messages listlengths 2 2 | meta dict |
|---|---|
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 6 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: for ... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 12 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: for... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 24 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: for... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 48 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: for... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 96 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: for... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Liu Hui (3rd century CE) computed π using regular polygons inscribed in and circumscribing a circle. For a regular polygon with 192 sides, compute the lower and upper bounds for π and state the error bound."
},
{
"role": "assistant",
"content": "Liu Hui's method: fo... | {
"tradition": "chinese",
"mathematician": "Liu Hui",
"century": "3rd CE",
"formula": "n·sin(π/n) < π < n·tan(π/n)",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Madhava of Sangamagrama (14th century CE, Kerala school, India) derived the series for π/4. Compute the approximation using 10 terms and state the error bound."
},
{
"role": "assistant",
"content": "Madhava's series (also known as the Leibniz series, which Leibniz d... | {
"tradition": "indian",
"mathematician": "Madhava",
"century": "14th",
"formula": "atan(1) = Σ(-1)^i/(2i+1)",
"certified": true,
"atlas_formula": "madhava_series"
} |
[
{
"role": "user",
"content": "Madhava of Sangamagrama (14th century CE, Kerala school, India) derived the series for π/4. Compute the approximation using 50 terms and state the error bound."
},
{
"role": "assistant",
"content": "Madhava's series (also known as the Leibniz series, which Leibniz d... | {
"tradition": "indian",
"mathematician": "Madhava",
"century": "14th",
"formula": "atan(1) = Σ(-1)^i/(2i+1)",
"certified": true,
"atlas_formula": "madhava_series"
} |
[
{
"role": "user",
"content": "Madhava of Sangamagrama (14th century CE, Kerala school, India) derived the series for π/4. Compute the approximation using 100 terms and state the error bound."
},
{
"role": "assistant",
"content": "Madhava's series (also known as the Leibniz series, which Leibniz ... | {
"tradition": "indian",
"mathematician": "Madhava",
"century": "14th",
"formula": "atan(1) = Σ(-1)^i/(2i+1)",
"certified": true,
"atlas_formula": "madhava_series"
} |
[
{
"role": "user",
"content": "Madhava of Sangamagrama (14th century CE, Kerala school, India) derived the series for π/4. Compute the approximation using 500 terms and state the error bound."
},
{
"role": "assistant",
"content": "Madhava's series (also known as the Leibniz series, which Leibniz ... | {
"tradition": "indian",
"mathematician": "Madhava",
"century": "14th",
"formula": "atan(1) = Σ(-1)^i/(2i+1)",
"certified": true,
"atlas_formula": "madhava_series"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 2/3 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n2... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 2/5 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n2... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 2/7 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n2... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 3/4 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n3... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 2/11 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Using the Egyptian method (Rhind Papyrus, ~1650 BCE), express 5/6 as a sum of distinct unit fractions."
},
{
"role": "assistant",
"content": "The Egyptian method uses the greedy algorithm: always take the largest unit fraction that doesn't exceed the remainder.\n\n5... | {
"tradition": "egyptian",
"mathematician": "Ahmes (Rhind Papyrus)",
"century": "~1650 BCE",
"formula": "greedy unit fraction decomposition",
"certified": true,
"atlas_formula": "TH_V18_04-egyptian-horus"
} |
[
{
"role": "user",
"content": "Solve x² + 4x = 12 using the Babylonian method of completing the square."
},
{
"role": "assistant",
"content": "The Babylonian method (from clay tablets like Plimpton 322, ~1800 BCE):\n\nx² + 4x = 12\n\nStep 1: Take half of the coefficient of x: 4/2 = 2.0\nStep 2: S... | {
"tradition": "babylonian",
"mathematician": "Babylonian scribes",
"century": "~1800 BCE",
"formula": "completing the square",
"certified": true,
"atlas_formula": "quadratic_completion"
} |
[
{
"role": "user",
"content": "Solve x² + 6x = 16 using the Babylonian method of completing the square."
},
{
"role": "assistant",
"content": "The Babylonian method (from clay tablets like Plimpton 322, ~1800 BCE):\n\nx² + 6x = 16\n\nStep 1: Take half of the coefficient of x: 6/2 = 3.0\nStep 2: S... | {
"tradition": "babylonian",
"mathematician": "Babylonian scribes",
"century": "~1800 BCE",
"formula": "completing the square",
"certified": true,
"atlas_formula": "quadratic_completion"
} |
[
{
"role": "user",
"content": "Solve x² + 3x = 10 using the Babylonian method of completing the square."
