Number Sets & Infinity

Activities to try

1. 1

   Decimal-signature classifier: repetition, not length, decides

   Why it works: A number is rational exactly when it can be written as a ratio of two integers — which shows up as a terminating OR repeating decimal. The deciding feature is whether the digits settle into a repeat, not how many digits there are: 0.333… is endless yet rational (=1/3). √2 is proved to be irrational, so its expansion never repeats. Checking a finite number of digits could never establish that on its own.

   1. Take a set of numbers: 0.75, 0.333…, 3/8, 0.25, √2, π, √9, 0.142857142857… (1/7).
   2. For each, write out or stream the decimal expansion and decide whether it TERMINATES or REPEATS. For √2 and π, note that a finite list of digits can never show that a decimal does NEITHER: some rationals have very long repeating blocks (1/97 repeats only after 96 digits). We know √2 and π are irrational because it has been proved, so their decimals never terminate or repeat.
   3. Mark the repeating block when there is one (e.g. box the 3 in 0.3̅, or the 142857 in 1/7) and write the equivalent fraction.
   4. State a rule in your own words for deciding rational vs irrational, then test it against √9 = 3 and 0.333…
   5. Keep √2 and π written exactly as symbols, noting that 1.414… and 3.14… are only approximations.
2. 2

   Squeeze another number in: density of the line

   Why it works: Between any two distinct numbers there are infinitely many others, so the number line is dense: there is no smallest gap and no 'next' number. This contrasts with the discrete, consecutive structure of the integers.

   1. Name two close numbers, e.g. 0.12 and 0.13 (or 1/4 and 1/3).
   2. Find a number strictly between them — averaging the two is one reliable method (e.g. 0.125).
   3. Now find a number between THAT one and the upper end, and again, and again.
   4. Argue why this process can never run out, and conclude how many numbers lie between the original pair.
   5. Explain why there is no 'next' number after a given one, unlike the integers.
3. 3

   Dialogue: why √2 cannot be a fraction

   Why it works: Some numbers genuinely cannot be written as a ratio of integers — this is a provable fact, not just 'it has lots of decimals'. Engaging with the proof-by-contradiction and the 'no next number' question deepens understanding of what irrationality and density actually mean.

   1. Work through the classic argument: suppose √2 = a/b in lowest terms, and reason about whether a and b can both be chosen with no common factor.
   2. Respond to Socratic prompts about what would have to be true of a² and b², and where a contradiction arises.
   3. Then tackle the intuition probe: 'Is there a number right after 0.12?' and explain your answer.
   4. Try to write π as a fraction of two whole numbers (e.g. 22/7 or 355/113) and check that these are only approximations. Note that mathematicians have proved π is irrational, though the proof is beyond this level, which is why its decimal never terminates or repeats.