Wikipedia:Reference desk/Archives/Mathematics/2007 November 2

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November 2

Currents

I am still having trouble with this concept. It is a different question though {{{Sailing with the current, a boat takes 3 hours to travel 48 miles. The return trip, against the current, takes 4 hours. Find the speed of the boad and the speed of the current}}}

How on earth do I solve that!? —Preceding unsigned comment added by Camillex3 (talk • contribs) 01:11, 2 November 2007

Speed = distance/time

What are the two speeds? Are they the same. If not why not? Theresa Knott | The otter sank 01:26, 2 November 2007 (UTC)

It's a linear system of two equations in two variables. If b=boat speed and c=current speed, then

${\begin{cases}b+c={\frac {48}{3}}\\b-c={\frac {48}{4}}\end{cases}}$

Keep in mind that b and c are speeds, which is miles/hour not just miles or just hours, so you have to recreate those units from our knowns. Looking at the system, the easiest method would seem to be Linear combination. A math-wiki 01:38, 2 November 2007 (UTC)

One essential fact must be understood to have any hope of solution. When the boat is sailing against the current, the speed of the current diminishes the speed of the boat; when the boat is sailing with the current, the speed of the current augments the speed of the boat. This is a property of the real world, well-known by any student of physics, and hopefully by pilots and swimmers, too. In a simple case like this, it is trivial to write a formula for the effective speed against the current, and one for that with the current. (With a cross current, the mathematics abruptly becomes more sophisticated.) Also in this case the current does not affect the distance traveled. (A cross current can.)

Now is a good time to review Pólya's suggestions. What is given? We have the distance traveled, 48 miles; we have the time with augmented speed, 3 hours; and we have the time with diminished speed, 4 hours. What are the unknowns? We want to know the speed of the current (which is unchanging); we also want to know the speed of the boat without any current. We may also want to find the two altered speeds, though the problem does not specifically ask for those.

We know that distance traveled is the product of speed and time. (This is again physics, and something we are expected to know.) Clearly we can write one equation in current and boat speeds for each direction of travel. If you have never seen such a pair of equations before you might have some difficulty in solving them. But it is hard for me to imagine that you would be asked to solve this problem if you had not been shown a method of solution, perhaps something like Gaussian elimination. So at this point you should study your text and your class notes. --KSmrq^T 02:31, 2 November 2007 (UTC)

Circles and hyperbolas, a "Complex" relationship???

I just though of an interesting relationship between Circles and Hyperbolas. Specifically the unit cases of both.

$x^{2}+y^{2}=1$ and $x^{2}-y^{2}=1$

If y is replaced by yi in $x^{2}+y^{2}=1$, then

$x^{2}+(yi)^{2}=1$

$x^{2}+y^{2}i^{2}=1$

$x^{2}+y^{2}(-1)=1$

$x^{2}-y^{2}=1$

A Hyperbola is just a Circle with y being a pure imaginary number. This is most intriguing, anyone know more about this relationship??? Thanks. A math-wiki 01:49, 2 November 2007 (UTC)

Yep, and in fact you've only proven that one family of hyperbolae are related to circles, but all the rest work if you go to a more general case of ellipses, because they're all conic sections and hence all fall under the same general formula. Confusing Manifestation 02:33, 2 November 2007 (UTC)

Yes I could also generalize the result, i was much more interested in the geometric implications of the comparison, dealing with compairing the Complex Cartesian plane with the Real Cartesian plane. A math-wiki 07:45, 2 November 2007 (UTC)

Have you seen Hyperbolic function#Hyperbolic functions for complex numbers? —Keenan Pepper 14:22, 2 November 2007 (UTC)

If you look at the article on conic sections, you will notice that the 2d conic section graphs are actually slices of a 3d figure. In fact, that 3d figure is a slice of a 4d figure and the substitution of $y=iy'$ that you did gives you an orthogonal transposition of the slice that you're looking at. Doing more complicated substitutions cam give different slices of the 4-d figure in 2-space. I'm not sure that Keenan's suggestion is directly relevant, although the interesting relationship between real and complex functions is fascinating and capable of great depth. If you have decent calculus, Tristan Needham's Visual Complex Analysis [1] is a good starting point for self-study (or better still, directed study). I would encourage anyone interested in complex numbers to buy this book. Donald Hosek 18:47, 2 November 2007 (UTC)

statistics calculator?

Is there a calculator button where I type in a z score like -1.6 and then press a magic button and then it gives me the area nuder the standard normal curve to the left of that value, in this case, 0.0548. Typing 0 and then this button would give 0.5 of course.

