Practice question · Multiple choice
Completeness is what separates ℝ from ℚ, and almost every theorem of calculus depends on it. Which of these fails over the rationals?
Hints
- The IVT promises a crossing. Ask where x² = 2 would have to cross over ℚ.
- Which of these four statements requires the line to have no holes?
Show the answer
D. The intermediate value theorem, with no rational root between 1 and 2
Why
The function changes sign and never hits zero, because the crossing point is missing from ℚ. That is completeness doing invisible work, the same property underwrites the extreme value theorem, convergence of bounded monotone sequences and the existence of maxima, which is why calculus is built on ℝ.
Practise Real Numbers and the Number Line
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More questions on Real Numbers and the Number Line
- Compute the value of |3 - 11| + |-4|.
- Sort each number by whether it lies in the interval from -2 to 3, including -2 but excluding 3.
- Complete the completeness property of the real numbers.
- Let S be the set of all rational numbers q with q squared less than 2. Select every TRUE statement.
- How many integers x satisfy |x - 4| < 3?
- When examining why the rational number line fails completeness, why does the bounded collection of fractions…