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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

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

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 〜
オープンセミナー2022@広島
2022/06/25(土)
#OSH2022

北䑓如法

June 26, 2022
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  1. ϏδϡΞϥΠθʔγϣϯͱ਺ֶ
    Φʔϓϯηϛφʔ2022@޿ౡ 2022/06/25(౔) ๺㝳೗๏ (͖͍ͨͩΏ͖ͷΓ)
    ʙ͢͏͕͘Ή͔͠͹ͳ͠ෛͷ਺ɾෳૉ਺ฤʙ

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  2. ࣗݾ঺հ

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  3. View Slide

  4. ͋Μͨ͊୭ͳΜ?
    • ๺㝳೗๏ (͖͍ͨͩΏ͖ͷΓ)


    • (ʮ೗๏ʯΛԻಡΈͯ͠) ʹΐ΄͏, Nyoho


    • ೔༵ϓϩάϥϚ


    • ਺ֶ (޿ౡେֶେֶӃਓؒࣾձՊֶݚڀՊ)


    • ϋΠύɾϝσΟΞɾτϥϯεϨʔλ


    • ޿ౡหਧ͖ସ͑γϦʔζ

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  5. ݕࡧ
    ৄ͘͠͸
    iPad ޿ౡห

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  6. ʮ͜Μͳײ͡ͷ͕ग़͖ͯ·͢ɻʯ

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  7. ݕࡧ
    ࠷ۙ͸
    ؟ڸࢢ৔ ޿ౡݶఆCM

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  8. ޿ౡΧʔϓ


    ಊྛᠳଠબख

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  9. ࠓ೔:


    ϏδϡΞϥΠθʔγϣϯͱ਺ֶ

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  10. ಥવͰ͕͢

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  11. ݟ͑ΔΑ͏ʹ͢Δ͜ͱ = ՄࢹԽ

    = visualization

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  12. Visualization
    • visual (ϏδϡΞϧ, ܗ༰ࢺ) ໨ʹݟ͑Δɺࢹ֮ͷ


    • visualize (ϏδϡΞϥΠζ, ಈࢺ) ໨ʹݟ͑ΔΑ͏ʹ͢Δɺࢹ֮Խ͢Δ


    • visualization (ϏδϡΞϥΠθʔγϣϯ, ໊ࢺ) ໨ʹݟ͑ΔΑ͏ʹ͢Δ͜ͱ

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  13. visualization

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  14. visualization

    /vìʒuəlaizéiʃən/

    [ͼ͡Ύ͋Β͍͍ͥ͠ΐΜ]

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  15. ՄࢹԽ

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  16. ࠓճ͸୯ͳΔϏδϡΞϥΠθʔ
    γϣϯͰ͸ͳ͘

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  17. ਺ֶͱ

    ϏδϡΞϥΠθʔγϣϯͱͷ࿩

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  18. ਺ͷࢢຽݖ֫ಘͱ

    ϏδϡΞϥΠθʔγϣϯ

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  19. ෛͷ਺
    ෳૉ਺
    ຊ೔ͷϝχϡʔ

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  20. View Slide

  21. ϏδϡΞϥΠθʔγϣϯͱ


    ෛͷ਺

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  22. ෛͷ਺
    • ํఔࣜͷʮ܎਺ʯͱͯ͠΋ʮղʯͱͯ͠΋ͳ͔ͳ͔ड͚ೖΕΒΕͳ͔ͬͨɻ


    • ܭࢉํ๏͸஌ΒΕ͍ͯͨͷʹɻ

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  23. ෛͷ਺ͷܭࢉ
    • ݱ୅Ͱ΋ɺ;ͱʮ͸ͯʯͱࢥ͏͜ͱ΋ɻ


    • (࣮͸͜͜΋໘ന͍ (ࠓ೔͸লུ))
    ( 1) ( 1) = +1

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  24. 3ੈل͝Ζ Τδϓτ
    • σΟΦϑΝϯτε 200–284


