Calkin-Wilf-Newman function or something

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In summary, the student is struggling with a problem that involves the Calkin-Wilf-Newman function and expressing pi as a rational plus a number. They have found that B = π-3, an exact number, but are unsure how to proceed with finding an "exact answer." Another student, who is possibly the creator of the function, suggests simply using π as the answer.
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1MileCrash
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Calkin-Wilf-Newman function... or something

Homework Statement


I have no idea what this has to do with our current chapter in Calc II but it seems simple enough.

[PLAIN]http://img253.imageshack.us/img253/2192/mathv.png


Homework Equations





The Attempt at a Solution



I managed fine for the first two, but I have no idea what to do for pi. The only way I can think to express pi as a rational plus a number is 3 + (pi - 3). Evaluating their function for that A and B can't give me an "exact answer" which is what they ask for.
 
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  • #2


1MileCrash said:

Homework Statement


I have no idea what this has to do with our current chapter in Calc II but it seems simple enough.

[PLAIN]http://img253.imageshack.us/img253/2192/mathv.png

Homework Equations


The Attempt at a Solution



I managed fine for the first two, but I have no idea what to do for pi. The only way I can think to express pi as a rational plus a number is 3 + (pi - 3). Evaluating their function for that A and B can't give me an "exact answer" which is what they ask for.

f(x)=floor(x)+(1-[x-floor(x)])
After, this change pi into it's series form to get infinite rational expressions although I'm not 100% sure if a change is required.
 
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  • #3


You don't need the series form for pi.
I'll show you how to do it for e = 2.7818... and you can see for yourself how
to do it for pi.
The integer part of e is 2, so here A=2, B=e-2.
Thus f(e) = A + (1-B) = 2 + (1-(e-2)) = 5-e=2.21...
(and yes, I'm the Calkin in Calkin-Wilf-Newman).
 
  • #4


calkin, welcome to PF.

I think people may be overcomplicating this problem. By "exact number", I think the professor just wants students to write π as π, not as "approx. 3.1415" or "3.1415...".
1MileCrash said:

The Attempt at a Solution



I managed fine for the first two, but I have no idea what to do for pi. The only way I can think to express pi as a rational plus a number is 3 + (pi - 3). Evaluating their function for that A and B can't give me an "exact answer" which is what they ask for.
You have found that B = π-3, which is an exact number. So what is 1-B here?
 

What is the Calkin-Wilf-Newman function?

The Calkin-Wilf-Newman function is a mathematical function that generates a sequence of rational numbers by iterating a simple algorithm. It was first introduced by mathematicians John Calkin and Herbert Wilf in 1991, and later expanded upon by mathematician Donald Newman in 1998.

What is the algorithm used in the Calkin-Wilf-Newman function?

The algorithm used in the Calkin-Wilf-Newman function is as follows: start with the rational number 1/1, then for each subsequent term, take the mediant (the fraction between two fractions a/b and c/d is (a+c)/(b+d)) of the previous term and its inverse. This results in a sequence of rational numbers that eventually includes every positive rational number exactly once.

What is the significance of the Calkin-Wilf-Newman function?

The Calkin-Wilf-Newman function has several applications in number theory, combinatorics, and computer science. It has been used to prove results about the distribution of rational numbers, and it has also been used to generate random numbers with specific properties. In computer science, it has been used in the design of efficient algorithms for generating fractions.

What is the connection between the Calkin-Wilf-Newman function and the Farey sequence?

The Calkin-Wilf-Newman function can be seen as a generalization of the Farey sequence, which is a sequence of rational numbers between 0 and 1 in lowest terms with denominators less than or equal to a given number. The Calkin-Wilf-Newman function also generates a sequence of rational numbers, but it includes all rational numbers and not just those between 0 and 1.

How is the Calkin-Wilf-Newman function related to continued fractions?

The Calkin-Wilf-Newman function can be used to generate continued fractions, which are a way of representing real numbers as an infinite sequence of rational numbers. The continued fraction of a real number can be found by applying the Calkin-Wilf-Newman function to the rational number representation of that real number. This connection has been used to prove results about the distribution of continued fractions.

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