How the square root of 2 became a number

Historical context of √2 and Pythagoreans

• Discussion centers on why √2 was historically shocking: it showed some right triangles have side ratios that are not rational, contradicting the belief that all magnitudes are commensurable.
• Clarification that only some right triangles have rational side ratios (Pythagorean triples); “most” do not.
• Debate over what Pythagoras himself actually did; earlier Babylonian knowledge of the theorem and triples is noted.
• The drowning-of-the-discoverer myth is flagged as likely untrue, though secrecy/anxiety around incommensurability is acknowledged.
• Ancient Greeks often preferred geometric reasoning and saw arithmetic as “dirty” or practical, not philosophical.

Nature of rationals, irrationals, and real numbers

• Commenters stress that irrationals are defined negatively (not expressible as a ratio of integers).
• Dense rationals versus uncountable reals leads to “most” real numbers being indescribable; only countably many can be named or encoded in finite text.
• Computable numbers are highlighted as a more “sane” subset of reals, though still with limitations (e.g., ordering arbitrary computable reals is not computable).

Cantor vs Dedekind and constructive mathematics

• One camp criticizes Cantor and uncountable sets as leading to “navel-gazing” and non-constructive horrors (e.g., Banach–Tarski, “almost everywhere” phenomena).
• Others defend classical set theory as extremely successful and central to modern science, noting no contradictions found and great practical utility.
• Intuitionism and constructive approaches (e.g., Dedekind cuts, reals as limits/approximations) are discussed as alternatives; some see them rising in relevance via proof assistants.

Irrationality and repeating decimals

• Explanation that any eventually repeating decimal is rational via algebraic manipulation (multiply by a power of 10 and subtract).
• Conversely, all rationals in base-10 must either terminate or repeat because there are finitely many possible remainders in long division.
• Proofs of irrationality (e.g., for √2, e, π) are emphasized as independent of computing digits.

Philosophy, teaching, and broader reflections

• Some wish basic math education covered formal definitions and the history of ideas more deeply.
• Others recommend history-of-science/math resources and warn of “rabbit holes.”
• Speculation appears on whether aliens or non-spatial AIs might develop very different foundational number systems.