Post on 12-Jul-2022
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
APEIROSTIKOS LOGISMOS I
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
APEIROSTIKOS LOGISMOS I
IstoselÐda tou Maj matoc
http://eclass.uoa.gr/courses/MATH130/
APEIROSTIKOS LOGISMOS I
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
APEIROSTIKOS LOGISMOS I
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
APEIROSTIKOS LOGISMOS I
RhtoÐ arijmoÐ
H diag¸nia mèjodoc tou Cantor kai h { 1- 1 } antistoiqÐa me
touc fusikoÔc arijmoÔc
APEIROSTIKOS LOGISMOS I
'Arrhtoi arijmoÐ
To sok tou Pujagìra - UpoteÐnousa orjog¸niwn trig¸nwn
APEIROSTIKOS LOGISMOS I
'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?
APEIROSTIKOS LOGISMOS I
'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?
APEIROSTIKOS LOGISMOS I
'Arrhtoi arijmoÐ
ArmonÐa sthn arqitektonik tou Parjen¸na, 438 p.Q.
APEIROSTIKOS LOGISMOS I
'Arrhtoi arijmoÐ
ArmonÐa sto èrgo tou Da Vinci - Vitruvian Man, 1487 m.Q.
APEIROSTIKOS LOGISMOS I
'Arrhtoi arijmoÐ
UperbatikoÐ arijmoÐ: Rht prosèggish tou π me 999 dekadik�
yhfÐa
APEIROSTIKOS LOGISMOS I
2. AkoloujÐec pragmatik¸n arijm¸n:
SugklÐnousec akoloujÐec
Monìtonec akoloujÐec
Kibwtismìc diasthm�twn
Anadromikèc akoloujÐec
APEIROSTIKOS LOGISMOS I
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?
APEIROSTIKOS LOGISMOS I
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?
APEIROSTIKOS LOGISMOS I
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?
APEIROSTIKOS LOGISMOS I
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?
APEIROSTIKOS LOGISMOS I
AkoloujÐec
Fragmènh mh sugklÐnousa akoloujÐa pragmatik¸n arijm¸n
�peirwn ìrwn
APEIROSTIKOS LOGISMOS I
3. Sunart seic:
BasikoÐ orismoÐ
Algebrikèc sunart seic
Trigwnometrikèc sunart seic
Ekjetik sun�rthsh
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma poluwnumik c sun�rthshc 4ou bajmoÔ
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma trigwnometrik c sun�rthshc hmitìnou
θ
sin θ
0
1
−1
2π
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma antÐstrofhc trigwnometrik c sun�rthshc hmitìnou
θ
arcsinθ
0
1
π/ 2
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma trigwnometrik c sun�rthshc efaptomènhc
θ
tan θ
0
π
2π
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma antÐstrofhc trigwnometrik c sun�rthshc efaptomènhc
θ
arctanθ
0
−π/ 2
π/ 2
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma ekjetik c sun�rthshc gia α > 1
x
y
0
y = ax
1
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
Gr�fhma logarijmik c sun�rthshc gia α > 1
x = logax
1 x
y
0
y
APEIROSTIKOS LOGISMOS I
Pragmatikèc sunart seic
H sun�rthsh f (x) =
{x , gia x ∈Q
−x , gia x /∈Q.
APEIROSTIKOS LOGISMOS I
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
APEIROSTIKOS LOGISMOS I
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
APEIROSTIKOS LOGISMOS I
Par�gwgoc sun�rthshc
H sun�rthsh f (x) =
{sin(1/x), x 6= 0
0 , x = 0
APEIROSTIKOS LOGISMOS I
Par�gwgoc sun�rthshc
H sun�rthsh g(x) =
{xsin(1/x), x 6= 0
0 , x = 0
APEIROSTIKOS LOGISMOS I
Par�gwgoc sun�rthshc
H sun�rthsh h(x) =
{x2sin(1/x), x 6= 0
0 , x = 0
APEIROSTIKOS LOGISMOS I
Par�gwgoc sun�rthshc
H sun�rthsh f (x) =
{x2, gia x ∈Q
−x2, gia x /∈Q.
APEIROSTIKOS LOGISMOS I
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.
APEIROSTIKOS LOGISMOS I