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human
Are some "irrational" numbers rational in other bases?

This is more AskMath but I thought someone here could help. 

First, I may need an explanation of how decimals in other bases work in general. 

I started thinking about binary, and how decimals should work there. 

I assumed the following (binary on the left, base 10 on the right): 

    .1n==.5n
    .01n==.5*.5n=.25n
    .11n=n(.1+.01)=.1n+.01n==.5n+.5*.5n=.75n

And so forth. 

However, aren't there many decimal numbers in base 10 that can't be easily represented with this system?  You run in to the old problem of "dividing in half forever" and never reaching your goal... But we should be able to represent the entire number line using binary, so I suppose this is just an issue of infinities... it's just super inconvenient to represent certain values?

This made me think that there may be numbers that are inconvenient to represent in base 10 that are easy to use in other bases, and I made the jump to rational/irrational numbers. 

However, my intuition is that stuff like PI wouldn't be rational no matter how big we made the base, right?  It's an infinite, non-repeating string.  This should be true for all irrationals, right?  They are defined as numbers that can't be represented as an integer ratio, so should have nothing to do with what base you are working in...

Any info/random musings on decimals in other bases, as well has the interplay of irrational numbers as bases change is appreciated. 

EDIT- sorry for the confusing terms, I know "decimal" is often used to mean "base 10" when talking about binary, but I here mean "numbers between 1 and 0"

EDIT 2: OK NEW QUESTION WTF are irrational bases?  Does that question even have meaning?  How do they work?  
IE: Do they include whole intigers up to the value of the irrational, or fractions of the irrational:

base Pi:
1, 2, 3, Pi, Pi1, Pi2...
OR
1 (pi/pi), Tao (pi/2..wait does 2 exist in this world???), Pi, Pi1, PiTao.. some other thing... I don't even know what this looks like.
gpt
You can have a rational number that terminates in one base but which repeats forever in another. In a base-3 system 1/3 (in base-10) terminates as 0.1 but in binary 1/3 is a repeating decimal, just as it is in base-10 (0.333333333.....).

But a rational number is rational in any base (i.e. can be expressed as the ratio of two integers) and an irrational number is a non-repeating infinite decimal in all bases. Integers are integers in all bases so the inability to find two integers whose ratio is, say, pi, in base-10 means that no such integers can be found in any base because all bases share the same integers just expressed differently.