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I've[1] recently become interested in discrete subrings containing 1 of the complex numbers. Being complex numbers these rings have all sorts of properties but my question may be formed in terms of the reals. The question is; when does a subring, say of the reals, ##\mathbb{R}##, becomes dense in the reals? I think the answer is when it contains an element, ##|x|<1##. Let's look at the ring generated by 1 and ##\sqrt{2}##, ##R=\langle 1,\sqrt{2}\rangle##. Any member of this ring, ##r\in R##, may be written,

##r = n+m\sqrt{2}##

where ##n,m\in \mathbb{Z}##. Okay, consider ##x = \sqrt{2}-1##. We have that ##|x| < 1## so the sequence, ##x_n = x^n## where ##n\in\mathbb{N}## accumulates at ##x=0##. Since the ring is closed under addition, every element of the ring is an accumulation point. The next step is showing that any real ##y## is also an accumulation point of some sequence in the ring. My thought was given ##y\notin R## there is always an ##r\in R## such that ##|y-r| < 1##. Given this ##r## we may always find a small ##x\in R## about 0 we can add so that, ##|y - r - x| < |y - r|##. So there is no smallest ##x## we can always find more ring elements near ##y##.

[1] My training is in physics.