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    "title": "Hermite’s proof that the number e is transcendental",
    "summary": "Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction.",
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        "title": "Sur la fonction exponentielle (ETH Rar22000; attributed undated offprint, catalogued [1873])",
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      "scope": "Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction. The assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].",
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        "description": "The assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].",
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      "statement": "# Hermite’s proof that the number e is transcendental\n\n## An endless decimal is not enough\n\nThe number e appears in continuous growth and in the exponential function. Its familiar decimal begins 2.71828, but inspecting more digits cannot establish the property Hermite proved. A number is algebraic if it solves some nonzero polynomial equation with integer coefficients. A number that solves none is transcendental. The word nonzero matters: the polynomial whose coefficients are all zero would otherwise make every number a solution.\n\n## Separate irrational from transcendental\n\nThe square root of two is irrational, yet it satisfies x squared minus two equals zero. It is therefore algebraic. Transcendence is a stronger statement than having no exact fractional representation. For e, Hermite had to rule out every nonzero integer-coefficient polynomial, with no fixed degree limit. Checking a million possible equations would still leave infinitely many possibilities. The theorem needs an argument that controls that entire family.\n\n## The arithmetic trap behind the proof\n\nA nonzero integer has absolute value at least one. Imagine a construction that, under an assumed polynomial relation, produces an integer N satisfying zero less than its absolute value less than one. That is impossible. This is a classroom explanation of the proof's arithmetic strategy, not its complete construction. Hermite uses carefully related approximations to exponential values, integrality and a nonvanishing determinant to make the contradiction work.\n\n## Why small error and nonvanishing both matter\n\nAn approximation error tending to zero is insufficient if the allegedly contradictory integer can be zero. The proof must secure both smallness and nonvanishing. Factorials supply powerful decay: for a fixed example, successive terms of 2 to the power n divided by n factorial have ratio 2/(n + 1). Eventually that ratio is below one half. This elementary illustration explains why factorial denominators can overwhelm fixed exponential growth.\n\n## One contribution, several historical versions\n\nThe record concerns transcendence of e and the associated arithmetic method. It does not prove algebraic independence of e and pi or every later exponential theorem. Four 1873 notes and the separately preserved offprint belong to one work lineage. The scanned offprint has an undated printed imprint, while institutional catalogues supply bracketed dating. A useful next task is to map a stated lemma to its exact version, without treating a handwritten inscription as authenticated authorship evidence.\n\n## Scientific scope\n\nThe assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].\n\n## Sources for this explanation\n\n- [Sur la fonction exponentielle (ETH Rar22000; attributed undated offprint, catalogued [1873])](https://www.e-rara.ch/download/pdf/27940137.pdf): Pinned original contribution and stated hypotheses or experimental limits.\n",
      "summary": "Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction.",
      "title": "Hermite’s proof that the number e is transcendental"
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    "title": "Hermite’s proof that the number e is transcendental",
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    "what_changed": "Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction.",
    "scope": "Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction. The assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].",
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      {
        "title": "Sur la fonction exponentielle (ETH Rar22000; attributed undated offprint, catalogued [1873])",
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      {
        "title": "An endless decimal is not enough",
        "text": "The number e appears in continuous growth and in the exponential function. Its familiar decimal begins 2.71828, but inspecting more digits cannot establish the property Hermite proved. A number is algebraic if it solves some nonzero polynomial equation with integer coefficients. A number that solves none is transcendental. The word nonzero matters: the polynomial whose coefficients are all zero would otherwise make every number a solution."
      },
      {
        "title": "Separate irrational from transcendental",
        "text": "The square root of two is irrational, yet it satisfies x squared minus two equals zero. It is therefore algebraic. Transcendence is a stronger statement than having no exact fractional representation. For e, Hermite had to rule out every nonzero integer-coefficient polynomial, with no fixed degree limit. Checking a million possible equations would still leave infinitely many possibilities. The theorem needs an argument that controls that entire family."
      },
      {
        "title": "The arithmetic trap behind the proof",
        "text": "A nonzero integer has absolute value at least one. Imagine a construction that, under an assumed polynomial relation, produces an integer N satisfying zero less than its absolute value less than one. That is impossible. This is a classroom explanation of the proof's arithmetic strategy, not its complete construction. Hermite uses carefully related approximations to exponential values, integrality and a nonvanishing determinant to make the contradiction work."
      },
      {
        "title": "Why small error and nonvanishing both matter",
        "text": "An approximation error tending to zero is insufficient if the allegedly contradictory integer can be zero. The proof must secure both smallness and nonvanishing. Factorials supply powerful decay: for a fixed example, successive terms of 2 to the power n divided by n factorial have ratio 2/(n + 1). Eventually that ratio is below one half. This elementary illustration explains why factorial denominators can overwhelm fixed exponential growth."
      },
      {
        "title": "One contribution, several historical versions",
        "text": "The record concerns transcendence of e and the associated arithmetic method. It does not prove algebraic independence of e and pi or every later exponential theorem. Four 1873 notes and the separately preserved offprint belong to one work lineage. The scanned offprint has an undated printed imprint, while institutional catalogues supply bracketed dating. A useful next task is to map a stated lemma to its exact version, without treating a handwritten inscription as authenticated authorship evidence."
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        "title": "Sur la fonction exponentielle (ETH Rar22000; attributed undated offprint, catalogued [1873])",
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