• Aucun résultat trouvé

Mathématiques récréatives. Éclairages historiques et épistémologiques

N/A
N/A
Protected

Academic year: 2022

Partager "Mathématiques récréatives. Éclairages historiques et épistémologiques"

Copied!
7
0
0

Texte intégral

(1)

Dominique Tournès

INTRODUCTION

'HSXLV O¶$QWLTXLWp GLYHUV SUREOqPHV D\DQW XQ FRQWHQX PDWKpPDWLTXH RQW pWpSUpVHQWpVSDUOHXUVDXWHXUVFRPPHSURSUHVj©ௗDPXVHUௗªWRXWHSHUVRQQH FXULHXVHTXLFRQVDFUHUDLWGXWHPSVjODUHFKHUFKHGHOHXUVVROXWLRQV&HVSUR- EOqPHV WUDQVPLV GH JpQpUDWLRQ HQ JpQpUDWLRQ RQW pWp SHX j SHX UDVVHPEOpV GDQVGHVUHFXHLOVSXLVSXEOLpVVRXVIRUPHGHOLYUHV&RQVXOWRQVWURLVGHFHV RXYUDJHVSDUPLFHX[TXLRQWFRQQXOHVXFFqVDXSRLQWG¶rWUHUpJXOLqUHPHQW UpLPSULPpVUppGLWpVRXDGDSWpVSDUIRLVMXVTX¶jQRVMRXUV 1(Q&ODXGH

*DVSDUG%DFKHWGH0p]LULDFSURSRVHjVRQSXEOLFGHVProblemes plaisans et delectables, qui se font par les nombres 2(QWrWHGHODVHFRQGHpGLWLRQGH RQWURXYHFHWWHGpGLFDFHj0RQVLHXUOHFRPWHGH7RXUQRQ©ௗ-HYRXVRIIUH GHVMHX[PDLVTXLVRQWjPRQDGYLVGLJQHVGHYRVWUHEHOHVSULWHWFDSDEOHV GHOXLIRXUQLUTXHOTXHVIRLVXQDJJUHDEOHGLYHUWLVVHPHQWௗª9LHQWHQVXLWHXQ VRQQHWTXH&KDUOHVOH*UDQGDYRFDWDXVLqJHSUpVLGLDOGH%UHVVHDHQYR\pj

%DFKHWVXU©ௗVRQOLYUHGHMHX[ௗªGDQVOHTXHOLOTXDOL¿HFHVGHUQLHUVGH©ௗSDVVH WHPSVHWUHFUHDWLRQVௗª-HXGLYHUWLVVHPHQWSDVVHWHPSVUpFUpDWLRQWRXVFHV PRWVVRQWGHQDWXUHjQRXVLQWULJXHUHWPpULWHQWG¶rWUHLQWHUURJpV)DLUHVXLYUH FKDFXQG¶HX[GHO¶DGMHFWLI©ௗPDWKpPDWLTXHௗªQ¶HVWFHSDVFUpHUDXWDQWG¶R[\- PRUHV DX[ \HX[ GH FHOOHV HW FHX[ HQFRUH WURS QRPEUHX[ DXMRXUG¶KXL TXL JDUGHQWXQPDXYDLVVRXYHQLUVFRODLUHGHFHWWHGLVFLSOLQHDVVRFLpHPDOJUpHOOH jODVRXIIUDQFHHWjO¶pFKHFௗ"(WSRXUWDQWGHSXLVOHXVIIeVLqFOHGHQRPEUHX[

OLYUHVRQWSRUWpOHWLWUHGHRécréations mathématiques(QTXRLFRQVLVWHGRQF FHWWH QRWLRQ GH ©ௗPDWKpPDWLTXHV UpFUpDWLYHVௗªௗ" 6¶DJLWLO VHXOHPHQW G¶RIIULU

1. 3RXUQHSDVDORXUGLUFHWWHLQWURGXFWLRQQRXVQHPHQWLRQQHURQVGDQVODVXLWHTXHOHV pGLWLRQVSXEOLpHVGXYLYDQWGHVDXWHXUV

2. /\RQ5LJDXGUHpGepG

(2)

Introduction

GHVPRPHQWVGHORLVLUGHGpWHQWHGHGpODVVHPHQWjTXHOTXHVULFKHVRLVLIVRX DPDWHXUVpFODLUpVௗ"

(QGDQVODSUpIDFHGHVHVRécréations mathématiques et physiques 3 -DFTXHV2]DQDPQRXVSHUPHWG¶DYDQFHUGDQVODUpÀH[LRQ©ௗ%LHQTXHOHVMHX[

