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?
Best Answer
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.