The rational number line

I ran across this diagram on another math site recently:


It's a good diagram, but a little misleading.  Between any two rational numbers is another rational number, so the rational number line would appear to be solid at any given magnification.  The fact that there is "room" for the irrationals continues to amaze me.  The fact that the irrationals are even more numerous than the rationals is inconceivable to me.


  1. "Fill in all the numbers to make a continuous line" is a bit glib, isn't it? If only constructing the real numbers were so easy...

  2. True, it is a bit 18th century, but if you don't know Augustin and his sequences, it may do.


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