G¶HVSULWGRQWMHSDUOHVRLHQWGHVDPXVHPHQWVLOVQHVRQWSHXWrWUHSDVPRLQV XWLOHVTXHOHVH[HUFLFHVDX[TXHOVRQDSSOLTXHOHVMHXQHVSHUVRQQHVGHTXDOLWp SRXUIDoRQQHUOHXUVFRUSVHWSRXUOHXUGRQQHUOHERQDLUௗª,O\DOjO¶LGpH TX¶HQPDWKpPDWLTXHVFRPPHGDQVWRXWDXWUHGRPDLQHOHVMHX[SHXYHQWVHUYLU GHPRWLYDWLRQHWGHVXSSRUWjO¶pGXFDWLRQGHVMHXQHV'HSXLVFHWWHLGpHDpWp SpULRGLTXHPHQWUpDI¿UPpH5DSSHORQVSDUH[HPSOHTXHOD©ௗ6WUDWpJLHPDWKp- PDWLTXHVௗªGXPLQLVWqUHGHO¶eGXFDWLRQQDWLRQDOHSXEOLpHHQGpFHPEUH UHPHWHQDYDQW©ௗODGLPHQVLRQOXGLTXHGHVPDWKpPDWLTXHVௗªHW©ௗODSODFHGX MHXGDQVOHVDSSUHQWLVVDJHVHQPDWKpPDWLTXHVௗª/HVQRXYHDX[SURJUDPPHV HQWUpV HQ DSSOLFDWLRQ j OD UHQWUpH QRWDPPHQW FHX[ GX F\FOH GpYH- ORSSHQW FHW REMHFWLI HQ OLHQ DYHF OD SODFH FURLVVDQWH TXH OHV SUREDELOLWpV HW O¶DOJRULWKPLTXH RFFXSHQW GDQV OHV PDWKpPDWLTXHV DFWXHOOHV OD SUDWLTXH GH MHX[SRXUOHVTXHOVLOIDXWGpYHORSSHUXQHVWUDWpJLHJDJQDQWHFRQWULEXHGHIDoRQ JpQpUDOHDXGpYHORSSHPHQWGHVIDFXOWpVGHUDLVRQQHPHQWௗO¶LQWHUSUpWDWLRQGHV GRQQpHVLVVXHVGHO¶pWXGHG¶XQMHXHWO¶pYDOXDWLRQGHVHVFKDQFHVGHJDLQVRQW XQSRLQWGHGpSDUWQDWXUHOSRXUO¶LQWURGXFWLRQGHVSUREDELOLWpVHWGHODVWDWLV- WLTXHௗO¶DQDO\VHODFRQFHSWLRQHWODSURJUDPPDWLRQGHMHX[VLPSOHVFRQVWL- WXHQWXQVXSSRUWPRWLYDQWSRXUO¶pWXGHGHO¶DOJRULWKPLTXH3OXVUpFHPPHQWOH UDSSRUW9LOODQL7RURVVLDQUHFRPPDQGHpJDOHPHQWOHSODLVLUSDUOHMHXFRPPH O¶XQHGHVHQWUpHVjSULYLOpJLHUSRXUDWWLUHUOHVMHXQHVYHUVOHVPDWKpPDWLTXHV

©ௗ(QWUDYDLOODQWOHVIRQGDPHQWDX[SDUXQHDSSURFKHGLIIpUHQWHOHMHXFRQWULEXH OXLDXVVLjODIRUPDWLRQPDWKpPDWLTXHGHVpOqYHV>/HVMHX[@VWLPXOHQWOHUDL- VRQQHPHQWORJLTXHHWFRQWULEXHQWjFUpHURXUHVWDXUHUOHSODLVLUGHIDLUHGHV PDWKpPDWLTXHVSRXUO¶pOqYHFRPPHSRXUVRQSURIHVVHXU 4ௗª'XSRLQWGHYXH SpGDJRJLTXHOHVUpFUpDWLRQVPDWKpPDWLTXHVMHX[pQLJPHVFRQFRXUVGp¿V HWKLVWRLUHVVRQWDXVVLPLVHVHQDYDQWSRXUOHXUDGpTXDWLRQDYHFODWULORJLH

©ௗPDQLSXOHUHWH[SpULPHQWHUYHUEDOLVHUDEVWUDLUHௗªSUpFRQLVpHSDUOHUDSSRUW /H WURLVLqPH RXYUDJH FODVVLTXH DXTXHO QRXV QRXV UpIqUHURQV GDQV FHWWH LQWURGXFWLRQDpWpSXEOLpSDUeGRXDUG/XFDVYHUVOD¿QGXXIXeVLqFOHHWSRUWH HQFRUHOHWLWUHGHRécréations mathématiques 52QOLWFHFLGDQVVDSUpIDFH

3. YRO3DULV-RPEHUW

4. &pGULF9LOODQLHW&KDUOHV7RURVVLDQ21 mesures pour l’enseignement des mathéma- tiquesUDSSRUWUHPLVOHIpYULHUDXPLQLVWUHGHO¶eGXFDWLRQQDWLRQDOHS 5. 3DULV*DXWKLHU9LOODUVUHpGYROepGYRO

(3)

