Revision #1586 → #1801 · back to history
modifiedReduction to constructing √πfc1481b9e7e0
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| anchors | [{"section":"Impossibility","snippet":"the length of the side of a square whose area equals that of a unit circle"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modified√π constructible implies π constructible52f63bcb9d18
| Field | From #1586 | To #1801 |
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| anchors | [{"section":"Impossibility","snippet":"it would follow from standard compass and straightedge constructions that"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedWantzel's theorem on constructible lengths3bcd845126c0
| Field | From #1586 | To #1801 |
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| anchors | [{"section":"Impossibility","snippet":"Pierre Wantzel showed that lengths that could be constructed with compass and straightedge"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedConstructible lengths are algebraic0b17abc862f3
| Field | From #1586 | To #1801 |
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| anchors | [{"section":"Impossibility","snippet":"constructible lengths must be algebraic numbers"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedSquarability implies π algebraic0f467186d803
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Impossibility","snippet":"If the circle could be squared using only compass and straightedge, then"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedTranscendence of π (Lindemann, 1882)5c7dc36671c5
| Field | From #1586 | To #1801 |
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| anchors | [{"section":"Impossibility","snippet":"Ferdinand von Lindemann proved the transcendence of"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedTranscendence of e (Hermite)06406159415a
| Field | From #1586 | To #1801 |
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| anchors | [{"section":"Impossibility","snippet":"the proof of transcendence of Euler's number"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
addedEuler's identitya4490be5706e
modifiedRational power of a transcendental is transcendental5ec282c1ee3f
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Impossibility","snippet":"a rational power of a transcendental number remains transcendental"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedLindemann–Weierstrass theorem9ed7655cdc2e
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Impossibility","snippet":"the Lindemann–Weierstrass theorem on linear independence of algebraic powers of"},{"type":"math_alttext","value":"{\\displaystyle e^{i\\pi }=-1.}"}] | — |
| provenance | ai | ai-moderated |
modifiedKochański's constructionc28c1afc2ebd
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Construction by Kochański","snippet":"a 1685 paper by Polish Jesuit Adam Adamandy Kochański"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\pi &\\approx {\\frac {|P_{3}P_{9}|}{|P_{1}P_{2}|}}={\\sqrt {{\\frac {40}{3}}-2{\\sqrt {3}}}}\\\\[3mu]&=3.141\\;5{\\color {red}33\\,338\\;\\ldots }\\end{aligned}}}"}] | — |
| provenance | ai | ai-moderated |
modifiedKochański's rational approximationsf548276bea23
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Construction by Kochański","snippet":"Kochański also derived a sequence of increasingly accurate rational approximations for"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\pi &\\approx {\\frac {|P_{3}P_{9}|}{|P_{1}P_{2}|}}={\\sqrt {{\\frac {40}{3}}-2{\\sqrt {3}}}}\\\\[3mu]&=3.141\\;5{\\color {red}33\\,338\\;\\ldots }\\end{aligned}}}"}] | — |
| provenance | ai | ai-moderated |
modifiedde Gelder's 355/113 constructionef2c1756d25d
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Constructions using 355/113","snippet":"Jacob de Gelder published in 1849 a construction based on the approximation"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\pi &\\approx {\\frac {355}{113}}\\\\[3mu]&=3.141\\;592{\\color {red}\\;920\\;\\ldots }\\end{aligned}}}"}] | — |
| provenance | ai | ai-moderated |
modifiedHobson's golden ratio constructionaf9cc50e756d
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Constructions using the golden ratio","snippet":"An approximate construction by E. W. Hobson in 1913"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\pi &\\approx {\\frac {6}{5}}\\cdot \\left(1+\\varphi \\right)\\\\[3mu]&=3.141\\;{\\color {red}640\\;\\ldots },\\end{aligned}}}"}] | — |
| provenance | ai | ai-moderated |
addedGolden ratio6c659b955ed2
modifiedRamanujan's second constructioneace4629cde5
| Field | From #1586 | To #1801 |
|---|
| anchors | [{"section":"Second construction by Ramanujan","snippet":"Ramanujan gave a construction which was equivalent to taking the approximate value for"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\pi &\\approx \\left(9^{2}+{\\frac {19^{2}}{22}}\\right)^{\\frac {1}{4}}={\\sqrt[{4}]{\\frac {2143}{22}}}\\\\[3mu]&=3.141\\;592\\;65{\\color {red}2\\;582\\;\\ldots }\\end{aligned}}}"}] | — |
| provenance | ai | ai-moderated |