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

1. geerky42

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

2. ganeshie8

that doesn't matter for definite integral right

3. geerky42

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

4. ganeshie8

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

5. ganeshie8

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

6. 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.

7. geerky42

maybe this problem is too advanced for me lol

8. geerky42

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

9. 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

10. geerky42

Exactly @freckles

11. freckles

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

12. chosenmatt

O.O

13. ganeshie8

is it given that b > a ?

14. geerky42

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

15. thomas5267

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

16. thomas5267

So b and a are positive integers right?

17. geerky42

Yeah

18. freckles

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

19. Kainui

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

"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

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

22. Kainui

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

23. thomas5267

This is the full range of the graph.

24. 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: $f_a^b(1) =f_a^b(1_a)$ $f_a^b(0) =f_a^b(0_a)$

25. thomas5267

What exactly is $$f_a^b(x_a)$$? For example: $0.5=0.1_2\\ f_2^b(0.1_2)=f_2^b(0.5)?\\ f_2^b(0.1_2)=f_2^b(0.1)?$

26. 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

27. thomas5267

The graph is wrong! This is the correct graph.

28. thomas5267

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

29. anonymous
30. anonymous