Wikipedia:Reference desk/Archives/Mathematics/2007 March 19

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March 19

Liouville-Roth constant of a logarithm

I was reading the article Liouville number, and it mentioned the Liouville-Roth constant associated with an irrational number as being a sort of irrationality measure. Does anyone know if this has been calculated for logarithms of integers to integer bases, e.g., ln(5)/ln(3)? GTBacchus^(talk) 00:33, 19 March 2007 (UTC)

derivative of the inverse of a logit function

I have gotten stuck on a derivative that I did not think would be too hard. Please help me out if you have the chance. The function is simply:

$f(y)=e^{-x}\left(e^{x}+1\right)\,$

I know that if f(z) = e^x / (1 + e^x) then:

amoooo a juannnn antoniooooooooo

$f'(z)={\frac {e^{x}}{\left(e^{x}+1\right)^{2}}}\,$

and if f(a) = (1 / z) then f'(a) = (-1 / z^2).

So for the initial equation of:

f(y) = 1 / [e^x / (1 + e^x)]

I thought that maybe:

f'(y) = -1 / [e^x / (1 + e^x)^2]^2

But that does not seem to be the case.

I believe I solved for the correct answer using the d method and a do-loop on a computer.

When f(y) = 1 / [e^x / (1 + e^x)] and x = -0.40 I believe that

f'(y) = -1.49182.

But I do not know the equation for f'(y). None that I've tried give -1.49182 when x = -0.40.

Thanks for any thoughts. Sorry I did not take the time to format this post. I am using ^ to mean "raised to the power of" and e = 2.71828. Also I posted this question on March 19, but it seems to be in the March 13 section. Sorry about that.

Mark W. Miller 05:38, 19 March 2007 (UTC)

There is something a bit strange about your notation. The value of an expression like, for instance, 1 – cos(x) depends on the value of the variable x. If, in mathematics, something depends like that on something else, it is known as a function. Let us give a name to the function in this example, say Jack. To express that Jack is the function of the example, you can write this:

Jack(x) = 1 – cos(x).

This then implies that Jack(0) = 1 – cos(0), Jack(π/3) = 1 – cos(π/3), and so on. The definition applies to all possible values of x, so if you substitute the same expression for x in the left-hand side an all occurrences of x in the right-hand side, the equation remains true. The x here is known as a bound variable. Just like the choice of the name of the function (Jack) is arbitrary, the name of a bound variable can be chosen arbitrarily. Without change of meaning, we could have defined

Jack(u) = 1 – cos(u).

In your equations you use different variable names in the two sides of the equations that define the functions. I assume that you want to find the derivative of the function h defined by

h(x) = 1 / (e^x / (1 + e^x))

The chain rule is the rule to use when determining the derivative of a composite function. Putting f(z) = 1/z and g(x) = e^x / (1 + e^x), we have h(x) = f(g(x)). Then

h'(x) = f'(g(x)) g'(x) = – 1 / (e^x / (1 + e^x))² · e^x / (1 + e^x)²

This can be simplified dramatically. Instead of doing that, here is a much easier way to determine h' for this specific case:

h(x) = 1 / (e^x / (1 + e^x)) = (1 + e^x) / e^x = (1 + e^x)e^–x = e^–x + 1.

Then

h'(x) = –e^–x.

By the way, the functional inverse of the logit function is not this function h, but the sigmoid function. Function h is the multiplicative inverse of the latter.  --Lambiam^Talk 06:45, 19 March 2007 (UTC)

Thanks much. This morning, after sleeping on it, I came up with the equation:

h'(x) = ((e^x)² - (1+e^x)e^x) / ((e^x)²).

It gives the same answer as the equation you provided. I'm sure the equation I typed reduces to the one you gave.

Thank you again. Thanks also for the information about the sigmoid function.

I feel I should mention for completeness that I really want the partial derivative of a function:

f(x,y,z) = x / (y*z)

where:

x = e^a / (1 + e^a),

y = e^b / (1 + e^b), and

z = e^c / (1 + e^c).

However, now I am pretty sure:

f'b(a,b,c) = –e^–b ((e^a / (1 + e^a)) / (e^c / (1 + e^c))) or

more simply:

f'a(a,b,c) = (e^a / (1+e^a)²) / (y z)

f'b(a,b,c) = –e^–b (x / z), and

f'c(a,b,c) = –e^–c (x / y).

I would like to copy much of this to my home page for future reference if that is not against policy.

Thanks again.

