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

by 北䑓如法

Slide 1

Slide 1 text

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

Slide 2

Slide 2 text

ࣗݾ঺հ

Slide 3

Slide 3 text

No content

Slide 4

Slide 4 text

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

Slide 5

Slide 5 text

ݕࡧ ৄ͘͠͸ iPad ޿ౡห

Slide 6

Slide 6 text

ʮ͜Μͳײ͡ͷ͕ग़͖ͯ·͢ɻʯ

Slide 7

Slide 7 text

ݕࡧ ࠷ۙ͸ ؟ڸࢢ৔ ޿ౡݶఆCM

Slide 8

Slide 8 text

޿ౡΧʔϓ ಊྛᠳଠબख

Slide 9

Slide 9 text

ࠓ೔: ϏδϡΞϥΠθʔγϣϯͱ਺ֶ

Slide 10

Slide 10 text

ಥવͰ͕͢

Slide 11

Slide 11 text

ݟ͑ΔΑ͏ʹ͢Δ͜ͱ = ՄࢹԽ = visualization

Slide 12

Slide 12 text

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

Slide 13

Slide 13 text

visualization

Slide 14

Slide 14 text

visualization /vìʒuəlaizéiʃən/ [ͼ͡Ύ͋Β͍͍ͥ͠ΐΜ]

Slide 15

Slide 15 text

ՄࢹԽ

Slide 16

Slide 16 text

ࠓճ͸୯ͳΔϏδϡΞϥΠθʔ γϣϯͰ͸ͳ͘

Slide 17

Slide 17 text

਺ֶͱ ϏδϡΞϥΠθʔγϣϯͱͷ࿩

Slide 18

Slide 18 text

਺ͷࢢຽݖ֫ಘͱ ϏδϡΞϥΠθʔγϣϯ

Slide 19

Slide 19 text

ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

Slide 20

Slide 20 text

No content

Slide 21

Slide 21 text

ϏδϡΞϥΠθʔγϣϯͱ ෛͷ਺

Slide 22

Slide 22 text

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

Slide 23

Slide 23 text

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

Slide 24

Slide 24 text

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

Slide 25

Slide 25 text

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

Slide 26

Slide 26 text

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

Slide 27

Slide 27 text

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

Slide 28

Slide 28 text

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

Slide 29

Slide 29 text

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

Slide 30

Slide 30 text

ͳ͔ͳ͔ղͱͯ͠ೝΊΒΕͳ͍

Slide 31

Slide 31 text

ͦΜͳத……

Slide 32

Slide 32 text

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

Slide 33

Slide 33 text

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

Slide 34

Slide 34 text

δϥʔϧ • ͨͿΜ͜͏ݴͬͨ Μ͡Όͳ͍͔ܶ৔

Slide 35

Slide 35 text

͔ͦ͠͠ͷޙ΋

Slide 36

Slide 36 text

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

Slide 37

Slide 37 text

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

Slide 38

Slide 38 text

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

Slide 39

Slide 39 text

ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

Slide 40

Slide 40 text

ෛͷ਺ ෳૉ਺ ຊ೔ͷϝχϡʔ

Slide 41

Slide 41 text

No content

Slide 42

Slide 42 text

ϏδϡΞϥΠθʔγϣϯͱ ෳૉ਺

Slide 43

Slide 43 text

ෳૉ਺ͱ͸?

Slide 44

Slide 44 text

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

Slide 45

Slide 45 text

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

Slide 46

Slide 46 text

ͦΜͳத……

Slide 47

Slide 47 text

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

Slide 48

Slide 48 text

ΞϧΨϯਤ Argand diagram • ෳૉ਺ͷϏδϡΞϥΠθʔγϣϯ

Slide 49

Slide 49 text

No content

Slide 50

Slide 50 text

No content

Slide 51

Slide 51 text

No content

Slide 52

Slide 52 text

No content

Slide 53

Slide 53 text

No content

Slide 54

Slide 54 text

No content

Slide 55

Slide 55 text

No content

Slide 56

Slide 56 text

No content

Slide 57

Slide 57 text

No content

Slide 58

Slide 58 text

No content

Slide 59

Slide 59 text

No content

Slide 60

Slide 60 text

No content

Slide 61

Slide 61 text

No content

Slide 62

Slide 62 text

No content

Slide 63

Slide 63 text

( 1) ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

Slide 64

Slide 64 text

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

Slide 65

Slide 65 text

͡Ό͋iഒͬͯʜ ·͔͞ʜ ( 1) =i2 ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

Slide 66

Slide 66 text

i ͟Θɾɾɾ ͟Θɾɾɾ ͟Θɾɾɾ ͟Θɾɾɾ ͡Ό͋iഒͬͯʜ ·͔͞ʜ ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

Slide 67

Slide 67 text

i i = 1 ͡Ό͋iഒͬͯʜ ·͔͞ʜ ※ ͜͜͸Ξχϝʔγϣϯ͍ͯͨ͠ɻ

Slide 68

Slide 68 text

i ഒ = 90౓ճస

Slide 69

Slide 69 text

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

Slide 70

Slide 70 text

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/