Use a constructed smaller object to refute existence of a least positive rational.
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Try
Try to name the smallest positive rational number. Whatever fraction you propose, halving it produces a smaller positive fraction. This construction targets leastness, not the existence of positive rationals.
Predict
Predict which properties must survive the construction. A candidate smaller number must remain rational, remain positive, and be strictly smaller. Omitting any one of these checks can break a contradiction proof.
Experiment
Start with q=3/5, then q=1/100, and calculate q/2. Now perform the same experiment inside positive integers, starting at 1. The output 1/2 exits that domain. The contrast shows why a construction must respect the universe of the original claim.
Understand
Worked argument: between any rational numbers a and b with a<b lies another rational. Let c=(a+b)/2. Rational numbers are closed under addition and division by the nonzero integer 2, so c is rational. Also c−a=(b−a)/2>0 and b−c=(b−a)/2>0. Therefore a<c<b. This verifies membership and both strict inequalities without relying on a picture.
An invalid proof of a least positive rational assumes a least q and proposes q−1 as a smaller positive rational. Rationality and smallness hold, but positivity can fail, for example at q=1/2. Halving fixes that missing condition.
Apply
Write the exact statement being assumed for contradiction. A contradiction could be an impossible equality or an object forbidden by the assumption. State why the contradiction rejects that assumption; do not conclude that all the objects involved fail to exist.
Continue
Depth: the positive rationals have lower bound 0 but no minimum. A lower bound need not belong to the set; a minimum must. Compare the nonnegative rationals, where 0 is a minimum. This distinction prepares the language of infima in real analysis.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For q=7/12, compute q/2.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What does the argument q/2<q show?
Prove there is no smallest positive rational number.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why does the same halving argument fail to prove that positive integers have no least element? Give the actual least element and justify it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