Mark W. Miller 16:56, 19 March 2007 (UTC)

I don't think it is against policy if you add where it is copied from. Instead of a copy you could also use a link to Wikipedia:Reference desk/Archives/Mathematics/2007 March 19#derivative of the inverse of a logit function, which right now is still a redlink but in a few days will become a link to this section.  --Lambiam^Talk 20:31, 19 March 2007 (UTC)

Fermat

That other day i was bored so i jotted Fermat's last theorem on a paper:$X^{n}+Y^{n}=Z^{n}$ Then i took the partial derivative first w.r.t to x....since y and z are treated as constants ,i got :$nX^{n-1}=0.$.

I did the same to y:$nY^{n-1}=0.$.

and to z:$nZ^{n-1}=0.$.

Knowing that n is different from zero (since n>2) it cancels in the three equations: $X^{n-1}=0.$ $Y^{n-1}=0.$ $Z^{n-1}=0.$

which gives us X=0, Y=0,or Z=0, the only solution in integers for the formula.

Ofcourse i couldn't have possibly proved Fermat's Last theorem .....of course not .....i am sure there's an error somewhere that i'm not finding. Could someone point it to me please? PS. Sorry if this is trivial or there's a principal error somewhere. I'm not a world class mathematician :) Hisham1987 17:47, 19 March 2007 (UTC)

The problem is in concluding that $nX^{n-1}=0$ etc. Suppose $x_{0},\ y_{0},\ z_{0}$ is a solution to the equation, so $x_{0}^{n}+y_{0}^{n}=z_{0}^{n}$. If you take $f(X)=X^{n}+y_{0}^{n}-z_{0}^{n}$, you will get $f(x_{0})=0$ for this particular $x_{0}$; you do not get $f(X)=0$ for every X, which has to hold if you want $f'(X)=0\,\!$. -- Meni Rosenfeld (talk) 19:41, 19 March 2007 (UTC)

If your (Hisham's) approach was valid, you would have succeeded in proving that the equation X¹ + Y¹ = Z¹ has no solutions except (0, 0, 0). Actually, also if n > 2, the equation has some other solutions in integers: (k, 0, k), (0, k, k), if n is even also (–k, 0, k) and (0, –k, k), and if n is odd also (k, –k, 0).  --Lambiam^Talk 20:21, 19 March 2007 (UTC)

Fermat's Last Theorem:

If an integer $n$ is greater than 2, then $a^{n}+b^{n}=c^{n}$ has no solutions in non-zero integers $a$, $b$, and $c$.

What you have shown is irrelevant to the theorem because the theorem already says that a, b, and c, must be nonzero. --Ķĩřβȳ♥♥♥ŤįɱéØ 23:36, 19 March 2007 (UTC)

It's not irrelevant. His conclusion is that there are no solutions besides (0,0,0), which if true would confirm the theorem. Black Carrot 00:26, 22 March 2007 (UTC)

Well no, that has nothing to do with the theorem at all. Even at solution to the equation, the derivatives need not agree at all—think of the solution to the equation 8x+5=4x+9. You've got to be kidding me if you think that the partial wrt x is true at the solution x=1. Last I checked, equating the partials gives 8=4, regardless of x. Baccyak4H (Yak!) 13:48, 23 March 2007 (UTC)

Latex program for beginners

Is there an easy program to work with LaTex? Ideally a program where we can just click icons to insert a mathematical symbol. 132.231.54.1 18:30, 19 March 2007 (UTC)

I think Lyx (official page) may be what you are looking for. -- Meni Rosenfeld (talk) 19:35, 19 March 2007 (UTC)

I thought perhaps something smaller would be nicier.132.231.54.1 21:11, 19 March 2007 (UTC)

I started with Texmaker, which has buttons, but there's still a learning curve. If you can get some sample documents, they're a great way to get started, because you can see how things are already done. -GTBacchus^(talk) 02:40, 20 March 2007 (UTC)

I Don't Understand This!!

I have neven been able to solve questions with loans. These are the two problems I can't solve. I don't want the answer, but I need to understand how to do them.

1. Bob borrowed $9,000 for ten years at an annual rate of 18%. What is his monthly payment?

2. Simon obtained a 2 year loan at a simple interest rate of 13% annually. At the end of two years, he has payed $1454.44. What amount of money was he loaned?

What does annually mean in this anyway?

