The decimal 0.142857142857… equals which fraction?

**Rational ↔ terminating OR non-terminating recurring decimal.**
**Irrational ↔ non-terminating non-recurring decimal.**

Example: $\frac{1}{4}=0.25$ (terminating → rational).
Example: $\frac{1}{3}=0.333$… (recurring → rational).
Example: $\sqrt{2}=1.41421356$… (non-terminating non-recurring → irrational).

**When does $\frac{p}{q}$ terminate?** Exactly when the denominator (in simplest form) has only 2's and 5's as prime factors. So $\frac{1}{8}(=1/2^3)$ terminates: 0.125. But $\frac{1}{6}(=1/(2\times 3))$ doesn't: 0.1666…