Ti e'inai sun'arthsh? - Mia pros'eggish sto Gumn'asio kai...

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn.Ti eÐnai sun�rthsh?Mia prosèggish sto Gumn�sio kai to LÔkeioEir nh Perusin�khEllhnikh Majhmatikh EtaireiaPararthma Hrakleiou16 FebrouarÐou 2008Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn.Perieqìmena1 Ti eÐnai sun�rthsh?Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.2 Efarmogèc twn Sunart sewn.BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.H mhqan thc sun�rthshc

Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.Perieqìmena1 Ti eÐnai sun�rthsh?Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.2 Efarmogèc twn Sunart sewn.BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.Pr¸tec prosp�jeiec orismoÔ thc ènnoiacTon ìro {sun�rthsh} eis gage o Leibnitz ìpwc kai tasÔmbola dy/dx kai ∫

dx .Sun�rthsh apoteleÐ mia èkfrash pou kataskeu�zetai meorismèno trìpo apì èna metablhtì mègejoc kai apì stajerèc.BernoulliSun�rthsh apoteleÐ mia analutik èkfrash kataskeuasmènhapì èna metablhtì mègejoc kai apì stajerèc . . .eÐnai dunatìn na onomasteÐ sun�rthsh {k�je eleÔjerasqediasmènh kampÔlh}. . .

EulerO Euler eis gage kai to sumbolismì f (x).Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.O sunolojewrhtikìc orismìcAut h genik ènnoia apaiteÐ me ton ìro sun�rthsh tou x naennooÔme ton arijmì pou dÐnetai gia k�je x kai o opoÐocmetab�lletai stadiak� mazÐ me to x . H tim miac sun�rthshcmporeÐ na orÐzetai apì k�poia analutik èkfrash apì miasunj kh pou parèqei th dunatìthta na dokim�soume ìlouc toucarijmoÔc kai na epilèxoume ènan apì autoÔc , tèloc, hex�rthsh endèqetai na up�rqei kai na paramènei �gnwsth.H genik morf thc jewrÐac dèqetai thn Ôparxh miac ex�rthshcmìnon upì thn ènnoia ìti ofeÐloume na nooÔme toucsqetizìmenouc arijmoÔc wc dedomènouc omoÔ.N.I. LobacevskiıEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.O sunolojewrhtikìc orismìcSun�rthsh me pedÐo orimoÔ to sÔnolo A kai pedÐo tim¸n tosÔnolo B eÐnai mia sqèsh kat� thn opoÐa se k�je stoiqeÐo xtou sunìlou A antistoiqeÐ èna monadikì stoiqeÐo y tou sunìlouB pou sumbolÐzoume me f (x).

f : A → B

x → f (x)

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.Metabolèc: Puretikì di�grammaProb�llontac ta pleonekt mata tou graf matoc.

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: Katano¸ntac th stajer sun�rthsh. . . PaÐzontac . . .ManteÔontac ènan arijmìB�le ènan arijmì sto noÔ sou.B�le kai gia ton fÐlo sou �lla tìsa.B�le kai gia mèna 10.RÐxe ta mis� sth j�lassa.D¸se pÐsw sto fÐlo sou ta dik� tou.5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: Katano¸ntac th stajer sun�rthsh. . . PaÐzontac . . .H majhmatik ermhneÐaB�le ènan arijmì sto noÔ sou.B�le kai gia ton fÐlo sou �lla tìsa.B�le kai gia mèna 10.RÐxe ta mis� sth j�lassa.D¸se pÐsw sto fÐlo sou ta dik� tou.5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: Katano¸ntac th stajer sun�rthsh. . . PaÐzontac . . .H majhmatik ermhneÐaxB�le kai gia ton fÐlo sou �lla tìsa.B�le kai gia mèna 10.RÐxe ta mis� sth j�lassa.D¸se pÐsw sto fÐlo sou ta dik� tou.5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: Katano¸ntac th stajer sun�rthsh. . . PaÐzontac . . .H majhmatik ermhneÐax

2xB�le kai gia mèna 10.RÐxe ta mis� sth j�lassa.D¸se pÐsw sto fÐlo sou ta dik� tou.5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

2x + 10RÐxe ta mis� sth j�lassa.D¸se pÐsw sto fÐlo sou ta dik� tou.5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

2x + 1012(2x + 10)D¸se pÐsw sto fÐlo sou ta dik� tou.