©ௗ6LFHVSDJHVSODLVHQWjTXHOTXHVVDYDQWVVLHOOHVLQWpUHVVHQWTXHOTXHVJHQV GXPRQGHVLHOOHVLQVSLUHQWjTXHOTXHVMHXQHVLQWHOOLJHQFHVOHJR€WGXUDL- VRQQHPHQW HW OH GpVLU GHV MRXLVVDQFHV DEVWUDLWHV MH VHUDL VDWLVIDLW 6ௗª$SUqV OHVJHQVGXPRQGHHWOHVFRPPHQoDQWVDSSDUDvWXQWURLVLqPHSXEOLFSRWHQ- WLHOFHOXLGHVVDYDQWVHWQRWDPPHQWGHVPDWKpPDWLFLHQVSURIHVVLRQQHOV(Q HIIHWFHVGHUQLHUVSHXYHQWWURXYHUGDQVOHVMHX[PDWKpPDWLTXHVGHVSUREOqPHV GLI¿FLOHVjUpVRXGUHHWGHVVXMHWVG¶LQVSLUDWLRQSRXUGpYHORSSHUGHVWKpRULHV QRXYHOOHV&¶HVWDLQVLTXHORQJWHPSVFRQVLGpUpHVFRPPHGHVHQIDQWLOODJHV GHVEDJDWHOOHVGHVDPXVHPHQWVGHVDORQVDQVJUDQGHSRUWpHOHVUpFUpDWLRQV PDWKpPDWLTXHVRQWSHXjSHXJDJQpHQUHVSHFWDELOLWpDXVHLQGHODFRPPX- QDXWp PDWKpPDWLTXH 'DQV OD GHUQLqUH YHUVLRQ GH ODMathematical Subject

&ODVVL¿FDWLRQDGRSWpHDXQLYHDXLQWHUQDWLRQDO06&LO\DXQHUXEULTXH JpQpUDOHRecreational mathematics$HWXQHLPSRVDQWHUXEULTXHVSp- FLDOLVpHGame theory$[[GRQWOHVVRXVUXEULTXHVIRQWHQWUHYRLUOHV QRPEUHX[OLHQVWLVVpVDYHFGLYHUVGRPDLQHVGHVPDWKpPDWLTXHVDXSUHPLHU UDQJGHVTXHOVODWKpRULHGHVJUDSKHVODFRPELQDWRLUHHWOHVSUREDELOLWpV3DU DLOOHXUVODWKpRULHGHVMHX[WLHQWXQU{OHGHSOXVHQSOXVLPSRUWDQWGDQVOHV VFLHQFHVVRFLDOHVO¶pFRQRPLHOHGRPDLQHPLOLWDLUHHWSOXVJpQpUDOHPHQWGDQV WRXWHVOHVVLWXDWLRQVRLOLPSRUWHG¶pODERUHUHWG¶RSWLPLVHUGHVWDFWLTXHVHW GHVVWUDWpJLHV5DSSHORQVHQ¿QTXHOHVUHFKHUFKHVSRXUFRQFHYRLUGHVSUR- JUDPPHVLQIRUPDWLTXHVFDSDEOHVGHEDWWUHOHVPHLOOHXUVMRXHXUVKXPDLQVDX MHXG¶pFKHFVDXMHXGHJRRXjG¶DXWUHVMHX[RQWFRQGXLWjGHVSURJUqVVLJQL-

¿FDWLIVHQDOJRULWKPLTXHSURJUDPPDWLRQHWLQWHOOLJHQFHDUWL¿FLHOOH

&RPSWHWHQXGHFHTXLSUpFqGHOHSUpVHQWOLYUHVHSURSRVHGRQFG¶LQWHU- URJHUODQRWLRQGH©ௗPDWKpPDWLTXHVUpFUpDWLYHVௗªGXSRLQWGHYXHpSLVWpPROR- JLTXHHWKLVWRULTXH4X¶HVWFHTXLHVWPDWKpPDWLTXHGDQVOHVGLIIpUHQWVW\SHV GHMHX[HQXVDJHjXQPRPHQWGRQQpGDQVXQHVRFLpWpGRQQpHௗ"4XDQGSRXU- TXRLHWFRPPHQWOHVPDWKpPDWLFLHQVVHVRQWLOVLQWpUHVVpVjO¶pWXGHGHFHUWDLQV MHX[ௗ"(QTXRLOHVMHX[RQWLOVFRQWULEXpjODFUpDWLRQRXDXGpYHORSSHPHQWGH FHUWDLQHVSDUWLHVGHVPDWKpPDWLTXHVHWGHO¶LQIRUPDWLTXHௗ"(QV¶LQVSLUDQWGH O¶KLVWRLUHFRPPHQWFRQFHYRLUGHVVLWXDWLRQVOXGLTXHVSHUWLQHQWHVSRXUO¶HQVHL- JQHPHQWGHVPDWKpPDWLTXHVG¶DXMRXUG¶KXLHQSDUWLFXOLHUGHVSUREDELOLWpVHW GHO¶DOJRULWKPLTXHௗ"

/HVGL[FKDSLWUHVUpXQLVLFLV¶HIIRUFHQWGHIDLUHOHWRXUGXWKqPHFRQGXF- WHXUGHVPDWKpPDWLTXHVUpFUpDWLYHVHQDGRSWDQWVXFFHVVLYHPHQWGLYHUVDQJOHV GHYXH

