An irrational number cannot be represented as a fraction composed of integer parts.

There are an infinite number of irrational numbers between each rational number, and there are an infinite number of rational numbers.  Which you might think means tehre are more irrational than rationals.

Any number whose decimal representation recurs at any point is [rational].  This is proved since:

Make PRE = the decimal representation up to where the number starts repeating, and replace the repeating part with 0000... then the number PRE can be represented as PRE*1000..../1000.... (ie a fraction, using as many digits in the multiplier as needed to make PRE an integer).

The repeating part can be made into a fraction by taking the repeat and dividing by 99999.... (as many 9s as are in the repeated part).

For example: .77777... = 7/9.  .7171.. = 71/99;  .142857142857... = 142857/999999 and so on.  So any number which has a recurring representation can be written as a sum of 2 fractions:
Val = PRE + rept/999....

Well that is a simple version of the proof, but largely correct.

[GWM]