Infinitely recurring decimals occur when you divide a number that is not a factor or share only common factors with the base into 1. In base 10, only 2 and 5 have this property. So 1/2 = 0.5 and 1/5 = 0.2. Also 1/4 = 0.25 and 1/8 = 0.125, because the only factor of those numbers is 2. 1/3=0.333r, 1/6=0.1666r 1/7 = 0.__142857__r and 1/9 = 0.111r. In base 3, however, 0.1 + 0.1 + 0.1 = 1.0, therefore 1/3 is 0.1. In base 9, 1/3 = 0.3. It's the other numbers which have infinite expansions.

Irrational numbers are numbers which cannot be represented by a fraction, so they have an infinite number of different digits in all bases. The best known example of an irrational is sqrt (2).

Transcendental numbers are numbers which cannot be represented or solved out of a simple (non-infinite) equation. sqrt(2) is not Transcendental because x<sup>2</sup> - 2 = 0 has the answer sqrt(2). This does not apply to pi or e, neither of which can be represented by equations that are not infinite series sums. Again, the commensurate corollary to this property is an infinite digit expansion in any number base.