6. Op. cit.epGYROSYLLL

(4)

Introduction

1RXVFRPPHQFHURQVSDUSUHQGUHXQSHXGHUHFXOSDUUDSSRUWDX[PDWKp- PDWLTXHVHOOHVPrPHVSRXUQRXVLQWpUHVVHUDX[FRQWH[WHVGDQVOHVTXHOVFHU- WDLQHVUpFUpDWLRQVPDWKpPDWLTXHVRQWSXYRLUOHMRXUHWVHGpYHORSSHU$LQVL GDQVOHSUHPLHUFKDSLWUHFRQVDFUpDX©ௗ-HXGHVTXLQ]HFUR\DQWVHWGHVTXLQ]H LQ¿GqOHVௗª 3LHUUH$JHURQ HW *pUDUG +DPRQ QRXV IRQW SUHQGUH FRQVFLHQFH G¶HQWUpH GH MHX TXH OHV UpFUpDWLRQV PDWKpPDWLTXHV FRQWUDLUHPHQW j FH TXH G¶DXFXQVSRXUUDLHQWFURLUHQDwYHPHQWQHVRQWSDVWRXMRXUVGHVMHX[GH©ௗERQQH VRFLpWpௗª/RLQG¶rWUHQHXWUHVHWDVHSWLVpVLOVUHÀqWHQWSDUIRLVFUXHOOHPHQWOH FRQWH[WHSROLWLTXHHWVRFLRFXOWXUHOGDQVOHTXHOLOVVRQWSUDWLTXpV'DQVFHMHX GHVFUR\DQWVHWGHVLQ¿GqOHVTXLFLUFXOHGDQVGHQRPEUHXVHVODQJXHVHWGH QRPEUHX[OLHX[GHSXLVOHIXeVLqFOHLOV¶DJLWWRXWVLPSOHPHQWGHVDXYHUOHV FUR\DQWVHWG¶pOLPLQHUOHVLQ¿GqOHVSDUGHVWHFKQLTXHVGHFRPSWDJHGDQVOH FDVSDUH[HPSOHRLOHVWQpFHVVDLUHG¶DOOpJHUXQEDWHDXHQSHUGLWLRQ6LJQH G¶XQHYLROHQFHRPQLSUpVHQWHOHVVRXUFHVTXLQRXVVRQWSDUYHQXHVPHWWHQWHQ VFqQHGHVFKUpWLHQVGHVMXLIVHWGHVPXVXOPDQVGDQVOHU{OHGHVFUR\DQWVWRXW DXWDQWTXHGDQVFHOXLGHVLQ¿GqOHVDYHFWRXWHVOHVFRPELQDLVRQVSRVVLEOHV/H VHFRQGFKDSLWUHLQWLWXOp©ௗ/¶H[SRQHQWLHOOHHQWUHMHXPDWKpPDWLTXHHWYLVLRQGX PRQGHௗªSHUPHWj%HQRvW5LWWDXGGHUHVWHUGDQVOHPrPHUHJLVWUH,OV¶DWWDFKH GHVRQF{WpjPRQWUHUHQTXRLODFURLVVDQFHH[SRQHQWLHOOHGHSXLVOHIDPHX[

FRQWHDQFLHQGXGRXEOHPHQWGHVJUDLQVGHEOpVXUFKDTXHFDVHGHO¶pFKLTXLHU HVWUpYpODWULFHGHQRVUHSUpVHQWDWLRQVFROOHFWLYHVHWGHQRVFUDLQWHVLUUDLVRQ- QpHV7RXWHQVHUYDQWGHSUpWH[WHHWGHVXSSRUWjGHVUpFUpDWLRQVQXPpULTXHV VXUSUHQDQWHVFHWWHFURLVVDQFHH[SRQHQWLHOOHSHUPHWGHPRGpOLVHUO¶pYROXWLRQ GHSRSXODWLRQVYpJpWDOHVDQLPDOHVRXKXPDLQHVGHVHOLYUHUjGHVH[pJqVHV ELEOLTXHVG¶pEDXFKHUGHVVFpQDULRVSROLWLTXHVRXpFRQRPLTXHVYRLUHG¶pOD- ERUHUGHVSURJUDPPHVGHSHXSOHPHQW'DQVVHVDYDWDUVOHVPRLQVDYRXDEOHV HOOHDDXVVLFRQGXLWFHUWDLQVjHQYLVDJHUO¶DUUrWGHWRXWHHVSqFHG¶DVVLVWDQFH DX[SOXVSDXYUHVRXHQFRUHG¶pWXGLHUODIDLVDELOLWpG¶XQUDSDWULHPHQWPDVVLI HQ$IULTXHGHVDQFLHQVHVFODYHVQRLUV

