Wikipedia:Reference desk archive/Mathematics/2006 July 5

From Wikipedia, the free encyclopedia

Humanities Science Mathematics Computing/IT Language Miscellaneous Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions at one of the pages linked to above.

< July 4 Mathematics desk archive July 6 >


Contents

[edit] intravital pressure unit of measurement

We would like to know if the table below concerning the unit of measure are still valild or has been changed:

                      Japan     USA  Germany  Holland France    UK      Sweden  Australia
Partial pressure of oxygen
in arterial blood      MmHg,Torr mmHg,Torr mmHg(kPa)kPa -        kPa     kPa     mmHg
Partial pressure of carbon dioxide
in arterial blood   MmHg,Torr mmHg,Torr mmHg(kPa) kPa   -       kPa     kPa     mmHg
Encephalon liquid pressure
                        mmH2O   mmH2O   mmHg    mmH2O   -       mmHg    (mmHg)  mmH2O
Intracranial pressure   mmHg    mmHg    mmHg    mmHg    -       (mmHg)  (mmHg)  mmHg
Intraocular pressure    mmHg    mmHg    mmHg    mmHg    -       mmHg    (mmHg)  mmHg
Central venous pressure cmH2O   cmH2O   cmH2O   cmH2O   -       cmH2O   (mmHg)  cmH2O or mmHg
Inner pulse pressure    cmH2O   cmH2O   mmHg    cmH2O   -       (mmHg)  (mmHg)  cmH2O
Pressure of rectum & anus cmH2O     cmH2O   cmH2O   cmH2O   -       (mmHg)  (mmHg)  cmH2O
Intravesical pressure   cmH2O   -       cmH2O   cmH2O   -       (mmHg)  (mmHg)  cmH2O or mmHg
Urethral pressure       cmH2O   cmH2O   cmH2O   cmH2O   -       (mmHg)  (mmHg)  cmH2O
You might have better luck posting this question to the Science Desk. StuRat 22:00, 6 July 2006 (UTC)

[edit] Calculating standard deviation

I was wondering if someone could explain to me the reason for using squared numbers in calculating standard deviation (mean distance from the mean).

Taking the example in http://en.wikipedia.org/wiki/Standard_deviation, we have 4 data points (5 6 8 & 9) with a mean of 7. If standard deviation is (as I understand it) the average distance from the mean, why is the calculation not simply 5 = 2 units from the mean 6 = 1 unit from the mean 8 = 1 unit from the mean 9 = 2 units from the mean totalling 6 units from the mean, divided by 4 data points = average 1.5 units from the mean? (rather than standard deviation which is 1.5811)?

Many thanks in advance TS

There are at least 2 reasons for that:
  • In many practical cases where the dispersion has some importance (quite often, a negative effect), the strength of the effect is greater when there are a few large changes than when there are many small changes. That is, in such a scenario, the data points (10, 10, 8, 12), where there are a few large changes, will have a worse effect than the data (9, 9, 11, 11), where there are many small changes. Squaring the deviations models this phenomenon more accurately, since larger deviations will have larger weights.
  • Since all the terms we add should be positive (otherwise they will cancel each other out), if we don't use squaring, we will have to use absolute values. The absolute value function is not differentiable, therefore expressions involving it are difficult to develop analytically. This limits our ability to build a statistical theory around this measure of dispersion, and ultimately we will not be really able to use it effectively.
This is why the most common measure of dispersion in usage is the standard deviation - it gives very reasonable results, and is easy to work with. -- Meni Rosenfeld (talk) 15:44, 5 July 2006 (UTC)

The quantity you mention is called the mean deviation. It is very rarely used because the standard deviation is much better behaved mathematically. One example is that the quantity \sum (x_i - y)^2 is minimised when y is the mean. However, \sum |x_i - y| does not have a unique minimum - consider two values to see why. McKay 16:09, 5 July 2006 (UTC)

[edit] 0 (Zero)

Me and my friends were sitting in class doing homework a couple months ago (schools out) and there was a question about the number 0. We quickly got through it and went on. One of my friends later presented something rather interesting to the rest of us. What he was saying was that there are infinite positive numbers and infinite negative numbers, therefore, 0 is the middle and therefore,
\infty/2=0
That means, following simple multipulcation and division rules, that
0*2=\infty
But we all know that any number times 0 equals 0. Is there a name for this paradox. I can't find anything doing a quick Google search that relates to my question. Thanks. schyler 19:39, 5 July 2006 (UTC)

Before even considering the correctness of "zero is in the middle": if it is in the middle, it doesn't follow that \infty/2=0, only that (-\infty+\infty)/2=0 (assuming you could do that; it's an undefined operation, see infinity for more info). --cesarb 19:52, 5 July 2006 (UTC)
Yeah, that "in the middle" stuff is wrong. Zero is also "in the middle" of −1 and 1, but that doesn't mean 1 / 2 = 0. If you want a real puzzler, check this out: \infty = {1\over0} = {1\over-0} = -{1\over0} = -\infty —Keenan Pepper 01:50, 6 July 2006 (UTC)

Exactly. It's a paradox, right? Is there a name for it? schyler 03:33, 6 July 2006 (UTC)