69.37.180.53 22:57, 19 March 2007 (UTC)

Just let me answer the part about what "annually" means. Obviously, in general it means once a year. When dealing with interest rates it could mean the payment is made once a year, or the interest compounds once a year, or that they are giving you either the nominal (no compounding) or effective (with compounding) annual rate. If they say there is an annual rate of 18%, and don't specify any other compounding, I would take that to mean the interest is compounded annually. I was a bit confused by the second question which states there is a 2 year loan with simple interest at a rate of 13% annually. The term "simple interest" means no compounding, they just give you the total interest over the entire period. But that evidently isn't what they really meant, due to the "13% annually" part. StuRat 23:30, 19 March 2007 (UTC)

If I understand the second question correctly, they mean that P dollars were borrowed, then, after 1 year, 13% interest was added, then, after 2 years, 13% interest was added again, including interest on the interest. At this point, the total (both principal and interest) was paid off. If my interpretation is correct, that means P(1.13)^2 = $1454.44, see if you can solve it from there. StuRat 23:37, 19 March 2007 (UTC)

P * 1.13^2 = 1454.44

sqrt(P) * 1.13 = sqrt(1454.44)

sqrt(P) = sqrt(1454.44) / 1.13

sqrt(P) = sqrt(1454.44) / sqrt(1.13*1.13)

sqrt(P) = sqrt(1454.44/(1.13*1.13))

P = 1454.44/(1.13*1.13)

P = 1454.44/1.13^2

202.168.50.40 01:58, 20 March 2007 (UTC)

Looks like you did that the long way. And aren't you going to solve for P ? StuRat 02:35, 20 March 2007 (UTC)

That would require me to use my calculator. I'm way too lazy for that! 202.168.50.40 03:28, 20 March 2007 (UTC)

Another possible interpretation than that offered by Stu is that returning the borrowed money is not considered paying, only the interest, in which case the equation to be solved is P(1.13²−1) = $1454.44. None of the interpretations gives a round number for P, which is suspicious. (Imagine applying for a $2377.03 loan.) Also, the qualification "simple" remains unexplained. Yet another interpretation is that the interest does not get compounded and simple Simon pays the lender twice 13% (once each year), or together 26%. Then the equation to be solved is 0.26P = $1454.44, which also has the virtue of having a solution that is a whole dollar amount – although still an unlikely amount to apply for.  --Lambiam^Talk 06:38, 20 March 2007 (UTC)

If you are interested in learning how to do <<compound interest rate calculations>>, you can find an very very old USENET message that explains how to do so.

• Goto http://groups.google.com
• Search for the string "surprise to me to learn that there are people who" . Make sure you type in the double quotes.

202.168.50.40 03:39, 20 March 2007 (UTC)

Why are teachers and textbook authors incapable of writing a clear interest rate problem ?

They rarely manage to state all the important facts, like whether payments are made at the beginning or end of each payment period, the compounding period for the interest, and whether the interest rate they give includes compounding or not. I'd say less than half of the interest rate questions I've seen are clearly stated. StuRat 13:32, 20 March 2007 (UTC)

I'd like to ask the same question for credit card companies. I have not been able to distill the algorithm used for determining the interest charged from the statement of the conditions for any of the credit cards I have, possibly mainly because I can't make much sense of the English.  --Lambiam^Talk 15:19, 20 March 2007 (UTC)

The warning my company gives on the consequence of not making full payment by a certain date certainly is unclear: "if your account is not fully cleared, interest will be charged on your daily balance, not just your outstanding balance". I can only guess that daily balance is the current one, day by day, and outstanding balance is the one at the time of issuing the statement, i.e. now fixed. But in my terms, "outstanding" would be whatever applied at any date, i.e. be exactly the same as what I presume they mean by "daily". If those of us who are intelligent and numerate find it hard, what chance do the rest have?86.132.166.51 16:09, 20 March 2007 (UTC)

Ask one of their help lines for the algorithm. Ask them for the best-case, worst-case, and average-case. Ask them if they use a Turing machine for calculating your payments. =) --Ķĩřβȳ♥♥♥ŤįɱéØ 17:20, 20 March 2007 (UTC)

The reason credit-card company rules are unclear is that they are intentionally trying to deceive consumers so they can charge higher fees and penalties than people think they can. If they honestly said "we will use every possible opportunity to overcharge you as much as we are legally allowed, and will even charge you illegal fees wherever we think we have sufficient politicians in our pockets to prevent an investigation", they might not get quite as many customers. StuRat 22:25, 22 March 2007 (UTC)

i have a question, witch equation from the derivate family, give you the same answer after derivating and befor derivationg?????