5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

2x + 1012(2x + 10)

12(2x + 10) − x

5 de sou èmeinan?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

2x + 1012(2x + 10)

12(2x + 10) − x

= 5 Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: SÔnjetec sunart seic.f (x) = hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

x → x2 → x2 + 1 → ex2+1 → hm(

ex2+1)

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.DiadikasÐec: Sumpl rwsh tetrag¸nou sto tri¸numo.f (x) = x2 + 2x + 3

x → x2 → x2 + 2x → x2 + 2x + 3. . .Enallaktik� . . .f (x) = (x + 1)2 + 2

x → x +1 → (x +1)2 → (x +1)2 +2

PSfrag replacementc0

−1 x

y = x2

y = (x + 1)2

y = (x + 1)2 + 2

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.'Apeirec diadikasÐec: H ekjetik . . . diaforetik�.ex = 1 +

4!+ · · ·. . . Epomènwc . . .

(ex )′ = exkaie0 = 1Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.'Apeirec diadikasÐec: H ekjetik . . . diaforetik�.ex = 1 +

4!+ · · ·. . . Epomènwc . . .

(ex )′ = exkaie0 = 1Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.O gnwstìc tÔpoc sun�rthshc.

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.Par�deigma: {eikonikì} estiatìrioO idiokt thc enìc estiatorÐou èkane touc akìloujouclogariasmoÔc pou deÐqnoun th sqèsh metaxÔ tou kajaroÔkèrdouc kai tou arijmoÔ twn pelat¸n.Arijmìc pelat¸n 30 40 50 60Kajarì kèrdoc se � 100 200 300 400 . . . . . . . . .Poioc eÐnai o arijmìc twn pelat¸n ìtan den up�rqeikèrdoc?Ti sumbaÐnei ìtan to pl joc twn pelat¸n eÐnai mikrìteroapì 20?Pìte arqÐzei h zhmi�?Up�rqei k�poia mèjodoc me thn opoÐa apant�me se ìlecautèc tic erwt seic?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.Isìthta sunart sewn: Istorikì sqìlio.H ζ sun�rthsh tou Euler orÐzetai gia k�je pragmatikì x wcex c:ζ(x) =

4x+ · · ·O Euler apèdeixe ìti h Ðdia sun�rthsh perigr�fetai kai apìènan deÔtero tÔpo:

ζ(x) =1

1 − (1/2)x× 1

1 − (1/3)x× 1

1 − (1/5)x× 1

1 − (1/7)x× · · ·O Riemann epèkteine ton orismì thc ζ sun�rthshc kai giamigadik� z . M�lista diatÔpwse thn eikasÐa ìti oi rÐzec thcbrÐskontai sth migadik eujeÐa Re(z) = 1/2 (upìjesh Riemann),er¸thma pou paramènei akìma kai s mera anoiqtì.Eir nh Perusin�kh Ti eÐnai sun�rthsh?

ζ(x) =1

1 − (1/2)x× 1

1 − (1/3)x× 1

1 − (1/5)x× 1

ζ(x) =1

1 − (1/2)x× 1

1 − (1/3)x× 1

1 − (1/5)x× 1

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.To sÔnolo tim¸n kai h 1�1 idiìthta.H arq tou Perister¸naPrìtashSe k�je kurtì polÔedro up�rqoun toul�qiston dÔo èdrec me tonÐdio arijmì koruf¸n.Upìdeixh: Jewr ste th sun�rthshèdra tou polÔedrou → arijmìc koruf¸n thc èdracEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.To sÔnolo tim¸n kai h 1�1 idiìthta.To par�doxo tou �peirou xenodoqeÐou.Se k�poio xenodoqeÐo up�rqoun �peira dwm�tia arijmhmèna 1,2, 3, 4, ... Sthn pìlh diex�getai èna festib�l kai ìla tadwm�tia eÐnai kateilhmmèna. Lìgw ìmwc twn meg�lwnapait sewn, o idiokt thc anagk�zetai na deqteÐ {�llouctìsouc} pel�tec. Poia ìmwc dwm�tia ja diajèsei stouckainoÔriouc pel�tec?Gr gora brÐskei thn ex c lÔsh: K�je {paliìc} pel�thcmetakineÐtai sto dwm�tio me noÔmero to dipl�sio tou arqikoÔtou dwmatÐou. 'Etsi, oi {palioÐ} pel�tec katalamb�noundwm�tia me zugoÔc arijmoÔc, en¸ ta dwm�tia me touc monoÔcarijmoÔc eÐnai sth di�jesh twn {nèwn} pelat¸n.f : N → 2N f (n) = 2nEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.SÔnjetoi kanìnec antistoÐqhshc.. . .Antimètwpoi me tic kladwtèc sunart seic . . .

f (x) =

x2 an x ≥ 0−x an x < 0

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.SÔnjetoi kanìnec antistoÐqhshc.. . .Antimètwpoi me tic kladwtèc sunart seic . . .