What's a good calculator for doing this kind of work and other stats work, like plugging in a data set and then seeing a histogram of it, or putting in coordinates to show correlation and even display scatterplot with least squares regression line, etc.--Sonjaaa 08:02, 2 November 2007 (UTC)

You may be better off using a computer rather than a calculator, for example R (programming language), which is a very powerful statistical system. --Salix alba (talk) 08:34, 2 November 2007 (UTC)

You may find something of use at our article, Standard deviation. In particular, I think Chebyshev's inequality, or Cumulative distribution function may apply here? - Rainwarrior 08:35, 2 November 2007 (UTC)

The area under the normal curve is intimately connected to the error function, or erf for short. Whatever calculator \ software you have access to, try looking for it (most calculators don't have it). -- Meni Rosenfeld (talk) 14:09, 2 November 2007 (UTC)

What brand and model is your calculator, if it's a TI-80series or TI-92 or TI-200 Voyage (what I have) then it can do it. I don't know how on any other calculators though. A math-wiki 20:11, 2 November 2007 (UTC)

cartesian product

assume the interval(0,1),now if A is the set of rational numbers of (0,1)and B is the set of irrational numbers;just irrational real numbers;of (0,1).the question is what (AхB)mean here?i mean how to find the cartesian product of (AхB)?86.62.27.222 11:53, 2 November 2007 (UTC)O`NEIL

It's simple: a cartesian product of A and B is a set of all such pairs (a,b) that a belongs to A and b belongs to B, i.e. all of pairs (<some rational number>, <some irrational number>) from the unit square (assuming that when you describe B as 'the set of irrational numbers', you meant 'the set of irrational numbers in (0,1)'. --CiaPan 12:02, 2 November 2007 (UTC)

Any clues as to this equation?

Hi all, does anyone have any idea what this represents? [2]

Many Thanks! --Chachu207 ^{talk to me} 11:54, 2 November 2007 (UTC)

Can't really help much, but it looks physics-related, so you may have better luck at the science desk. -- Meni Rosenfeld (talk) 13:43, 2 November 2007 (UTC)

Try Maxwell's equations. --KSmrq^T 13:50, 2 November 2007 (UTC)

Thanking you kindly --Chachu207 ^{talk to me} 19:56, 3 November 2007 (UTC)

basic question

Are these equivalent?

• (1/2)^(1/2)
• 1/Sqr(2)
• 1/2+1/4+1/8+1/16+...

And how do you call them? —Preceding unsigned comment added by 83.57.66.241 (talk) 12:42, 2 November 2007 (UTC)

The first 2 are equal to each other because (1/2)^(1/2) is 1^(1/2) divided by 2^(1/2). They are not equal to the 3rd one (the sum of the first three terms is more than 1/Sqr(2)). Zain Ebrahim 12:56, 2 November 2007 (UTC)

I'm not sure what your second question means.Zain Ebrahim 12:59, 2 November 2007 (UTC)

Thanks for your prompt answer. My second question is if there is a name for a series like: /2+1/4+1/8+1/16+... . I suppose that the first two have no name, there are plain normal fractions, right? —Preceding unsigned comment added by 83.57.66.241 (talk) 13:07, 2 November 2007 (UTC)

Its a Geometric Series with common ratio equal to 1/2. It actually sums to 1. I'm not sure about calling them plain normal fractions. I'd call them radicals but I'm probably wrong there.Zain Ebrahim 13:14, 2 November 2007 (UTC)

Yes, I´ll take a look at these links. Thank you very much. —Preceding unsigned comment added by 83.57.66.241 (talk) 13:24, 2 November 2007 (UTC)

You're welcome. Just a hint: it's generally considered good manners to sign your posts by typing ~~~~ at the end.Zain Ebrahim 13:33, 2 November 2007 (UTC)

what's this?

Wat's d/dx, what does it do? --142.132.6.4 19:16, 2 November 2007 (UTC)

It is a differential operator. It differentiates the function of x to which it is applied. See differential calculus for what seems to be the least technical article on these matters we have. Algebraist 20:00, 2 November 2007 (UTC)

system matrix

What is a system matrix? It is used here and there but I could not find any general definition. My best guess it that it refers to a matrix that, together with 2 vectors, gives a linear system of equations. Is this correct? Arthena^(talk) 19:33, 2 November 2007 (UTC)

I think from the term System Matrix, that it is the matrix which represents a system of linear equations. All n variable n equation systems can be represented by an nXn matrix. And it it not hard with a calculator capable of matrix math to solve the system via said matrix. I think our article on Linear systems may be of some help. A math-wiki 20:17, 2 November 2007 (UTC)

That link didn't go where I though it would, here use this link System of linear equations A math-wiki 20:24, 2 November 2007 (UTC)

It is impossible to say without more context. Many parts of applied mathematics describe some kind of system using linearity, which means the system can be described by a matrix. Quantum mechanics, linear filters, process control, and many other applications may refer to a "system matrix". --KSmrq^T 21:44, 2 November 2007 (UTC)