Wikipedia:Reference desk/Archives/Mathematics/2007 January 4

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January 4

Nanosecands

How many nanosecands are there in a 986 billion years?

I assume you mean nanoseconds? That depends on the length of your year: the common year of 365 days, the Julian year of 365.25 days, the Gregorian year of 365.2425 days, the sidereal year of 365.256 363 051 days, or some other year entirely. --Carnildo 06:39, 4 January 2007 (UTC)

You can find this out with Google calculator. Try Googling for 986 billion years in nanoseconds and your answer will be right there. Note that Google appears to use a Gregorian year. For more information, check out the Year article, which lists all the different year lengths in days. Once you've chosen a year, multiply its length in days by 986 billion (for the number of years) and then multiply that by 864 x 10¹³, which is the number of nanoseconds in a day. Maelin (Talk | Contribs) 07:14, 4 January 2007 (UTC)

Why 986 billion years? That's an awfully long time. Where does the number come from? The universe may not last nearly that long; see Big Crunch and Big Rip. --Lambiam^Talk 09:40, 4 January 2007 (UTC)

Apparently because of the news that came out last year saying the Universe could be 986 billion years old. Why nanoseconds, then? I think it was just one of those "lol science" or "lol big numbers" question. — Kieff 09:54, 4 January 2007 (UTC)

LOL. If you read that article carefully it says "986 billion years older than physicists thought", because somone had come up with a multi-big-bang theory in which the universe is at least 1 trillion (10¹²) years old, and 986 billion is the difference between 1 trillion and the "orthodox" age of 14 billion years. —The preceding unsigned comment was added by Gandalf61 (talk • contribs) 15:50, 4 January 2007 (UTC).Who came back to add his sig and found some darn bot had beaten him to it. Gandalf61 15:54, 4 January 2007 (UTC)

Heh. – b_jonas 19:24, 5 January 2007 (UTC)

Realistically, it depends on where you take your calculation from. Say, if you took it from on or after 1AD (or 1CE), then all you need to remember is that a leap year occurs 97 times in 400 years (you have to take into consideration the fact that a leap century occurs every 400 years; if a century is not divisible by 400, like 1900, then it does not have the 366th day); similar is the case if you finish your 986 billion years on or before 1BC (1BCE). However, if your period of time crosses between the two eras (since there is no year 0), then you need to take into consideration the six years without a leap year between 3BC and 3AD (3BCE and 3CE).

However, I'll keep this really simplistic, it's basically such:

${\begin{array}{rl}{\frac {986\times 10^{9}}{400}}\times 146097\times 24\times 60\times 60\times 10^{9}&=2465\times 10^{17}\times 146097\times 864\\&=146097\times 212976\times 10^{18}\\&=31115154672\times 10^{18}\end{array}}$

--JB Adder | Talk 00:01, 14 January 2007 (UTC)

When addition maps A×B->A but not A×A->A

In physical applications, if one is pedantic there is a difference between duration and time, and between absolute and relative temperature. For example, the difference between 400K and 300K is −100K even though a temperature of −100K is meaningless. In mathematical terms, it looks like this:

addition $:(A\times B)\cup (B\times A)\to A$ (E.g., the sum of a position and a displacement is a position.)

subtraction $:(A\times A)\cup (B\times B)\to B$ (E.g., the difference of positions is a displacement, as is the difference of displacements.)

Is there mathematics that describes this? It appears that relative position, relative temperature, etc. is a vector space but absolute position is not. —Ben FrantzDale 15:34, 4 January 2007 (UTC)

A quick study suggests affine space. — Arthur Rubin | (talk) 15:52, 4 January 2007 (UTC)

That looks right. Thanks! —Ben FrantzDale 16:13, 4 January 2007 (UTC)

See also torsor. Dave Rosoff 00:11, 5 January 2007 (UTC)

A related question

In physics, the vector space of displacements (measured in units of distance) is distinct from the vector space of velocities (measured in units of speed), however we can define an angle between a velocity vector and a distance vector. Is there mathematical language to describe this relationship? Thanks again. —Ben FrantzDale 18:23, 4 January 2007 (UTC)

Thinking simply - at a given instance - both are just directions - ie vectors in 3 dimensional space - so just use the formala that gives the angle between two lines eg the dot product#Geometric interpretation87.102.8.102 19:45, 4 January 2007 (UTC)

Your 'vector space of velocities' is really just a 'vector space of distance' where the distances are functions of time (an external variable). So you don't really have two 'vector spaces' at all. 87.102.8.102 20:15, 4 January 2007 (UTC)

Since d(distance)/d(time) = velocity you could say that the 'mathematical language to describe this relationship' is one of differentiation and integration.87.102.8.102 20:20, 4 January 2007 (UTC)

I guess {displacements} × {durations} forms a 4-D vector space, but there seems to be some additional structure allowing for division by duration. It seems that velocity is either in a different vector space from displacement in that you can't add them, but they obviously have some relation to one another since you can take angles between them. (That is, I feel like group theory has something to say about this.) —Ben FrantzDale 20:37, 4 January 2007 (UTC)

Good observation; we cheat. (But not much.) Suppose we have an ice hockey rink marked off in cartesian coordinates x and y, measured in metres. The displacement of a puck on the surface would be a vector such as (3 m,4 m), where the units of displacement are metres. If the puck traverses that distance in 0.5 seconds, its velocity is a different vector, (6 m/s,8 m/s), where the units of velocity are metres per second. Physics training tells us to pay attention to the difference between units, for "we must not mix apples and oranges". But the velocity has a direction which is a pure displacement, and the angle between two nonzero vectors does not depend on the length of the vectors. So we quietly convert from velocity to displacement.

