geerky42
  • geerky42
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 \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"
Mathematics
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SOLVED
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chestercat
  • chestercat
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geerky42
  • geerky42
Well, is \(f_a^b(x)\) even continuous function?
ganeshie8
  • ganeshie8
that doesn't matter for definite integral right
geerky42
  • geerky42
It doesn't? I'm not sure haha. I'm interesting in solution.

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ganeshie8
  • ganeshie8
:) recall the definite integrals of famous discontinuous functions, for example, greatest integer function
ganeshie8
  • ganeshie8
the problem looks perfectly fine to me somehow if we could represent the integrand as a series..
geerky42
  • geerky42
Well, 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
  • geerky42
maybe this problem is too advanced for me lol
geerky42
  • geerky42
I believe we should start by finding "formula" for \(f_a^b(x)\) before evaluating integral.
freckles
  • freckles
like 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
geerky42
  • geerky42
Exactly @freckles
freckles
  • freckles
though it does seem for ever x near .5 we have the output close to .1 so it could be continuous
chosenmatt
  • chosenmatt
O.O
ganeshie8
  • ganeshie8
is it given that b > a ?
geerky42
  • geerky42
No, but yeah I think we should resist to \(b\ge a\).
thomas5267
  • thomas5267
\(b\geq a\) seems necessary to me. I do not know how to interpret 9 in base 5 for example.
thomas5267
  • thomas5267
So b and a are positive integers right?
geerky42
  • geerky42
Yeah
freckles
  • freckles
\[0
Kainui
  • Kainui
This 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
  • ikram002p
"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
  • thomas5267
The graph is horrifying! This graph is \(f_2^{10}(x)\) from 0 to 1.
1 Attachment
Kainui
  • Kainui
Well, I can try solving it for when a=b.
thomas5267
  • thomas5267
This is the full range of the graph.
1 Attachment
Kainui
  • Kainui
Is 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
  • thomas5267
What 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
  • Kainui
\(x_a\) 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
  • thomas5267
The graph is wrong! This is the correct graph.
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thomas5267
  • thomas5267
For a=b, it seems that \(f_a^b(x)=x\).
anonymous
  • anonymous
cough https://en.wikipedia.org/wiki/Cantor_function
anonymous
  • anonymous
https://en.wikipedia.org/wiki/Cantor_function#Generalizations

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