    • ਖ਼ͷ༗ཧ਺ͷΈΛղͱͯ͠ೝΊΔɻ


    • ଞ͸ࣺͯΔɻ

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  25. 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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  26. 9ੈل Ξϥϒ
    • Πϯυ਺ֶͷෛͷ਺ͷԋࢉنଇ͸ਫ਼௨͍ͯͨ͠ɻ


    • ͔͠͠ɺෛͷ਺Λड͚ೖΕͳ͔ͬͨɻ

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  27. 17ੈل σΧϧτ
    René Descartes (1596—1650)
    • Ҽ਺ఆཧͰଟ߲ࣜͷ࣍਺ΛԼ͛ΔͨΊʹෛͷղ΋ར༻


    • ͔͠͠ෛͷ਺ͷղ͸ false root (ؒҧͬͨղ) ͱ͢Δ

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  28. 17ੈل ύεΧϧ
    Blaise Pascal (1623—1662)
    • ʮ0 ͔Β 4 ΛҾ͘ͳΜͯφϯηϯεͩʯ
    0 4 = 4

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  29. Ξʔϊϧυ(ύεΧϧͷ༑ୡ)
    • ʮൺʯΛߟ͑ͯΈΔͱ


    • ʮେ͖͍਺ : খ͍͞਺ʯͱ͍͏ൺͱʮখ͍͞਺ : େ͖͍਺ʯͱ͍͏ൺ͕౳͍͠
    ͳΜ͓͔͍ͯ͠ by ύεΧϧ
    1 : 1 = 1 : 1

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  30. ͳ͔ͳ͔ղͱͯ͠ೝΊΒΕͳ͍

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  31. ͦΜͳத……

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  32. 16ੈل຤ εςϏϯ
    Simon Stevin (1548–1620)
    • ෛͷ਺΋܎਺ͱͯ͠࢖͏


    • ෛͷ਺ͷղ΋ड͚ೖΕΔ


    • ʮฏํࠜɺແཧ਺ɺෛͷ਺ͳͲ΋͢΂ͯʮ਺ʯͱͯ۠͠ผͤͣѻΘΕΔ΂
    ͖ʯ

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  33. 17ੈل δϥʔϧ
    Albert Girard (1595 — 1632)
    • ෛͷ਺͸ʮޙΖʹ໭Δ͜ͱʯΛද͢ɻ


    • ʮϓϥεه߸Ͱલਐ͢Δͱ͖ɺϚΠφεه߸Ͱ͸໭Δɻʯ


    • ෛͷ਺ͷزԿֶతͳղऍ


    • ͭ·Γෛͷ਺ͷϏδϡΞϥΠθʔγϣϯ


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  34. δϥʔϧ
    • ͨͿΜ͜͏ݴͬͨ
    Μ͡Όͳ͍͔ܶ৔

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  35. ͔ͦ͠͠ͷޙ΋

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  36. 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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  37. 19ੈل υŋϞϧΨϯ
    • ෕͕56ࡀɺଉࢠ29ࡀɻԿ೥લʹ2ഒʹ?


    • υŋϞϧΨϯʮ͜ΕͳΒOKʯ
    56 x = 2(29 x)
    x = 2

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  38. 17ੈل δϥʔϧ
    Albert Girard (1595 — 1632)
    • ෛͷ਺͸ʮޙΖʹ໭Δ͜ͱʯΛද͢


    • ෛͷ਺ͷزԿֶతͳղऍ


    • ͭ·Γෛͷ਺ͷϏδϡΞϥΠθʔγϣϯ


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  39. ෛͷ਺
    ෳૉ਺
    ຊ೔ͷϝχϡʔ

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  40. ෛͷ਺
    ෳૉ਺
    ຊ೔ͷϝχϡʔ

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  41. View Slide

  42. ϏδϡΞϥΠθʔγϣϯͱ


    ෳૉ਺

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  43. ෳૉ਺ͱ͸?

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

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  45. 2৐ͯ͠ϚΠφε1??
    • Ͳ͏΋͓͔͠ͳ਺ͩͳɻ


    • ͜ͷੈͷ΋ͷͰ͸ͳ͍ͳɻ


    • ͦΜͳ਺ߟ͍͍͑ͯͷ?


