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 \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"

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- geerky42

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- geerky42

Well, is \(f_a^b(x)\) even continuous function?

- ganeshie8

that doesn't matter for definite integral right

- geerky42

It doesn't? I'm not sure haha. I'm interesting in solution.

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## More answers

- ganeshie8

:) recall the definite integrals of famous discontinuous functions,
for example, greatest integer function

- ganeshie8

the problem looks perfectly fine to me
somehow if we could represent the integrand as a series..

- 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

maybe this problem is too advanced for me lol

- geerky42

I believe we should start by finding "formula" for \(f_a^b(x)\) before evaluating integral.

- 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

Exactly @freckles

- freckles

though it does seem for ever x near .5 we have the output close to .1
so it could be continuous

- chosenmatt

O.O

- ganeshie8

is it given that b > a ?

- geerky42

No, but yeah I think we should resist to \(b\ge a\).

- thomas5267

\(b\geq a\) seems necessary to me. I do not know how to interpret 9 in base 5 for example.

- thomas5267

So b and a are positive integers right?

- geerky42

Yeah

- freckles

\[0

- 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

"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

The graph is horrifying! This graph is \(f_2^{10}(x)\) from 0 to 1.

##### 1 Attachment

- Kainui

Well, I can try solving it for when a=b.

- thomas5267

This is the full range of the graph.

##### 1 Attachment

- 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

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

\(x_a\) 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

The graph is wrong! This is the correct graph.

##### 1 Attachment

- thomas5267

For a=b, it seems that \(f_a^b(x)=x\).

- anonymous

cough https://en.wikipedia.org/wiki/Cantor_function

- anonymous

https://en.wikipedia.org/wiki/Cantor_function#Generalizations

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