APEIROSTIKOS LOGISMOS I

APEIROSTIKOS LOGISMOS I

Tet�rth, 30 SeptembrÐou 2015

PerÐgramma

1 PlhroforÐec gia to m�jhma - katanom foitht¸n

2 Hlektronik t�xh

3 Perigraf tou Maj matoc

PragmatikoÐ arijmoÐ

AkoloujÐec pragmatik¸n arijm¸n

Sunart seic

'Oria kai Sunèqeia Sun�rthshc

Par�gwgoc sun�rthshc

4 BibliografÐa


Kwdikìc Maj matoc: 101

'Etoc didaskalÐac: 2015-2016, Qeimerinì Ex�mhno

Hmèrec: Deut. - Tet. - Par., 'Wra: 13:00-15:00

Did�skontec

Tm ma 1o (AM pou l gei se 1,2,3) Amf 24,

Zaqari�dhc Jeodìsioc, GrafeÐo: 217, thl. 210-7276380,

http://noether.math.uoa.gr/Academia/didaktiki/tzaharia_gr

Tm ma 2o (AM pou l gei se 4,5,6,7) Amf 22,

Euaggel�tou-D�lla Le¸nh

GrafeÐo: 207, thl. 210-7276375,

http://noether.math.uoa.gr/Academia/analysis/ldalla_gr

Tm ma 3o (AM pou l gei se 8,9,0) Amf 23,

PapatriantafÔllou MarÐa

GrafeÐo: 203, thl. 210-7276349,

http://noether.math.uoa.gr/Academia/analysis/papatriantafylloy-maria

http://noether.math.uoa.gr/Academia/didaktiki/tzaharia_gr

http://noether.math.uoa.gr/Academia/analysis/ldalla_gr

http://noether.math.uoa.gr/Academia/analysis/papatriantafylloy-maria


IstoselÐda tou Maj matoc

http://eclass.uoa.gr/courses/MATH130/

http://eclass.uoa.gr/courses/MATH130/


Perigraf tou Maj matoc

1 PragmatikoÐ arijmoÐ

2 AkoloujÐec pragmatik¸n arijm¸n

3 Sunart seic

4 'Orio kai Sunèqeia Sunart sewn

5 Par�gwgoc


1. PragmatikoÐ arijmoÐ:

Axiwmatik jemelÐwsh twn Pragmatik¸n arijm¸n

FusikoÐ, Akèraioi kai RhtoÐ arijmoÐ

AxÐwma plhrìthtac

'Uparxh tetragwnik c rÐzac

'Arrhtoi arijmoÐ

Akèraio mèroc

Puknìthta twn rht¸n kai twn arr twn stouc pragmatikoÔc

arijmoÔc

Klassikèc anisìthtec


RhtoÐ arijmoÐ

H diag¸nia mèjodoc tou Cantor kai h { 1- 1 } antistoiqÐa me

touc fusikoÔc arijmoÔc


'Arrhtoi arijmoÐ

To sok tou Pujagìra - UpoteÐnousa orjog¸niwn trig¸nwn


'Arrhtoi arijmoÐ

Qrus tom : φ = 1+√5

2 = 1.6180339887...

Gewmetrik� prokÔptei mèsw thc analogÐac: a+ba = a

b ≡ φ

Algebrik� prokÔptei wc jetik rÐza thc exÐswshc:

x2−x−1 = 0

Ti ekfr�zei pragmatik� autìc o arijmìc?


'Arrhtoi arijmoÐ

Qrus tom : φ = 1+√5

2 = 1.6180339887...

Gewmetrik� prokÔptei mèsw thc analogÐac: a+ba = a

b ≡ φ

Algebrik� prokÔptei wc jetik rÐza thc exÐswshc:

x2−x−1 = 0

Ti ekfr�zei pragmatik� autìc o arijmìc?


'Arrhtoi arijmoÐ

ArmonÐa sthn arqitektonik tou Parjen¸na, 438 p.Q.


'Arrhtoi arijmoÐ

ArmonÐa sto èrgo tou Da Vinci - Vitruvian Man, 1487 m.Q.


'Arrhtoi arijmoÐ

UperbatikoÐ arijmoÐ: Rht prosèggish tou π me 999 dekadik�

yhfÐa


2. AkoloujÐec pragmatik¸n arijm¸n:

SugklÐnousec akoloujÐec

Monìtonec akoloujÐec

Kibwtismìc diasthm�twn

Anadromikèc akoloujÐec


AkoloujÐec

Ti eÐnai mia akoloujÐa pragmatik¸n arijm¸n?

EÐnai mia apeikìnish apì to sÔnolo N sto R.Mèsw tÔpou:

1 αn = n√n, limn→∞

n√n = 1

2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...

Mèsw anadromik c sqèshc:

1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .

