A community for students.
Here's the question you clicked on:
 0 viewing
geerky42
 one year ago
Repost from Brilliant: Can anyone solve this?
"Define \(f_a^b(x)\) as a function which converts \(x\) into base \(a\) and then interprets it as a number in base \(b\).
For example, \(f_2^{10}(0.5)\) will mean first changing \(0.5\) to base \(2\) i.e. \(0.1\) and then interpreting \(0.1\) as a base \(10\) number. That's it!
So, find a general formula for \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"
geerky42
 one year ago
Repost from Brilliant: Can anyone solve this? "Define \(f_a^b(x)\) as a function which converts \(x\) into base \(a\) and then interprets it as a number in base \(b\). For example, \(f_2^{10}(0.5)\) will mean first changing \(0.5\) to base \(2\) i.e. \(0.1\) and then interpreting \(0.1\) as a base \(10\) number. That's it! So, find a general formula for \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"

This Question is Closed

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0Well, is \(f_a^b(x)\) even continuous function?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0that doesn't matter for definite integral right

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0It doesn't? I'm not sure haha. I'm interesting in solution.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0:) recall the definite integrals of famous discontinuous functions, for example, greatest integer function

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0the problem looks perfectly fine to me somehow if we could represent the integrand as a series..

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0Well, what I'm saying is that \(f_a^b(x)\) may be discontinuous at any value of x, given that we are converting bases. Hard to imagine "area" under it.

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0maybe this problem is too advanced for me lol

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0I believe we should start by finding "formula" for \(f_a^b(x)\) before evaluating integral.

freckles
 one year ago
Best ResponseYou've already chosen the best response.0like something like this: dw:1441131850218:dw @geerky42 ? this is kinda what I'm seeing too in my mind maybe the points aren't exactly where they should be or whatever but something like this

freckles
 one year ago
Best ResponseYou've already chosen the best response.0though it does seem for ever x near .5 we have the output close to .1 so it could be continuous

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.0is it given that b > a ?

geerky42
 one year ago
Best ResponseYou've already chosen the best response.0No, but yeah I think we should resist to \(b\ge a\).

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0\(b\geq a\) seems necessary to me. I do not know how to interpret 9 in base 5 for example.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0So b and a are positive integers right?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[0<x<1 \\ f_2^{10} (x)=\cdots \frac{x_4}{2^4}+\frac{x_3}{2^3}+\frac{x_2}{2^2}+\frac{x_1}{2} \ \text{ where } x=0.x_1x_2x_3x_4\cdots\] can we used a fixed a and b first maybe ... I don't know

Kainui
 one year ago
Best ResponseYou've already chosen the best response.0This integral comes up a lot in chemistry not so surprisingly where the pH needs to be balanced, so if the pH is too low, this integral sums up all the changes in the bases... **cough bad joke cough**

ikram002p
 one year ago
Best ResponseYou've already chosen the best response.0"Define \(f_a^b(x)\) as a function which converts \(x\) into base \(a\) and then interprets it as a number in base \(b\). For example, \(f_2^{10}(0.5)\) will mean first changing \(0.5\) to base \(2\) i.e. \(0.1\) and then interpreting \(0.1\) as a base \(10\) number. That's it! So, find a general formula for \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0The graph is horrifying! This graph is \(f_2^{10}(x)\) from 0 to 1.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.0Well, I can try solving it for when a=b.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0This is the full range of the graph.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.0Is this identity legitimate? \[\int_0^1f_a^b(x) dx= \int_0^1f_a^b(x_a)dx\] That is to say, instead of taking x in base 10, converting it to base a, can we just assume that over the interval [0,1] that we are looking at x in base a already and and then graphing it? After all, it is true for the end points: \[f_a^b(1) =f_a^b(1_a)\] \[f_a^b(0) =f_a^b(0_a)\]

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0What exactly is \(f_a^b(x_a)\)? For example: \[0.5=0.1_2\\ f_2^b(0.1_2)=f_2^b(0.5)?\\ f_2^b(0.1_2)=f_2^b(0.1)? \]

Kainui
 one year ago
Best ResponseYou've already chosen the best response.0\(x_a\) just means x is already written in base a to begin with, it's still the same point, I guess it doesn't really matter, it's more a way of thinking of the problem

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0The graph is wrong! This is the correct graph.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0For a=b, it seems that \(f_a^b(x)=x\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0https://en.wikipedia.org/wiki/Cantor_function#Generalizations
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.