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

Transcript

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

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2. ࣗݾ঺հ
   

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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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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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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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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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