    • ͦΜͳ਺ʮ͋Δʯͷ?


    • Ͳ͜ʹ͋Δͷ?


    • ܭࢉ͸Ͱ͖Δ͚Ͳ…
    i2 = 1

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  46. ͦΜͳத……

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  47. ΞϧΨϯ
    Jean Robert Argand (1768 — 1822)
    • ͜ͷෳૉ਺ͷҧ࿨ײΛݟࣄʹ෷১ɻ


    • ͦΕ·ͰͳΜ͔Α͘Θ͔Βͳ͍Ծ૝తͳ਺ͩͱࢥΘΕ͍ͯͨෳૉ਺͕Ұؾʹ
    ࢖ΘΕΔΑ͏ʹͳΔͷʹଟେͳӨڹΛ༩͑ͨɻ


    • ΞϧΨϯਤ (Argand diagram)

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  48. ΞϧΨϯਤ
    Argand diagram
    • ෳૉ਺ͷϏδϡΞϥΠθʔγϣϯ

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  49. View Slide

  50. View Slide

  51. View Slide

  52. View Slide

  53. View Slide

  54. View Slide

  55. View Slide

  56. View Slide

  57. View Slide

  58. View Slide

  59. View Slide

  60. View Slide

  61. View Slide

  62. View Slide

  63. ( 1)
    ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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  64. ( 1) ( 1)
    =1
    ͱ͍͏͜ͱ͸ʮϚΠφεഒʯͬͯ
    ౓ճసͬͯ͜ͱͳΜ͡Όʜ
    ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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  65. ͡Ό͋iഒͬͯʜ
    ·͔͞ʜ
    ( 1)
    =i2
    ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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  66. i
    ͟Θɾɾɾ
    ͟Θɾɾɾ
    ͟Θɾɾɾ
    ͟Θɾɾɾ
    ͡Ό͋iഒͬͯʜ
    ·͔͞ʜ
    ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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  67. i i
    = 1
    ͡Ό͋iഒͬͯʜ
    ·͔͞ʜ
    ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

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  68. i ഒ = 90౓ճస

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  69. ࠓ೔ͷ·ͱΊ
    • ਺ֶͷ֓೦ͷϏδϡΞϥΠθʔγϣϯ


    • ֓೦͕ड͚ೖΕΒΕΔ͔ɺࢢຽݖΛಘΔ͔


    • ७ਮͳ਺ֶͷੈքͰ͢Βɺ֓೦͕ʮड͚ೖΕΒΕΔʯ͜ͱʹϏδϡΞϥΠ
    θʔγϣϯɺՄࢹԽ͕ॏཁɻ


    • ʮΘ͔ͬͨؾ͕͢ΔʯʮΘ͔Δʯͱ͸Կ͔


    • ਺ֶͬͯʮ࿦ཧతʹਖ਼͍͠ʯ͚ͩͰ͸ͳ͍


    • ͕͕ͪͪͷ࿦ཧͰ͸ͳ͍਺ֶͷ࢟

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  70. 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/

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  71. ͋Γ͕ͱ͏͍͟͝·ͨ͠ʔ

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  72. ࠓ೔ͷ·ͱΊ
    • ਺ֶͷ֓೦ͷϏδϡΞϥΠθʔγϣϯ


    • ֓೦͕ड͚ೖΕΒΕΔ͔ɺࢢຽݖΛಘΔ͔


    • ७ਮͳ਺ֶͷੈքͰ͢Βɺ֓೦͕ʮड͚ೖΕΒΕΔʯ͜ͱʹϏδϡΞϥΠ
    θʔγϣϯɺՄࢹԽ͕ॏཁɻ


    • ʮΘ͔ͬͨؾ͕͢ΔʯʮΘ͔Δʯͱ͸Կ͔


    • ਺ֶͬͯʮ࿦ཧతʹਖ਼͍͠ʯ͚ͩͰ͸ͳ͍


    • ͕͕ͪͪͷ࿦ཧͰ͸ͳ͍਺ֶͷ࢟

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