Such quiet conversions are everywhere, even in pure mathematics. (Computer programming languages do it so often they have a formal term to describe it in reference manuals: "coercion".) For example, when we write 3 + 0.1415 we appear to be adding an integer and a decimal fraction. In fact, we quietly use the canonical injection of the integers into the rationals (or the reals) to make both numbers rational (or real) before we add. --KSmrq^T 02:12, 5 January 2007 (UTC)

The thing is, I'm sure this can be done without this sort of hand waving. I just haven't seen it done. My intuition tells me that velocity is in a separate vector space from displacement but they are somehow closely linked in that, like the anonymous person above said, velocity is in the space of time derivatives. I just haven't seen rigorous math to flesh out that intuition. —Ben FrantzDale 14:26, 5 January 2007 (UTC)

I'm sorry if I did not make myself clear enough. Far from hand waving, I told you what the formal interpretation must be for this to work. We are cheating, in that (normally) we do not explicitly mention the mapping from the vector space of velocities to the vector space of displacements; but, again, brevity is standard practice in many circumstances. Sometimes a text in mathematics will acknowledge such elision with the phrase "by abuse of notation". We write to communicate to humans, not computers, and we do not include all the formally necessary notations when they hinder more than they help. --KSmrq^T 10:00, 7 January 2007 (UTC)

I write to communicate with computers as well as humans. :-) —Ben FrantzDale 00:22, 8 January 2007 (UTC)

1 2 3...?

what number is after three? —The preceding unsigned comment was added by 74.102.217.142 (talk • contribs) 4 January 2007.

Difficult question - could you give some more contextual information.87.102.8.102 20:16, 4 January 2007 (UTC)

See above question (1 plus 1) by same anon user, who appears to be trolling. -- NorwegianBlue ^talk 20:27, 4 January 2007 (UTC)

Also, take a look at the user's contributions. — Kieff 20:41, 4 January 2007 (UTC)

The sequence you have described seems to be F₂ to F₄ of the Fibonacci series. Therefore your answer is F₅, which is 5. Readro 20:35, 4 January 2007 (UTC)

The fourth number can be any number n that you care to chose, if the numbers are the first four values of the sequence given by $f(x)={\frac {(x-2)(x-3)(x-4)}{-6}}+(x-1)(x-3)(x-4)+{\frac {3(x-1)(x-2)(x-4)}{-2}}+{\frac {(x-1)(x-2)(x-3)n}{6}}$ This function can be adapted to generate any arbitary finite sequence that you care to suggest, and is fundamentally uninteresting. -- AJR | Talk 00:56, 5 January 2007 (UTC)

For these questions, the right resource is the OEIS. If you search for 1, 2, 3, you'll find that the next number is most likely one between 1 and 6 inclusive: the first few hits give 5, 5, 6, 3, 1, 2, 2, 4, 5, 4, 2, 6, 2, 4, 4, 4, 4, 4, 4, 4, 4, 2, 3, 3, 3, 3, 3, 3, 4, ... as the next single number after 1, 2, 3 —The preceding unsigned comment was added by B jonas (talk • contribs) 19:21, 5 January 2007 (UTC).

Philosophical question

This question may be more suitable for the humanities page but here goes..

I notice that my world experience (outside imagination) can be mostly explained by simple mathematical concepts - euclidean space, mechanics, statistics, fractals etc. (ie basic science and engineering type maths)

My question for any more advanced mathematicians is - does knowledge and understanding of other more 'obscure' mathematical ideas cause an increase in the level of experience of the physical world - eg does an understanding of hyperbolic geometry, greater than 3 dimensional space, number theory etc enhance your experience of the world? (I'm working on the principle that 'inner knowledge' (in the 'brain') is reflected in 'outer experience' (in the world) that so far has seemed to be true.)

I hope someone can understand what I'm asking and I would really appreciate any replies you can give. Thanks.87.102.8.102 20:43, 4 January 2007 (UTC)

To turn the question on its head, consider if it is possible for any accurate knowledge to degrade or detract from your experience of the physical world. The answer there is that all knowledge, if true, at worst is irrelevant to your set of outside world experiences, in which case there would be no impact positive or negative. If knowledge is relevant, though, it can not help but to enhance the level of an experience. The situations where increased knowledge fails to enhance experience are those where knowledge is faulty or incomplete or misleading, all of which can lead to a correspondingly flawed experience or flawed understanding.

