Let *θ* be an irrational number, so that for any integers *r* and *s* ≥ 1 we have *θ* − *r*/*s* ≠ 0 (one usually takes *p*, *q* here, but these letters have been used up in earlier chapters). It is now natural to ask how small |*θ* − *r*/*s*| > 0 can be, and this can be answered by an inequality

(8.1) |

for all such *r* and *s*, where *ϵ*(*r*, *s*) > 0 is an easily calculated function only mildly dependent (if at all) on the numerical value of *θ*. Such a function is usually called an irrationality measure (see for example Exercise 2.5 with *θ* = *e* and We will see a very important example in Chapter 12.

If *θ* is not real, say with imaginary ...

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