&RPPH QRXV O¶DYRQV GLW SOXV KDXW OH JHQUH GHV UpFUpDWLRQV PDWKpPD- WLTXHVDGRQQpQDLVVDQFHjXQQRPEUHLPSRUWDQWG¶RXYUDJHVGDQVWRXWHVOHV ODQJXHV 0DLV TXL VRQW OHV DXWHXUV GH FHV RXYUDJHVௗ" 4XHOV VRQW OHXUV SDU- FRXUVOHXUVFHQWUHVG¶LQWpUrWOHXUVPRWLYDWLRQVௗ"&RPPHQWFROOHFWHQWLOVGHV pQLJPHVGHVGHYLQHWWHVGHVSUREOqPHVSURSUHVjpYHLOOHUODFXULRVLWpGHOHXUV OHFWHXUVௗ"7URLVFKDSLWUHVGHFHOLYUHYRQWV¶DWWDFKHUjPLHX[FHUQHUFHVDXWHXUV G¶XQ W\SH SDUWLFXOLHU TXH QRXV DSSHOOHURQV ©ௗUpFUpDWHXUVௗª HQ EURVVDQW GHV SRUWUDLWVKDXWVHQFRXOHXU$LQVLOHWURLVLqPHFKDSLWUHV¶HIIRUFHGHSHUFHUOHV VHFUHWVGH©ௗ'LGLHU+HQULRQFRPSLODWHXUGHUpFUpDWLRQVPDWKpPDWLTXHVGHV DQQpHVௗª¬SDUWLUGHVRXUFHVDUFKLYLVWLTXHVQRXYHOOHV)UpGpULF0pWLQ

(5)

SUpFLVHODELRJUDSKLHREVFXUHGXSHUVRQQDJHHQpWDEOLVVDQWQRWDPPHQWTXH VRQSUpQRPHVWELHQ'LGLHUHWQRQ'HQLVFRPPHRQOHFUR\DLWjWRUWHWTXH F¶HVWXQDXWHXUGLVWLQFWGH3LHUUH+pULJRQHHWGH&\ULDTXHGH0DQJLQDYHFTXL RQO¶DYDLWSDUIRLVFRQIRQGX/HFKDSLWUHVHSRXUVXLWSDUXQHpWXGHGpWDLOOpHGHV

©ௗ4XHVWLRQVLQJHQLHXVHVHWUHFUHDWLYHVௗªTXLRFFXSHQWSDJHVGHVDCollec- tion mathematiqueSXEOLpHHQ)DLVDQWXQVDXWGHSOXVGHGHX[VLqFOHV 6\OYLDQH6FKZHUVHSHQFKHVXUXQDXWUHUpFUpDWHXURULJLQDOGDQVOHTXDWULqPH FKDSLWUHLQWLWXOp©ௗ5HYHQLUDX[PDWKpPDWLTXHVSDUOHVUpFUpDWLRQVO¶H[HPSOH GH+HQUL$XJXVWH'HODQQR\ௗª$SUqVXQHORQJXHFDUULqUHPLOLWDLUHOHSRO\- WHFKQLFLHQ+HQUL$XJXVWH'HODQQR\V¶LQWpUHVVHjQRXYHDXDX[PDWKpPDWLTXHV jSDUWLUGHSDUO¶LQWHUPpGLDLUHGHVUpFUpDWLRQV/HVSUREOqPHVGHJpRPp- WULHGHVLWXDWLRQTXHO¶RQSHXWUpVRXGUHjO¶DLGHG¶XQpFKLTXLHUOHFRQGXLVHQW jODSXEOLFDWLRQGHRQ]HDUWLFOHVGHPDWKpPDWLTXHVSRUWDQWVXUGHVTXHVWLRQV GH SUREDELOLWpV GLVFUqWHV 8Q WHO SDUFRXUV HVW H[HPSODLUH GX U{OH PLOLWDQW GHV UpFUpDWLRQV PDWKpPDWLTXHV GDQV O¶HQWUHGHX[JXHUUHV SRXU ODUHFRQVWLWXWLRQG¶XQHpOLWHVFLHQWL¿TXHIUDQoDLVHDSUqVODGpIDLWHGH6HGDQ HWODFKXWHGX6HFRQG(PSLUH&¶HVWGDQVFHPrPHFRQWH[WHTX¶DpWpFUppH HQ O¶$VVRFLDWLRQ IUDQoDLVH SRXU O¶DYDQFHPHQW GHV VFLHQFHV 'HODQQR\