A beauty of mathematics, then, is that it possesses a particularly high degree of accuracy within it, such that knowledge based on mathematical principles has a correspondingly high probability of enhancing understanding. At worst, an understood mathematical principle will have no practical application to the outside world, but when it does possess relevancy it can not help but to enhance the personal understanding of the situation. (Not sure if that relates to your question, but just a thought.) Dugwiki 21:51, 4 January 2007 (UTC)

I do think that knowing obscure mathematics can improve your way of perceiving the world, but not always in the direct sense of understanding it more, but of getting more awe out of it, of seeing more beauty or interesting things on it. By knowing about the homeomorphic equivalence between a donut and a coffee mug, you can have a nice feeling of surprise during breakfast. That sort of thing. — Kieff 22:01, 4 January 2007 (UTC)

Just choosing one area of higher math as an example, I find that abstract algebra is reflected in my day-to-day experience, both on a literal level and on a metaphorical one. I certainly find groups in various places, particularly anytime I'm dealing with permutations. I can't arrange the billiard balls in the rack for 8-ball without thinking about permutation groups, and decomposition into disjoint cycles. (Well, maybe I can, but I prefer not to!)

On a different level, I remember finding it tremendously satisfying to learn the Fundamental theorem on homomorphisms for groups, and to apply it to human interactions. The theorem basically says that, to have a nice mapping from one group to another, they have to have certain correspondences in their internal structures (namely, the first group has to have a normal subgroup that induces a quotient group isomorphic to some subgroup of the second group). I like to think of that in terms of people - in order to interact with another human, you have to have something inside you that's common with something in that person, and that can form the basis for a relationship. A bit whimsical perhaps, but that's the kind of stuff I think about. -GTBacchus^(talk) 23:26, 4 January 2007 (UTC)

Yes thanks for all those answers - I have similar experiences. But I'd really be interested in the point of view of someone who understands say hyperbolic geometry or 4d space or something else I don't know much about. (Though I probably wouldn't understand what they were saying) - and if they can find examples of that in the world we inhabit. Thanks to you above though.87.102.23.224 23:45, 4 January 2007 (UTC)

When I studied Lobachevsky, I remember a professor making an analogy between hyperbolic geometry and love. Parallel lines, in a space with negative curvature, don't remain the same distance apart, but continually approach one another, becoming more and more aligned the closer they get. This professor used it as a metaphor for marriage, which I thought was sweet.

As for higher dimensional spaces, those are easy to apply to situations if we don't assume that all dimensions have to be spatial dimensions. A lot of things can be described by more than 3 mutually independent variables, so they could be thought of as inhabiting points in an n-dimensional space, where n is just about any number you like. One very concrete example is the phase space of a particle, like Planck used in his development of the idea of entropy. One particle is described by six numbers - three for position and three for velocity. The air in a room can be described as a point in (6*n)-dimensional space, where n is the number of molecules of air in the room. This is a useful tool for understanding how some aspects of thermodynamics work, and can be extended to many other areas, literally or by analogy. -GTBacchus^(talk) 23:56, 4 January 2007 (UTC)

Thanks again - it seems that hyperbolic geometry (like love) will remain in the realms of the mind for now.87.102.19.164 01:12, 5 January 2007 (UTC)

Ha, ha. Lobachevsky's "Theory of Parallels" - his work where he develops hyperbolic geometry - is quite accessible, actually. If you follow along with some paper and a pencil, you can get quite a ways into it. Good luck to you. -GTBacchus^(talk) 01:17, 5 January 2007 (UTC)

We could get into theories of mind and explore philosophy and recall the history of mathematics and physics. But perhaps one or two examples will be more informative and inspiring. Euclidean geometry, by rejecting the parallel postulate split into three choices: Euclidean, spherical, and hyperbolic. All of these have one thing in common, namely constant curvature (zero, positive, and negative, respectively). Although this mathematics began as an abstract inquiry, surfaces with these properties exist tangibly. Then Gauss and Riemann went one step further, and began the study of differential geometry, where the curvature could vary depending on location. This more accurately describes most surfaces. Einstein realized that a synthesis of varying curvature with higher dimensions provided exactly the tools he needed to extend special relativity to include gravity. He, and all modern physicists, could "see" curved higher-dimensional space in the physical universe.

Mathematics is sometimes described as the formal description of patterns, and the more patterns we know about the more we "see". If you would like to experience this yourself, read "Caustic (mathematics)" and then "Caustic (optics)". Working in reverse, computer graphics generates images simulating the real world out of pure mathematics. When trained mathematician and Academy-award winner Alvy Ray Smith looks at a plant, what he "sees" is surely shaped by his study of L-systems. Visit the pages of Przemyslaw Prusinkiewicz for beautiful examples of what that vision can accomplish. --KSmrq^T 02:46, 5 January 2007 (UTC)

The manager of a small shop is an example of using higher dimensional vector spaces involving the quantity, the price for apple, banana, cherry, donut, etc. Actually the administrators use muti-dimensional product sets including our name, age, sex, income brackets, height, weight, etc to approximate a person. Twma 01:33, 10 January 2007 (UTC)