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It is simple (I think) to demonstrate that the series of rationals is continuous, since, for any two rational numbers, X=a/b, and Y=c/d, you can always find at least one rational number between them.

[tex]\frac{X+Y}{2} = \frac{ad+bc}{2bd}[/tex]

This derivation works for *any* pair of rational numbers, no matter how close, so the series of rationals is continuous.

But, doesn't this mean that there is nowhere -- no gaps -- in which we can fit irrational numbers?

Does this mean that, logically speaking, pi doesn't really exist between 3.14 and 3.15 on rational number line?

Do the irrationals exist in a completely separate series to rationals (similarly to the way in which reals and imaginary numbers are on different series)?

Perhaps I should put my copy of Russell's "Principles of Mathematics" away, and stop pretending I'm a mathematician? ;)