f (x) =

x2 an x ≥ 0−x an x < 0

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.SÔnjetoi kanìnec antistoÐqhshc.DrasthriìthtaPSfrag replacementcx

y = −x

y = x2y = x2

y = −x2y = −x2

1© 2© 3© 4©

5© 6© 7© 8©

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.SÔnjetoi kanìnec antistoÐqhshc.DrasthriìthtaPSfrag replacementcx

y = −x

y = x2y = x2

y = −x2y = −x2

1© 2© 3© 4©

5© 6© 7© 8©

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.SÔnjetoi kanìnec antistoÐqhshc.DrasthriìthtaPSfrag replacementc

y = −x

y = −xy = x2

y = x2

y = −x2 y = −x2

1© 2©3©

4©5©

7©8©

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.'Otan o kanìnac antistoÐqhshc de mporeÐ na dojeÐ mek�poio {tÔpo}.f (x) =

du. . .Prosèggish . . .Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.'Otan o kanìnac antistoÐqhshc de mporeÐ na dojeÐ mek�poio {tÔpo}.f (x) =

du. . .Prosèggish . . .Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.'Otan o kanìnac antistoÐqhshc de mporeÐ na dojeÐ mek�poio {tÔpo}.Pìsoi pr¸toi arijmoÐ up�rqoun an�mesa stouc N arqikoÔcfusikoÔc {1, 2, 3, . . . ,N}?DN =

NEikasÐa Gauss (1791)

DN ≈ 1

lnNJe¸rhma twn Pr¸twn Arijm¸nHadamard—de la Valee Poussin (1896)

limN→+∞

lnN · P(N)

N= 1Eir nh Perusin�kh Ti eÐnai sun�rthsh?

DN ≈ 1

limN→+∞

lnN · P(N)

DN ≈ 1

limN→+∞

lnN · P(N)

DN ≈ 1

limN→+∞

lnN · P(N)

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacPerieqìmena1 Ti eÐnai sun�rthsh?Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.2 Efarmogèc twn Sunart sewn.BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacAgor� kinhtoÔ.To prìblhmaAc upojèsoume ìti jèlete na agor�sete kinhtì thlèfwno kaimet� apì èreuna sthn agor� katal xate metaxÔ tri¸naxiìpistwn etairi¸n...A-B-G.H A qre¸nei to mhniaÐo p�gio 15 � kai 0, 30 � to k�jeleptì thlefwnik c kl shc.H B qre¸nei to mhniaÐo p�gio 12 � kai 0, 40 � to k�jeleptì thlefwnik c kl shc.H G qre¸nei to mhniaÐo p�gio 9 � kai 0, 50 � to k�je leptìthlefwnik c kl shc.Poia etaireÐa sac sumfèrei na epilèxete telik�?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacAgor� kinhtoÔ.H grafik anapar�stashPSfrag replacementc

0 10 20 30 40 50

y = 9 + 0, 5xy = 12 + 0, 4xy = 15 + 0, 3xEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacπ

e eπ?

Poio eÐnai megalÔtero? To πe to eπ?Poioc apì touc arijmoÔc π1/π kai e1/e eÐnai o megalÔteroc?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacπ

e eπ?

Poio eÐnai megalÔtero? To πe to eπ?Poioc apì touc arijmoÔc π1/π kai e1/e eÐnai o megalÔteroc?Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacPerieqìmena1 Ti eÐnai sun�rthsh?Istorik anadrom .H sun�rthsh wc metabol .H sun�rthsh wc diadikasÐa.Kanìnec antistoÐqhshc.2 Efarmogèc twn Sunart sewn.BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacH gewmetrik èkfrash thc epimeristik c idiìthtac.GiatÐ α(−β) = −αβ ?Algebrikì epiqeÐrhmaα · 0 = 0

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

Gewmetrikì epiqeÐrhma.PSfrag replacementc0 x

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

⇔ α [β + (−β)] = 0

⇔ αβ + α(−β) = 0

⇔ α(−β) = −αβ

α(−β)

(β, αβ)

(−β, α(−β))

y = αx

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacH gewmetrik èkfrash thc epimeristik c idiìthtac.PrìtashOi epìmenec duo prot�seic eÐnai isodÔnamec.1 Gia opoiousd pote pragmatikoÔc arijmoÔc α, β kai γisqÔeiα(β + γ) = αβ + αγ2 Gia opoiod pote pragmatikì arijmì α, h kampÔlh y = αxeÐnai eujeÐa gramm .Apìdeixh (2. ⇒ 1.):'Estw α, β kai γ tuqaÐoi pragmatikoÐ arijmoÐ. Ja apodeÐxoumeìti an h f (x) = αx eÐnai grammik , tìte isqÔei