},
{
"role": "assistant",
"content": "The Babylonian method (from clay tablets like Plimpton 322, ~1800 BCE):\n\nx² + 3x = 10\n\nStep 1: Take half of the coefficient of x: 3/2 = 1.5\nStep 2: S... | {
"tradition": "babylonian",
"mathematician": "Babylonian scribes",
"century": "~1800 BCE",
"formula": "completing the square",
"certified": true,
"atlas_formula": "quadratic_completion"
} |
[
{
"role": "user",
"content": "Using Archimedes' method of exhaustion (3rd century BCE), approximate the area of a unit circle with an inscribed regular polygon of 4 sides."
},
{
"role": "assistant",
"content": "Archimedes' method of exhaustion:\n\nA regular 4-gon inscribed in a unit circle has a... | {
"tradition": "greek",
"mathematician": "Archimedes",
"century": "3rd BCE",
"formula": "method of exhaustion",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Using Archimedes' method of exhaustion (3rd century BCE), approximate the area of a unit circle with an inscribed regular polygon of 8 sides."
},
{
"role": "assistant",
"content": "Archimedes' method of exhaustion:\n\nA regular 8-gon inscribed in a unit circle has a... | {
"tradition": "greek",
"mathematician": "Archimedes",
"century": "3rd BCE",
"formula": "method of exhaustion",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Using Archimedes' method of exhaustion (3rd century BCE), approximate the area of a unit circle with an inscribed regular polygon of 16 sides."
},
{
"role": "assistant",
"content": "Archimedes' method of exhaustion:\n\nA regular 16-gon inscribed in a unit circle has... | {
"tradition": "greek",
"mathematician": "Archimedes",
"century": "3rd BCE",
"formula": "method of exhaustion",
"certified": true,
"atlas_formula": "liu_hui_pi"
} |
[
{
"role": "user",
"content": "Explain how the Andean khipu (quipu) recording system worked and why it is the first known system of verified records."
},
{
"role": "assistant",
"content": "The khipu (Quechua for 'knot') was the Andean civilization's recording system, used from ~3000 BCE until the... | {
"tradition": "andean",
"mathematician": "Khipukamayuq (knot-keepers)",
"century": "~3000 BCE - 1532 CE",
"formula": "positional decimal knot encoding",
"certified": true,
"atlas_formula": "khipu_merkle_root"
} |
[
{
"role": "user",
"content": "State and prove the Kraft inequality for prefix-free codes, and explain its connection to the estate's governance."
},
{
"role": "assistant",
"content": "Kraft-McMillan inequality: for any prefix-free (instantaneous) code with codeword lengths ℓ₁, ℓ₂, ..., ℓₙ over a... | {
"tradition": "modern",
"mathematician": "Kraft (1949), McMillan (1956)",
"century": "20th",
"formula": "Σ D^(-ℓᵢ) ≤ 1",
"certified": true,
"atlas_formula": "TH_V18_03-kraft"
} |
SZL Ancient Math Reasoning SFT v1
24 chat-format SFT examples teaching mathematical reasoning through the traditions that invented certified computation — each example pairs a problem with its ancient method, a verified computation, and an explicit error bound.
Traditions
| Tradition | Examples | Mathematician | Contribution |
|---|---|---|---|
| Chinese | 6 | Liu Hui (3rd CE) | Polygon π with error bound |
| Indian | 4 | Madhava (14th CE) | Series with convergence rate |
| Egyptian | 6 | Ahmes (~1650 BCE) | Greedy unit fractions |
| Babylonian | 3 | Scribes (~1800 BCE) | Quadratic by completing the square |
| Greek | 3 | Archimedes (3rd BCE) | Method of exhaustion |
| Andean | 1 | Khipukamayuq | The knot-record system (the estate's namesake) |
| Modern | 1 | Kraft/McMillan | Prefix-code resource bound |
Why this corpus matters
Every example teaches the same principle: mathematics is certified computation with an error bound. Liu Hui, Madhava, Archimedes, and the Babylonian scribes all produced results that came with their own accuracy guarantees — the first known instances of what we now call certified computation. The Khipu is the first tamper-evident record system. The estate is the modern continuation.
Provenance
- Every example's
metacarries the tradition, mathematician, century, formula, and the corresponding formula-atlas entry - All computations verified (the corpus is generated with exact arithmetic)
- Connection to the 21-formula atlas: each example references its atlas formula
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