Rational Irrational Numbers I know that a rational number can always be expressed as a fraction, but can't we also say that it is a number that follows a definite pattern? Like one-third for example; it is never ending as a decimal, but it is predictable. This is observed in most if not all rational numbers, as far as I know.
So would it be possible for a number to be partially predictable?
Could a number have the exact same digits as pi, but instead have a periodic, predictable digit in it?
Is there a category of Rational irrational numbers (or irrational rational numbers)? For instance, could a number with digits like pi or the square root of 2 have rational parts?
 A: It is possible for an irrational number to have a predictable pattern; consider $0.1101001000100001...$. It is also possible to have an irrational number that is another irrational number away from a rational; i.e. $x-y = r $, where $x,y$ irrational and $r $ rational; in fact the equivalence classes of such numbers are dense in the reals. So you can subtract some irrational number from $\pi $ and get a number with a repeating pattern... in fact any pattern that you want.
A: A rational number is one that is the ratio of two integers.  Rational numbers will have a predictable pattern in their decimal representations but that is not the definition of a rational number.  As others have said, this pattern is that, eventually,a block of digits will start to repeat.  In a very simple case, e.g. $1/3$ this will just be the single digit $3$ repeating.  Slightly more complex is $1/7000 = 0.000 142857 142857 ...$ which has three $0$s followed by repeating blocks of $142857$.  As others have said, this includes the terminating case by considering the repeating block to be $0$.  
All sorts of other predictable and interesting patterns are possible but they will not be rational numbers. Read about Liouville numbers.  These have a very predictable pattern to their digits but they are not rational, in fact they were the first known transcendental numbers.  
So, no - you cannot say that a rational is one whose digits have a pattern.  It is only rational if the digits follow the quite precise pattern of ending with a repeating block of digits. 
