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

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  1. geerky42
    • one year ago
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    Well, is \(f_a^b(x)\) even continuous function?

  2. ganeshie8
    • one year ago
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    that doesn't matter for definite integral right

  3. geerky42
    • one year ago
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    It doesn't? I'm not sure haha. I'm interesting in solution.

  4. ganeshie8
    • one year ago
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    :) recall the definite integrals of famous discontinuous functions, for example, greatest integer function

  5. ganeshie8
    • one year ago
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    the problem looks perfectly fine to me somehow if we could represent the integrand as a series..

  6. geerky42
    • one year ago
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    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.

  7. geerky42
    • one year ago
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    maybe this problem is too advanced for me lol

  8. geerky42
    • one year ago
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    I believe we should start by finding "formula" for \(f_a^b(x)\) before evaluating integral.

  9. freckles
    • one year ago
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    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

  10. geerky42
    • one year ago
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    Exactly @freckles

  11. freckles
    • one year ago
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    though it does seem for ever x near .5 we have the output close to .1 so it could be continuous

  12. chosenmatt
    • one year ago
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    O.O

  13. ganeshie8
    • one year ago
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    is it given that b > a ?

  14. geerky42
    • one year ago
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    No, but yeah I think we should resist to \(b\ge a\).

  15. thomas5267
    • one year ago
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    \(b\geq a\) seems necessary to me. I do not know how to interpret 9 in base 5 for example.

  16. thomas5267
    • one year ago
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    So b and a are positive integers right?

  17. geerky42
    • one year ago
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    Yeah

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

  19. Kainui
    • one year ago
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    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**

  20. ikram002p
    • one year ago
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    "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 \(\displaystyle \int_0^1 f_a^b(x)~\mathrm dx\)"

  21. thomas5267
    • one year ago
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    The graph is horrifying! This graph is \(f_2^{10}(x)\) from 0 to 1.

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  22. Kainui
    • one year ago
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    Well, I can try solving it for when a=b.

  23. thomas5267
    • one year ago
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    This is the full range of the graph.

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  24. Kainui
    • one year ago
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    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: \[f_a^b(1) =f_a^b(1_a)\] \[f_a^b(0) =f_a^b(0_a)\]

  25. thomas5267
    • one year ago
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    What exactly is \(f_a^b(x_a)\)? For example: \[0.5=0.1_2\\ f_2^b(0.1_2)=f_2^b(0.5)?\\ f_2^b(0.1_2)=f_2^b(0.1)? \]

  26. Kainui
    • one year ago
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    \(x_a\) 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

  27. thomas5267
    • one year ago
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    The graph is wrong! This is the correct graph.

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  28. thomas5267
    • one year ago
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    For a=b, it seems that \(f_a^b(x)=x\).

  29. anonymous
    • one year ago
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    cough https://en.wikipedia.org/wiki/Cantor_function

  30. anonymous
    • one year ago
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    https://en.wikipedia.org/wiki/Cantor_function#Generalizations

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