ビジュアライゼーションと数学〜すうがくむかしばなし負の数・複素数編〜 / Visualization-and-mathematics-OSH2022

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ϏδϡΞϥΠθʔγϣϯͱ਺ֶ Φʔϓϯηϛφʔ2022@޿ౡ 2022/06/25(౔) ๺㝳೗๏ (͖͍ͨͩΏ͖ͷΓ) ʙ͢͏͕͘Ή͔͠͹ͳ͠ෛͷ਺ɾෳૉ਺ฤʙ

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ࣗݾ঺հ

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͋Μͨ͊୭ͳΜ? • ๺㝳೗๏ (͖͍ͨͩΏ͖ͷΓ) • (ʮ೗๏ʯΛԻಡΈͯ͠) ʹΐ΄͏, Nyoho • ೔༵ϓϩάϥϚ • ਺ֶ (޿ౡେֶେֶӃਓؒࣾձՊֶݚڀՊ) • ϋΠύɾϝσΟΞɾτϥϯεϨʔλ • ޿ౡหਧ͖ସ͑γϦʔζ

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ݕࡧ ৄ͘͠͸ iPad ޿ౡห

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ʮ͜Μͳײ͡ͷ͕ग़͖ͯ·͢ɻʯ

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ݕࡧ ࠷ۙ͸ ؟ڸࢢ৔ ޿ౡݶఆCM

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޿ౡΧʔϓ ಊྛᠳଠબख

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ࠓ೔: ϏδϡΞϥΠθʔγϣϯͱ਺ֶ

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ಥવͰ͕͢

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ݟ͑ΔΑ͏ʹ͢Δ͜ͱ = ՄࢹԽ = visualization

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Visualization • visual (ϏδϡΞϧ, ܗ༰ࢺ) ໨ʹݟ͑Δɺࢹ֮ͷ • visualize (ϏδϡΞϥΠζ, ಈࢺ) ໨ʹݟ͑ΔΑ͏ʹ͢Δɺࢹ֮Խ͢Δ • visualization (ϏδϡΞϥΠθʔγϣϯ, ໊ࢺ) ໨ʹݟ͑ΔΑ͏ʹ͢Δ͜ͱ

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visualization

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visualization /vìʒuəlaizéiʃən/ [ͼ͡Ύ͋Β͍͍ͥ͠ΐΜ]

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ՄࢹԽ

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ࠓճ͸୯ͳΔϏδϡΞϥΠθʔ γϣϯͰ͸ͳ͘

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਺ֶͱ ϏδϡΞϥΠθʔγϣϯͱͷ࿩

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਺ͷࢢຽݖ֫ಘͱ ϏδϡΞϥΠθʔγϣϯ

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ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

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ϏδϡΞϥΠθʔγϣϯͱ ෛͷ਺

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ෛͷ਺ • ํఔࣜͷʮ܎਺ʯͱͯ͠΋ʮղʯͱͯ͠΋ͳ͔ͳ͔ड͚ೖΕΒΕͳ͔ͬͨɻ • ܭࢉํ๏͸஌ΒΕ͍ͯͨͷʹɻ

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ෛͷ਺ͷܭࢉ • ݱ୅Ͱ΋ɺ;ͱʮ͸ͯʯͱࢥ͏͜ͱ΋ɻ • (࣮͸͜͜΋໘ന͍ (ࠓ೔͸লུ)) ( 1) ( 1) = +1

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3ੈل͝Ζ Τδϓτ • σΟΦϑΝϯτε 200–284 • ਖ਼ͷ༗ཧ਺ͷΈΛղͱͯ͠ೝΊΔɻ • ଞ͸ࣺͯΔɻ

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7ੈلɾ12ੈل Πϯυ • 7ੈل • ਖ਼ͷ਺Λࢿ࢈ɺෛͷ਺ΛआۚΛද͢ͷʹ࢖͏ɻ • (आۚʹ͸࢖ΘΕ͍ͯͨ!) • 12ੈل όʔεΧϥ • ਖ਼ͷ਺ͷฏํࠜʹ͸ϓϥεϚΠφε྆ํ͋Δͱؾ෇͘ • ʮෛͷ਺ͷղ͸ෆద੾ɻղͱͯ͠࠾୒͞Εͳ͍ɻਓʑ͸ෛͷ਺ͷղΛೝΊ ͳ͍ɻʯ

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9ੈل Ξϥϒ • Πϯυ਺ֶͷෛͷ਺ͷԋࢉنଇ͸ਫ਼௨͍ͯͨ͠ɻ • ͔͠͠ɺෛͷ਺Λड͚ೖΕͳ͔ͬͨɻ

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17ੈل σΧϧτ René Descartes (1596—1650) • Ҽ਺ఆཧͰଟ߲ࣜͷ࣍਺ΛԼ͛ΔͨΊʹෛͷղ΋ར༻ • ͔͠͠ෛͷ਺ͷղ͸ false root (ؒҧͬͨղ) ͱ͢Δ

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17ੈل ύεΧϧ Blaise Pascal (1623—1662) • ʮ0 ͔Β 4 ΛҾ͘ͳΜͯφϯηϯεͩʯ 0 4 = 4

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Ξʔϊϧυ(ύεΧϧͷ༑ୡ) • ʮൺʯΛߟ͑ͯΈΔͱ • ʮେ͖͍਺ : খ͍͞਺ʯͱ͍͏ൺͱʮখ͍͞਺ : େ͖͍਺ʯͱ͍͏ൺ͕౳͍͠ ͳΜ͓͔͍ͯ͠ by ύεΧϧ 1 : 1 = 1 : 1