ODIUpTXHQWHDVVLGXPHQWWRXWFRPPHG¶DXWUHVLQJpQLHXUVHWPDWKpPDWLFLHQV pJDOHPHQWVHQVLEOHVjO¶LQWpUrWGHVUpFUpDWLRQVSRXUODUHFKHUFKHHWO¶HQVHLJQH- PHQW8QDXWUHPHPEUHGHFHPLOLHXIRLVRQQDQWIDLWMXVWHPHQWO¶REMHWGXFLQ- TXLqPHFKDSLWUHpFULWSDU-pU{PH$XYLQHW©ௗ/HVUpFUpDWLRQVPDWKpPDWLTXHV FKH]&KDUOHV$QJH/DLVDQWGHODJpRPpWULHGHVLWXDWLRQjO¶Initiation mathé- matiqueௗª'¶RULJLQHSRO\WHFKQLFLHQQHHWPLOLWDLUHFRPPH'HODQQR\/DLVDQW HVWpOXGpSXWpGHjDYDQWGHGHYHQLUHQVHLJQDQWHQFODVVHVSUpSDUD- WRLUHVDX[JUDQGHVpFROHV$\DQWODYRORQWpGHSRSXODULVHUOHVPDWKpPDWLTXHV HWGHUHQRXYHOHUOHXUHQVHLJQHPHQWLOSHUoRLWOHVVLWXDWLRQVUpFUpDWLYHVFRPPH XQHVRXUFHG¶LQQRYDWLRQVSpGDJRJLTXHVHWHQIDLWODVXEVWDQFHG¶XQOLYUHQRYD- WHXU SRXU O¶pGXFDWLRQ GHV MHXQHV HQIDQWV O¶Initiation mathématique SXEOLpH HQ/HVWURLVFKDSLWUHVVXLYDQWVQHVRQWSOXVFHQWUpVSULQFLSDOHPHQWVXUGHVSHU- VRQQDJHVPDLVVXUGHVW\SHVSDUWLFXOLHUVGHMHX[RXGHUpFUpDWLRQVDYHFO¶DP- ELWLRQGHOHVUHSODFHUGDQVOHXUFRQWH[WHKLVWRULTXHG¶HQSUpFLVHUOHFRQWHQX PDWKpPDWLTXH VRXVMDFHQW HW GH VXJJpUHU OHXUV SRWHQWLDOLWpV SpGDJRJLTXHV 7RXW G¶DERUG )UDQoRLV *RLFKRW VH FRQFHQWUH GDQV OH VL[LqPH FKDSLWUH VXU

©ௗ/DULWKPRPDFKLHXQMHXSpGDJRJLTXHGXXIeDXXVIeVLqFOHௗª,QYHQWpSDU OHPRLQH$VLORQDXXIeVLqFOHODULWKPRPDFKLHRX©ௗFRPEDWGHVQRPEUHVௗª VHYRXODLWXQMHXSRXUGpODVVHUOHVVDYDQWVHWSHUPHWWUHDX[SOXVMHXQHVG¶DS- SUHQGUHDJUpDEOHPHQWO¶DULWKPpWLTXHGH%RqFH3OXVTX¶XQFRPEDWGHQRPEUHV

(6)

Introduction

LOV¶DJLVVDLWG¶XQFRPEDWGHUDSSRUWVGHQRPEUHVIDLVDQWDOOXVLRQjO¶LQWHUSUp- WDWLRQPXVLFDOHGHO¶DULWKPpWLTXH/HFKDSLWUHSUpVHQWHG¶DERUGTXHOTXHVXQV GHVQRPEUHX[WH[WHVTXLRQWÀHXULHQWUHHWVXUFHMHXWUqVFRP- SOH[HSXLVHQGRQQHOHVUqJOHVHWHQpYRTXHXQHYHUVLRQVLPSOL¿pHDGDSWpHj XQHXWLOLVDWLRQHQFROOqJH'DQVOHVHSWLqPHFKDSLWUHeYHO\QH%DUELQDERUGH HQVXLWHOD©ௗ*pRPpWULHFRPELQDWRLUHHWDOJRULWKPHVGHVFDUUpVPDJLTXHVௗª/H SUREOqPHIDVFLQDQWGHVFDUUpVPDJLTXHVFRQVLVWHjUHPSOLUOHVFDVHVG¶XQFDUUp GHnOLJQHVHWGHnFRORQQHVjO¶DLGHGHVQRPEUHVGHjnGHVRUWHTXHOD VRPPHGHVQRPEUHVVRLWODPrPHVXUFKDTXHOLJQHFKDTXHFRORQQHHWFKDTXH GLDJRQDOH/HFKDSLWUHDQDO\VHOHVWUDYDX[GH%HUQDUG)UHQLFOHGH%HVV\TXL SXEOLH HQ XQH WDEOH GHV FDUUpV PDJLTXHV GH F{Wp SXLV FHX[ GH 0LFKHO )URORY HW eGRXDUG /XFDV TXL j OD ¿Q GXXIXeVLqFOH UHSUHQQHQW OH GpQRPEUHPHQWGH)UHQLFOHSRXUHQJDJHUGHVLQYHVWLJDWLRQVFRPSOpPHQWDLUHV GH QDWXUH JpRPpWULTXH HW FRPELQDWRLUH ,O VH WHUPLQH SDU XQH UpÀH[LRQ VXU OHVH[SORLWDWLRQVSRVVLEOHVGHVFDUUpVPDJLTXHVGDQVO¶HQVHLJQHPHQWSRXUTXH FHX[FLWRXWHQUHVWDQWGLYHUWLVVDQWVVRLHQWSRUWHXUVGHFRQQDLVVDQFHVDXWKHQ- WLTXHPHQWPDWKpPDWLTXHV(Q¿Q/LVD5RXJHWHWWUDLWHG¶XQHDXWUHFDWpJRULH GH MHX[ GDQV OH KXLWLqPH FKDSLWUH ©ௗ/HV MHX[ FRPELQDWRLUHV RX FRPPHQW WLVVHU XQ OLHQ HQWUH PDWKpPDWLTXHV DOJRULWKPLTXH HW SURJUDPPDWLRQௗª &HV MHX[TXLVHFDUDFWpULVHQWSDUXQHDOWHUQDQFHGHFRXSVHQWUHGHX[MRXHXUVXQH LQIRUPDWLRQFRPSOqWHHWO¶DEVHQFHGHKDVDUGVRQWGpWHUPLQpVLOHVWSRVVLEOH HQWKpRULHGHGpQRPEUHUWRXWHVOHVSRVLWLRQVSRXYDQWVHSUpVHQWHUGXUDQWXQH SDUWLHHWGHFDUDFWpULVHUFHOOHVTXLVRQWJDJQDQWHVSRXUO¶XQGHVMRXHXUV(Q LOOXVWUDQWVHVSURSRVSDUO¶pWXGHGpWDLOOpHGXMHXGH1LPHWGXMHXGH.D\OHV O¶DXWHXUHPRQWUHTXHFHVMHX[SUHQQHQWOHXUVVRXUFHVGDQVGHVRXYUDJHVGH UpFUpDWLRQV PDWKpPDWLTXHV GqV OH GpEXW GXXVIIeVLqFOH TX¶LOV VRQW j O¶RUL- JLQHGHGpYHORSSHPHQWVPDWKpPDWLTXHVHWDOJRULWKPLTXHVWRXWjIDLWDFWXHOV HWTX¶LOHVWSHUWLQHQWGHOHVXWLOLVHUHQFODVVHSRXUDERUGHUFHUWDLQHVQRWLRQVGX SURJUDPPHGHPDWKpPDWLTXHVGXF\FOH