2 AkoloujÐa Fibonacci: αn = αn−1+αn−2, α0 = 0, α1 = 1,limn→∞ αn =+∞, limn→∞

αn+1αn

= φ .

AkoloujÐa twn pr¸twn arijm¸n:

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...

Up�rqei sun�rthsh anadromikìc tÔpoc pou na dÐnei thn

parap�nw akoloujÐa?


AkoloujÐec


EÐnai mia apeikìnish apì to sÔnolo N sto R.

Mèsw tÔpou:


n√n = 1

2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...


1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .


αn+1αn

= φ .


2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...




AkoloujÐec




n√n = 1

2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...


1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .


αn+1αn

= φ .


2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...




AkoloujÐec




n√n = 1

2 αn = (1+1/n)n, limn→∞(1+1/n)n = e = 2,71828...


1 αn+1 =√1+αn, n ≥ 1, α1 = 1, limn→∞ αn = φ .


αn+1αn

= φ .


2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53, ...




AkoloujÐec

Fragmènh mh sugklÐnousa akoloujÐa pragmatik¸n arijm¸n

�peirwn ìrwn


3. Sunart seic:

BasikoÐ orismoÐ

Algebrikèc sunart seic

Trigwnometrikèc sunart seic

Ekjetik sun�rthsh


Pragmatikèc sunart seic

Gr�fhma poluwnumik c sun�rthshc 4ou bajmoÔ



Gr�fhma trigwnometrik c sun�rthshc hmitìnou

θ

sin θ

0

1

−1

2π



Gr�fhma antÐstrofhc trigwnometrik c sun�rthshc hmitìnou

θ

arcsinθ

0

1

π/ 2



Gr�fhma trigwnometrik c sun�rthshc efaptomènhc

θ

tan θ

0

π

2π



Gr�fhma antÐstrofhc trigwnometrik c sun�rthshc efaptomènhc

θ

arctanθ

0

−π/ 2

π/ 2



Gr�fhma ekjetik c sun�rthshc gia α > 1

x

y

0

y = ax

1



Gr�fhma logarijmik c sun�rthshc gia α > 1

x = logax

1 x

y

0

y



H sun�rthsh f (x) =

{x , gia x ∈Q

−x , gia x /∈Q.


4. 'Oria kai Sunèqeia Sun�rthshc:

H ènnoia tou orÐou sun�rthshc - Sunèqeia

Arq thc metafor�c

Sunèqeia gnwst¸n sunart sewn

Sunèqeia kai topik sumperifor�

Je¸rhma endi�meshc tim c

'Uparxh megÐsthc kai elaqÐsthc tim c gia suneqeÐc

sunart seic orismènec se kleist� diast mata - Monìtonec

sunart seic

SuneqeÐc kai {1-1} sunart seic

AntÐstrofec trigwnometrikèc sunart seic

Logarijmik sun�rthsh


5. Par�gwgoc:

Eisagwg : ParadeÐgmata apì th GewmetrÐa kai th Fusik

H ènnoia thc parag¸gou

Kanìnec parag¸gishc

Par�gwgoi basik¸n sunart sewn

Je¸rhma mèshc tim c

Je¸rhma Darboux

Krit ria monotonÐac sun�rthshc

Krit ria topik¸n akrot�twn

Genikeumèno je¸rhma mèshc tim c

Kanìnec De L’Hospital

Kurtèc kai koÐlec sunart seic - ShmeÐa kamp c

Melèth sunart sewn




{sin(1/x), x 6= 0

0 , x = 0



H sun�rthsh g(x) =

{xsin(1/x), x 6= 0

0 , x = 0



H sun�rthsh h(x) =

{x2sin(1/x), x 6= 0

0 , x = 0




{x2, gia x ∈Q

−x2, gia x /∈Q.


BibliografÐa

S. Negrepìnthc, S. Giwtìpouloc, E. GiannakoÔliac:{ Apeirostikìc Logismìc I }, Ekdìseic SummetrÐa.

L. TsÐtsac: { Efarmosmènoc Apeirostikìc Logismìc }, EkdìseicSummetrÐa.

M. Spivak: “ Calculus ”, Benjamin (kukloforeÐ se Ellhnik met�frash me tÐtlo: { Diaforikìc kai Oloklhrwtikìc Logismìc}, Panepisthmiakèc Ekdìseic Kr thc.)

R. Courant and F. John: “ Introduction to Calculus and Analysis ”,Vol. I, Interscience.

G. H. Hardy: “ A Course in Pure Mathematics”, CambridgeUniversity Press.

S. Salas and E. Hille: “ Calculus ”, John Wiley.

R. Bartle and D. Sherbert: “ Introduction to Real Analysis ”, JohnWiley.


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APEIROSTIKOS LOGISMOS I