α(β + γ) = αβ + αγ.Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacH gewmetrik èkfrash thc epimeristik c idiìthtac.PrìtashOi epìmenec duo prot�seic eÐnai isodÔnamec.1 Gia opoiousd pote pragmatikoÔc arijmoÔc α, β kai γisqÔeiα(β + γ) = αβ + αγ2 Gia opoiod pote pragmatikì arijmì α, h kampÔlh y = αxeÐnai eujeÐa gramm .Apìdeixh (2. ⇒ 1.):'Estw α, β kai γ tuqaÐoi pragmatikoÐ arijmoÐ. Ja apodeÐxoumeìti an h f (x) = αx eÐnai grammik , tìte isqÔei

α(β + γ) = αβ + αγ.Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacH gewmetrik èkfrash thc epimeristik c idiìthtac.f (β) + f (γ) = αβ + αγ

2α · 2β +

2α · 2γ

2f (2β) +

2f (2γ) orismìc thc f

22β +

) grammikìthta thc f

= f (β + γ)DeÐxame loipìn ìtif (β) + f (γ) = f (β + γ)

⇔ αβ + αγ = α(β + γ)Eir nh Perusin�kh Ti eÐnai sun�rthsh?

2α · 2β +

2α · 2γ

2f (2β) +

22β +

2α · 2β +

2α · 2γ

2f (2β) +

22β +

2α · 2β +

2α · 2γ

2f (2β) +

22β +

2α · 2β +

2α · 2γ

2f (2β) +

22β +

2α · 2β +

2α · 2γ

2f (2β) +

22β +

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐac√

= α √α

2= |α|?

Anazht¸ntac thn ìmoia kampÔlh sthn y = |x |.PSfrag replacementc000 xxx

y = |x |y = xy = x2

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐac√

= α √α

2= |α|?

x2 = θ2 ⇔ |x | = θ, . . .PSfrag replacementc000 xxx

y = |x |y = x

y = x2

θθθ

−θ−θ

Tri¸numo Apìlutax2 = θ2 ⇔ x = θ x = −θ |x | = θ ⇔ x = θ x = −θ

x2 < θ2 ⇔ −θ < x < θ |x | < θ ⇔ −θ < x < θ

x2 > θ2 ⇔ x < −θ x > θ |x | > θ ⇔ x < −θ x > θEir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacKampÔlec pou xeqwrÐzoun.Logarijmik speÐra.PSfrag replacementc x

(x , y) =(

αθ sun θ, αθ hm θ)

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacKampÔlec pou xeqwrÐzoun.GewmetrikoÐ metasqhmatismoÐ.PSfrag replacementc

y = sun x

y = hm x

y = sun x

y = hm x

y = 2sun x

y = 2 hm x

y = sun x

y = hm x

y = sun x

y = hm x

y = sun 2x

y = hm 2x

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacKampÔlec pou xeqwrÐzoun.Ekjetik sun�rthsh

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

Logarijmik sun�rthshf (x) = logαx

f (c x) = logα(c x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

f (x) = αx

c f (x) = c αx

= αlogαc · αx

= αC · αx

= αC+x

= f (C + x)

= logαc + logαx

= C + logαx

= C + f (x)

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacKeith Devlin, The Millennium Problems - The seven greatest

unsolved mathmatical puzzles of our time, Granta Books,2005.

M. Gardner, About three types of spirals and how to

construct them, Mathematical Games, Scientific American,Apr. 1962, V 206, No 4.

Richard Mankiewicz, H istorÐa twn majhmatik¸n, ekdìseicAlex�ndreia, 2002.V.M. Tikhomirov, IstorÐec gia mègista kai el�qista,ekdìseic K�toptro, 1999.Carol Vorderman, AnakalÔptw ta Majhmatik�, ekdìseicEreunhtèc, 1998.Eir nh Perusin�kh Ti eÐnai sun�rthsh?

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacSwkr�thc Ntri�nkoc, Sunart seic,http://users.auth.gr/˜lemonidi/sde yliko/paradigmata.htmMp�mphc Toum�shc, P¸c na energopoi soume ta paidi�sto m�jhma twn Majhmatik¸n, ekdìseic Kwstìgiannoc,QalkÐda, 1999.

Ti eÐnai sun�rthsh?Efarmogèc twn Sunart sewn. BeltistopoÐhsh-Grammikìc Programmatismìc.O g�moc thc 'Algebrac kai thc GewmetrÐacsac euqarist¸!

http://users.ira.sch.gr/iriniper

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