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ͳ͔ͳ͔ղͱͯ͠ೝΊΒΕͳ͍

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ͦΜͳத……

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16ੈل຤ εςϏϯ Simon Stevin (1548–1620) • ෛͷ਺΋܎਺ͱͯ͠࢖͏ • ෛͷ਺ͷղ΋ड͚ೖΕΔ • ʮฏํࠜɺແཧ਺ɺෛͷ਺ͳͲ΋͢΂ͯʮ਺ʯͱͯ۠͠ผͤͣѻΘΕΔ΂ ͖ʯ

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17ੈل δϥʔϧ Albert Girard (1595 — 1632) • ෛͷ਺͸ʮޙΖʹ໭Δ͜ͱʯΛද͢ɻ • ʮϓϥεه߸Ͱલਐ͢Δͱ͖ɺϚΠφεه߸Ͱ͸໭Δɻʯ • ෛͷ਺ͷزԿֶతͳղऍ • ͭ·Γෛͷ਺ͷϏδϡΞϥΠθʔγϣϯ

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δϥʔϧ • ͨͿΜ͜͏ݴͬͨ Μ͡Όͳ͍͔ܶ৔

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͔ͦ͠͠ͷޙ΋

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19ੈل υŋϞϧΨϯ Augustus De Morgan (1806–1871) • ෕͕56ࡀɺଉࢠ29ࡀɻԿ೥ޙʹ2ഒʹ? • (2೥લ!) • υŋϞϧΨϯʮෆ߹ཧͩ!ʯ 56 + x = 2(29 + x) x = 2

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19ੈل υŋϞϧΨϯ • ෕͕56ࡀɺଉࢠ29ࡀɻԿ೥લʹ2ഒʹ? • υŋϞϧΨϯʮ͜ΕͳΒOKʯ 56 x = 2(29 x) x = 2

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17ੈل δϥʔϧ Albert Girard (1595 — 1632) • ෛͷ਺͸ʮޙΖʹ໭Δ͜ͱʯΛද͢ • ෛͷ਺ͷزԿֶతͳղऍ • ͭ·Γෛͷ਺ͷϏδϡΞϥΠθʔγϣϯ

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ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

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ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

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ϏδϡΞϥΠθʔγϣϯͱ ෳૉ਺

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ෳૉ਺ͱ͸?

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ෳૉ਺ i2 = 1 i ڏ਺୯Ґ x2 = 1 (x2 + 1 = 0) ͷղ Λ࣮਺ʹ෇͚Ճ͑ͨ਺ମܥ i ͸

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2৐ͯ͠ϚΠφε1?? • Ͳ͏΋͓͔͠ͳ਺ͩͳɻ • ͜ͷੈͷ΋ͷͰ͸ͳ͍ͳɻ • ͦΜͳ਺ߟ͍͍͑ͯͷ? • ͦΜͳ਺ʮ͋Δʯͷ? • Ͳ͜ʹ͋Δͷ? • ܭࢉ͸Ͱ͖Δ͚Ͳ… i2 = 1

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ͦΜͳத……

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ΞϧΨϯ Jean Robert Argand (1768 — 1822) • ͜ͷෳૉ਺ͷҧ࿨ײΛݟࣄʹ෷১ɻ • ͦΕ·ͰͳΜ͔Α͘Θ͔Βͳ͍Ծ૝తͳ਺ͩͱࢥΘΕ͍ͯͨෳૉ਺͕Ұؾʹ ࢖ΘΕΔΑ͏ʹͳΔͷʹଟେͳӨڹΛ༩͑ͨɻ • ΞϧΨϯਤ (Argand diagram)

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ΞϧΨϯਤ Argand diagram • ෳૉ਺ͷϏδϡΞϥΠθʔγϣϯ

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( 1) ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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( 1) ( 1) =1 ͱ͍͏͜ͱ͸ʮϚΠφεഒʯͬͯ ౓ճసͬͯ͜ͱͳΜ͡Όʜ ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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͡Ό͋iഒͬͯʜ ·͔͞ʜ ( 1) =i2 ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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i ͟Θɾɾɾ ͟Θɾɾɾ ͟Θɾɾɾ ͟Θɾɾɾ ͡Ό͋iഒͬͯʜ ·͔͞ʜ ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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i i = 1 ͡Ό͋iഒͬͯʜ ·͔͞ʜ ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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i ഒ = 90౓ճస

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ࠓ೔ͷ·ͱΊ • ਺ֶͷ֓೦ͷϏδϡΞϥΠθʔγϣϯ • ֓೦͕ड͚ೖΕΒΕΔ͔ɺࢢຽݖΛಘΔ͔ • ७ਮͳ਺ֶͷੈքͰ͢Βɺ֓೦͕ʮड͚ೖΕΒΕΔʯ͜ͱʹϏδϡΞϥΠ θʔγϣϯɺՄࢹԽ͕ॏཁɻ • ʮΘ͔ͬͨؾ͕͢ΔʯʮΘ͔Δʯͱ͸Կ͔ • ਺ֶͬͯʮ࿦ཧతʹਖ਼͍͠ʯ͚ͩͰ͸ͳ͍ • ͕͕ͪͪͷ࿦ཧͰ͸ͳ͍਺ֶͷ࢟

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ࢀߟจݙ • ໦ଜढ़Ұ, ʰఱ࠽਺ֶऀ͸͜͏ղ͍ ͨɺ͜͏ੜ͖ͨʱ, ߨஊࣾબॻϝν Τ, 2001 • Morris Kline, Mathematical Thought from Ancient to Modern Times, Vol. 1, Oxford University Press, 1990. • The MacTutor History of Mathematics archive, http://www- history.mcs.st-andrews.ac.uk/