Practice question · Multiple choice
Between any two rationals lies another rational, so they leave no gaps you can point to. Yet they have gaps. How can both be true?
Hints
- The sequence 1, 1.4, 1.41, 1.414… is entirely rational. Where is it heading?
- Ask whether a gap has to lie between two neighbours to exist.
Show the answer
C. Because density and completeness are different properties
Why
Density says there is no first rational after any other; completeness asks whether every convergent sequence has somewhere to land. A rational sequence can march toward √2 and find nothing there, the hole is at a limit, which is a place density was never looking.
Practise Rational and Irrational Numbers
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