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    "title": "Roth: optimal rational-approximation exponent for algebraic irrationals",
    "summary": "Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions.",
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        "title": "Rational approximations to algebraic numbers (Mathematika2,1–20,1955)",
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      "scope": "Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions. This is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.",
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        "description": "This is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.",
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      "statement": "# Roth: optimal rational-approximation exponent for algebraic irrationals\n\n## How good can a fraction be?\n\nThe fraction 7/5 is close to the square root of two, but it is not exact. Using a larger denominator can sometimes make an approximation much better. The interesting question is how often that improvement can beat a chosen power of the denominator. A single excellent fraction and an endless supply of excellent fractions are very different claims.\n\n## The boundary Roth settles\n\nFor a fixed real algebraic irrational and any positive epsilon, Roth proves that only finitely many reduced fractions p/q have error smaller than q to the power minus two minus epsilon. Algebraic means the number solves a nonzero polynomial equation with integer coefficients. The square root of two qualifies because its square is two. At exponent two, infinitely many approximations remain possible.\n\n## Check one fraction carefully\n\nHere is a classroom calculation, not evidence for the full theorem. The square root of two lies between 1.4142 and 1.4143. Therefore the error of 7/5 lies between 0.0142 and 0.0143. The quantity 1/5 squared is 0.04, while 1/5 cubed is 0.008. This fraction beats the exponent-two threshold but misses the exponent-three threshold. Changing the exponent changes the question even when the fraction stays fixed.\n\n## Finite does not mean we can list the exceptions\n\nA statement that there are only finitely many exceptions need not tell you where the last one occurs. Checking denominators up to a million would be a finite search; it would not rule out a much larger exceptional denominator. Roth builds on earlier approximation methods and closes a sharp exponent gap. His rational-approximation result should remain separate from a practical algorithm for listing every exception and from claims about arbitrary transcendental numbers.\n\n## A useful next step\n\nMake a small table of fractions near one fixed algebraic irrational. Record the denominator, an interval containing the error, and the threshold being tested. If rounding makes the comparison uncertain, leave it unresolved or increase precision. Explain why your finite table is an illustration rather than a proof of finiteness. A reusable calculation with explicit error bounds can help another learner understand the theorem without pretending to rediscover it.\n\n## Scientific scope\n\nThis is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.\n\n## Sources for this explanation\n\n- [Rational approximations to algebraic numbers (Mathematika2,1–20,1955)](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EFB89E2873019B246F004FAA06E05A7F/S0025579300000644a.pdf/rational-approximations-to-algebraic-numbers.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions.",
      "title": "Roth: optimal rational-approximation exponent for algebraic irrationals"
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    "what_changed": "Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions.",
    "scope": "Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions. This is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.",
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        "title": "How good can a fraction be?",
        "text": "The fraction 7/5 is close to the square root of two, but it is not exact. Using a larger denominator can sometimes make an approximation much better. The interesting question is how often that improvement can beat a chosen power of the denominator. A single excellent fraction and an endless supply of excellent fractions are very different claims."
      },
      {
        "title": "The boundary Roth settles",
        "text": "For a fixed real algebraic irrational and any positive epsilon, Roth proves that only finitely many reduced fractions p/q have error smaller than q to the power minus two minus epsilon. Algebraic means the number solves a nonzero polynomial equation with integer coefficients. The square root of two qualifies because its square is two. At exponent two, infinitely many approximations remain possible."
      },
      {
        "title": "Check one fraction carefully",
        "text": "Here is a classroom calculation, not evidence for the full theorem. The square root of two lies between 1.4142 and 1.4143. Therefore the error of 7/5 lies between 0.0142 and 0.0143. The quantity 1/5 squared is 0.04, while 1/5 cubed is 0.008. This fraction beats the exponent-two threshold but misses the exponent-three threshold. Changing the exponent changes the question even when the fraction stays fixed."
      },
      {
        "title": "Finite does not mean we can list the exceptions",
        "text": "A statement that there are only finitely many exceptions need not tell you where the last one occurs. Checking denominators up to a million would be a finite search; it would not rule out a much larger exceptional denominator. Roth builds on earlier approximation methods and closes a sharp exponent gap. His rational-approximation result should remain separate from a practical algorithm for listing every exception and from claims about arbitrary transcendental numbers."
      },
      {
        "title": "A useful next step",
        "text": "Make a small table of fractions near one fixed algebraic irrational. Record the denominator, an interval containing the error, and the threshold being tested. If rounding makes the comparison uncertain, leave it unresolved or increase precision. Explain why your finite table is an illustration rather than a proof of finiteness. A reusable calculation with explicit error bounds can help another learner understand the theorem without pretending to rediscover it."
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