6L OHV KXLW SUHPLHUV FKDSLWUHV VXJJqUHQW WRXV GHV SLVWHV FRQFUqWHV SRXU H[SORLWHU GDQV O¶HQVHLJQHPHQW OHV UpFUpDWLRQV TX¶LOV pWXGLHQW OHV GHX[ GHU- QLHUVYRQWSOXVORLQGDQVFHWWHGLUHFWLRQGDQVODPHVXUHRLOVVRQWFRQVWUXLWV VXUO¶DQDO\VHGLGDFWLTXHG¶H[SpULPHQWDWLRQVUpDOLVpHVHQFODVVH$ODLQ%HUQDUG HW(PPDQXHOOH5RFKHUpFULYHQWjTXDWUHPDLQVXQQHXYLqPHFKDSLWUHLQWLWXOp

©ௗ(QWUHKLVWRLUHHWPDWKpPDWLTXHVYDULDWLRQVSpGDJRJLTXHVDXWRXUGHVSUR- EOqPHVG¶$OFXLQௗª/HVXSSRUWKLVWRULTXHHVWLFLXQUHFXHLOG¶XQHFLQTXDQWDLQH GHSUREOqPHVTXLRQWFLUFXOpHQWUHOHIXe et le XIIeVLqFOH%LHQTX¶LOVRLWDWWULEXp j$OFXLQG¶<RUNXQ$QJODLVGHIDPLOOHQREOHD\DQWMRXpXQU{OHLPSRUWDQWj ODFRXUGH&KDUOHPDJQHFHUHFXHLOHVWXQHQVHPEOHGLVSDUDWHGRQWRQQHSHXW

(7)