No, there is no paradox, only a little fun at your expense.
But as long as we're determined to muddy the waters, we might as well meet some higher mathematics. Before we go any further, we must understand that infinity is not a standard integer, nor rational number, nor real number. We cannot get it as an answer, and we cannot use it in a computation. Since we're about to do both of these things, we're leaving the familiar numbers behind.
  • In projective geometry, great simplification of theorems and proofs is possible by using "points at infinity". Thus we can define a projective line, where each point is represented by a pair of coordinates, essentially treated as a ratio. Points on the usual (non-projective) line might take the form (x:w), with w ≠ 0, while the "point at infinity" takes the form (x:0). Similarly, the projective plane has, not just a point, but a line at infinity. Points are ratios (x:y:w), and the equation of the line at infinity is w = 0. Computer graphics makes heavy use of projective space, with coordinates (x:y:z:w). As an example of a simplification so obtained, in the projective plane any two distinct lines intersect in a unique point, with parallel lines intersecting at a point at infinity. However, the topology of a projective line is the same as a circle, and that of a projective plane more closely resembles a sphere. On the projective line we can define the reciprocal of every number (x:w) to be (w:x), so that 1/0 = ∞.
  • In the complex plane, we often use a one-point compactification, adding a single point at infinity. This gives something different from the projective plane, which has multiple points at infinity. This new topology is exactly that of a sphere. One of the fun things we can do in the extended complex plane is a Möbius transformation. For example, we can turn a circle "inside out", which has the curious property that circles and lines remain circles and lines (though a circle may become a line and vice versa), and angles are unchanged. Inversion in a circle exchanges the center of the circle with infinity, so we again have a sense in which the inverse of zero (the complex number 0+i0) can be infinity.
  • In non-standard analysis, sophisticated theorems from mathematical logic are used to embed the standard real numbers in a larger non-standard model that includes many infinities as well as their reciprocals, infinitesimals. Unlike the projective line, there is no "wrapping around", and we prohibit taking the reciprocal of zero. Although we again have multiple infinities, this is also quite different from the projective plane.
  • In the IEEE floating-point standard (IEEE 754), numerical algorithms are simplified by the inclusion of distinct values representing positive and negative infinity. It also attaches a sign to zero, and includes different kinds of "not-a-number" (NaN). The standard includes rules for arithmetic with these values. This is arithmetic specifically designed for computer use; it is not the mathematicians' idea of real arithmetic.
These are just a few of the diverse possibilities for giving infinity, or infinities, a formal meaning, and of interpreting 1/0. --KSmrqT 06:33, 6 July 2006 (UTC)
You've left out Conway's surreal numbers which contain some infinitely biggies too. -lethe talk +
I got tired of writing! I left out cardinals, and topos-based reals, and … You might say I was exhausted, rather than exhaustive! :-) --KSmrqT 08:27, 6 July 2006 (UTC)
To clarify: Your original argument had no paradox, just a fallacy. There's no reason at all why ∞ / 2 should be 0. About Keenan's ∞ = -∞ argument: No paradox there either. It just shows that division by zero can only be sensibly defined in structures where ∞ = -∞ - For example, the real projective line and Riemann sphere which KSmrq mentioned. -- Meni Rosenfeld (talk) 08:31, 6 July 2006 (UTC)


Your friend's logic is wrong. --Proficient 09:41, 6 July 2006 (UTC)
There is another possibility to consider. Since infinitesimal is the inverse of infinite and is "a number that is smaller in absolute value than any positive real number" and \begin{matrix}\frac{1}{10}&=&.1;\quad\frac{1}{10+\frac{1}{10}}&=&\!\!\!\!\!.09900990099...;\qquad\qquad\qquad\qquad\qquad\\\frac{1}{100}&=&.01;\quad\frac{1}{100+\frac{1}{100}}&=&\!\!\!.0099990000999900009999...;\qquad\qquad\quad\\\frac{1}{1000}&=&.001;\quad\frac{1}{1000+\frac{1}{1000}}&=&.000999999000000999999000000999999...;\end{matrix}\,\!
it follows that (\iota\,\! is infinitesimality or iota) \frac{1}{\infty}=\iota;\quad-\frac{1}{\infty}=-\iota;\quad0=\sqrt{\frac{1}{\infty+\iota}\times\frac{1}{-\infty-\iota}}=\frac{i}{\sqrt{\infty+\iota}};\,\!
This solution takes into account the +/- dilemma! P=) ~Kaimbridge~16:30, 6 July 2006 (UTC)

[edit] Proof?

How does one prove that if 1) a≠0 2) b≠0 3) a+b=ab, then a or b =2? Are those the only possible values? --Tuvwxyz 23:54, 5 July 2006 (UTC)

As a first step, one might try subtracting a from equation (3). I would also note that the assertion is only true for a and b integers, and that the reference desk is not a place for homework problems. Tesseran 01:34, 6 July 2006 (UTC)
There are infinitely many possible values. —Keenan Pepper 01:46, 6 July 2006 (UTC)
A proof would depend on how much mathematics is familiar, and on whether a and b are supposed to be integers, or real numbers or integers modulo n or whatever. Here's a strategy for the integers: Let a be some value other than 2, such as 3. Consider possible values of b and look for a systematic argument about their success or failure. Then take another value for a and do the same thing, and so on. It should not take too many experiments to begin to understand the possibilities. Now turn that understanding into a proof. A strategy for reals might be different, taking advantage of algebraic manipulation not available with integers. This is such an elementary problem that the most valuable thing we can do for you is encourage you to solve it for yourself. If you get stuck formalizing a proof, write down for us your informal understanding and we can see how you're doing. --KSmrqT 01:53, 6 July 2006 (UTC)
Almost forgot: working with integers modulo 9, two solutions are a = 5, b = 8 and a = 3, b = 6. (Challenge: Are there others?) But I suggest you not mention these in a homework solution, as it will be obvious that you didn't do the work yourself. :-) --KSmrqT 03:31, 6 July 2006 (UTC)
Here's an approach. First, substitute x=a-1, y=b-1 to get a simpler equation in x and y. It should be obvious that the only integer solutions to this simpler equation are (x,y) = (-1,-1) or (1,1). Convert back to (a,b) values and you are done. Gandalf61 13:17, 6 July 2006 (UTC)