, as we can only state the probability of obtaining any particular end. mislih 23:08, 21 April 2009 (UTC)

As a matter of fact we are in a position to compute precisely the evolution of the universe. The only thing is that, so many are the data, so high the precision needed, so complicated the equations, that the simplest computer able to do that, is the universe itself. It is reasonably fast: it takes not more than 24 hours to predict what will happen tomorrow ;) pma (talk) 07:44, 22 April 2009 (UTC)

Cutesie, small-texted answers aside the question at hand is whether quantum mechanics and, if QM is a valid description of reality, is deterministic. Many take the stand that the only reason that it is treated as a probabilistic is that we have incomplete knowledge of the universe, and if we knew the exact state of the universe now we could predict it at some future date. My understanding of the uncertainty principle is that complete knowledge is not possible, and that the universe is therefore not deterministic. Maybe this question would get more expert attention if it were at the science desk. mislih 17:56, 22 April 2009 (UTC)

I think even without the uncertainty principle, QM is non-deterministic. Even if you have perfect information about an atomic nucleus, say, you can't predict when it will decay because that is inherently random. --Tango (talk) 18:11, 22 April 2009 (UTC)

QM is nondeterministic; that doesn't mean the universe is. I'm not even sure what it means to say that the universe is deterministic or nondeterministic. Among things that didn't happen, how do you distinguish those that couldn't have happened from those that could have and merely didn't? Newtonian determinism doesn't really buy you much in this regard. Newtonian physics would appear to permit the existence of a universe in which Keanu Reeves is never born, yet we don't live in that universe. Is his existence logically necessary, or is there an element of randomness in the world? Those are the only two possibilities. Even without QM most people would probably guess that he's random.

I wouldn't be at all surprised if there turned out to be a deterministic layer underneath QM. For example, as the original poster said, "everything could be predestined" in the sense that there is some kind of constraint on the final conditions of the universe as well as the initial conditions. When a measurement takes place the outcome that is realized is the one that leads to the necessary final state of the universe; if the final condition is specific enough and the second law of thermodynamics holds, then there can only ever be one such outcome. The outcomes could easily look random, at least at times far from the beginning and end of the universe. We've invented deterministic processes that no one, not even their designers, can figure out how to distinguish from "true" (i.e. quantum) randomness. I'm not saying I believe everything is predestined in this way, just that you shouldn't conclude on the basis of QM that the universe is truly random (any more than you should conclude on the basis of Newton that it truly isn't). -- BenRG (talk) 20:46, 22 April 2009 (UTC)

I would define a deterministic universe as one where someone with complete knowledge of the state of the universe at a given time and complete knowledge of the laws of physics could not accurately and reliably predict the state of the universe at a later time - I think that's a standard definition. If the universe were governed by Newtonian physics, it would be deterministic. Someone with a complete knowledge of Newtonian physics and the state of the universe at any time prior to Keanu Reeves' birth would be able to work out whether he will be born or not. There is no randomness, although there is chaos which can appear random - that's why you need complete knowledge of the state of the universe, not just almost complete knowledge. If there universe were governed by QM, then it would be non-deterministic and no-one, however knowledgeable, could know for certain whether or not Reeves will be born until it happens. Now, we know the universe isn't actually governed by either of those sets of laws, so whether or not it is deterministic is unknown - the evidence certainly suggests that it isn't, though. --Tango (talk) 21:27, 22 April 2009 (UTC)