pWDEOLUO¶RULJLQHDYHFSUpFLVLRQ/HVDXWHXUVSURSRVHQWXQHWUDGXFWLRQHWXQ EUHIFRPPHQWDLUHGHTXHOTXHVXQVGHVSUREOqPHVSXLVIRQWOHFRPSWHUHQGX GpWDLOOpG¶XQDWHOLHULQWHUGLVFLSOLQDLUHPDWKpPDWLTXHVKLVWRLUHTXLV¶HVWGpURXOp HQFODVVHGHVHFRQGH3DUJURXSHVGHTXDWUHRXFLQTOHVpOqYHVRQWHXjUpDOLVHU XQJUDQGSDQQHDXFDOOLJUDSKLpHWHQOXPLQpGDQVOHVW\OHPpGLpYDOFRPSRU- WDQWXQHSDUWLHKLVWRULTXHHWODVROXWLRQGHFHUWDLQVGHVSUREOqPHVG¶$OFXLQ (Q¿QGDQVOHGL[LqPHHWGHUQLHUFKDSLWUH0DUF0R\RQDGRSWHXQHGpPDUFKH VLPLODLUHHQSXLVDQWjGHV¿QVGLGDFWLTXHVGHV©ௗ5pFUpDWLRQVPDWKpPDWLTXHVHW DOJRULWKPLTXHGDQVOHLiber abaciGH)LERQDFFLXIIIeVLqFOHௗª/HSULQFLSDO SUREOqPHUHWHQXHVWFHOXLGXYHUJHUSRXUVRUWLUG¶XQYHUJHULOIDXWGRQQHU VXFFHVVLYHPHQWjVHSWJDUGLHQVODPRLWLpGHVHVIUXLWVSOXVXQHWLOQHUHVWH TX¶XQIUXLWௗFRPELHQHQDYDLWRQDXGpSDUWௗ"&HSUREOqPHWUqVULFKHTXLSHXW VHUpVRXGUHSDUIDXVVHSRVLWLRQSDUO¶DOJqEUHRXHQDSSOLTXDQWjO¶HQYHUVO¶DO- JRULWKPHGHO¶pQRQFpDIDLWO¶REMHWGHGHX[VpDQFHVG¶XQHKHXUHHQFODVVHGH WURLVLqPH /¶DXWHXU GpFULW DYHF SUpFLVLRQ OHV FDUDFWpULVWLTXHV GH FHWWH H[Sp- ULHQFHSpGDJRJLTXHOHVGLIIpUHQWHVSURFpGXUHVVXLYLHVSDUOHVpOqYHVSXLVOD JpQpUDOLVDWLRQTXLHQDpWpWLUpHHWTXLDGpERXFKpVXUO¶pFULWXUHG¶XQDOJRULWKPH GHUpVROXWLRQDYHFOHORJLFLHO6FUDWFK/¶REMHFWLIDI¿FKpHVWTXHFKDFXQHRX FKDFXQSXLVVHV¶HPSDUHUG¶DXWUHVSUREOqPHVGXLiber abaciSRXUFRQVWUXLUHj VRQWRXUVHVSURSUHVVpDQFHVG¶HQVHLJQHPHQW

/H SUpVHQW RXYUDJH HVW LVVX GHV WUDYDX[ GX eFROORTXH LQWHU,5(0 G¶pSLVWpPRORJLH HW G¶KLVWRLUH GHV PDWKpPDWLTXHV TXL V¶HVW WHQX j O¶8QLYHU- VLWp*UHQREOH$OSHVGXHUDXMXLQ6HVDXWHXUVDSSDUWLHQQHQWjGHV ,5(0,QVWLWXWVGHUHFKHUFKHVXUO¶HQVHLJQHPHQWGHVPDWKpPDWLTXHVHWRQW OHVRXFLG¶LQWURGXLUHXQHSHUVSHFWLYHKLVWRULTXHGDQVO¶HQVHLJQHPHQWjWRXV OHV QLYHDX[ /HXU DPELWLRQ Q¶HVW SDV G¶HQVHLJQHU O¶KLVWRLUH GHV PDWKpPD- WLTXHVHQWDQWTXHWHOOHPDLVGHV¶LQVSLUHUGHFHWWHKLVWRLUHSRXUFRQFHYRLUHW H[SpULPHQWHUHQFODVVHGHVVLWXDWLRQVULFKHVGHVHQVVXVFHSWLEOHVGHIDYRULVHU OHVDSSUHQWLVVDJHVPDWKpPDWLTXHVGHVpOqYHV/HWKqPHGHVPDWKpPDWLTXHV UpFUpDWLYHVVHPEODQWSDUWLFXOLqUHPHQWSHUWLQHQWjFHWWH¿QQRXVHVSpURQVTXH OHVGL[FKDSLWUHVGHFHOLYUHGRQWQRXVDYRQVWHQWpGHPHWWUHHQpYLGHQFHOD YDULpWpHWODULFKHVVHIRXUQLURQWDX[HQVHLJQDQWVHWDX[IRUPDWHXUVGHVUHV- VRXUFHVpSLVWpPRORJLTXHVKLVWRULTXHVHWSpGDJRJLTXHVXWLOHVSRXUUHQRXYHOHU OHXUSUDWLTXH«WRXWHQV¶DPXVDQW

Références

Documents relatifs

In addition to assessing the glacier limit uncertainty when derived from surface features observed on datasets acquired through remote sensing measurements, we consider the

Récréations mathématiques et algorithmique dans le Liber abaci. de Fibonacci ( XIII e siècle)

Notamment, l'expression du retard thermique, utilisé en milieu sédimentaire pour prédire le transport de chaleur à partir de tests de traçage de soluté, n'a jamais été

La postérité ne retiendra que le terme axiome, qui d’ailleurs ne traduit plus la conception d’Aristote : de nos jours, une science formelle comme les mathématiques pose, pour

aux négociations internes (dans et entre les classes) et externes (entre les professeurs et la société) du système éducatif à propos des curriculums d'enseignement. Problèmes

...A Choquet integral is a nondecreasing Lov´ asz extension (vanishing at 0).... Generalization: Quasi-Lov´

We describe the function class axiomatized by these properties and we show that, up to certain regularity conditions (based on those we usually add to the Cauchy functional equation

Depuis le 18 octobre 2019, les États-Unis ont mis en place une taxe de 25 % sur une partie des exportations de vin tranquille en bouteille de moins de 2 litres dont le