M.Se
Matk REALANALYSIS
CH#
Wumbes Systems.
Real Anaysis:
Tebhanch f'malhemathes
shich dals wih sudy frpesta
nubers and the funechons fpm et ef
ralrumbers to set
Aea
Ten numbers.
Real Nuwnbers
sfem tnumbexs T from- to +iSCalled
fead number System denoted by R=]-a,qL
Infaet,e number Sustem is a linenumbers
-g
Flous Chont f Number Systems:
R-SetofealNumbers.
Q-Set raienal Numbers
-Set e Rationa Nanbers
Z Set Tateye Ys
W Set a whoe rumbes
N Set of Natwra Nawbera
Nambers wih maqinary
Set d all real numbers is a set Compex
Past Zeyos R -C-{a,b)} aeR ad bro
, Paopertes oReal Number ystem:
) Field Propexties:
w.rt, Addition .T.t. MuHipliealian
A: Closur Propeat M,: Closure Propesty
Y a,b e R Va,beR
a + b eR a.be R
A: Commutative fropert M:ommuthtive Popety
Y a,beR a,be
a b = bta a.b = b a
A Associahive Popes M:Amociathe Pmpen:
a,b,ceR
a+(btc) =(a+b)+tc ab-c)= ab).c
AIelentiExistance: My: Identi Enistance:
VaeR 1 oeR YaeR 1e R
such that
Sueh thot
a+oe o+a = Qa
a.L=1 a 2 a
"'o Called Additie iletity
i
Ts called
A: Inerse Existance: muthpicahie idenl.
M: Invere Eristance:
Y aeR -atR VaeR 3 LeR
Such hat Such That
a+-a) =-a +a=0
-a" is called Additie
a.
a
=1
inverse a" 6 Caled mulhiplèahive
inverse a"
Disthibutie Propest:
V a,b,cE R
a.Cbte) = a-bta.c
a+b)C = a.c+b.c Left Distaibutive La
A
Set that satisfres Riht Distibuhe Lud
all te
above 11
Field ad is
denoted as
propesties is clue
Set of rationa) (F+,
(F+,)-)
numbers
numbe rs, Set of rea) numbers avd Set
are 3
fields. ofamplex
Matk REALANALYSIS
CH#
Wumbes Systems.
Real Anaysis:
Tebhanch f'malhemathes
shich dals wih sudy frpesta
nubers and the funechons fpm et ef
ralrumbers to set
Aea
Ten numbers.
Real Nuwnbers
sfem tnumbexs T from- to +iSCalled
fead number System denoted by R=]-a,qL
Infaet,e number Sustem is a linenumbers
-g
Flous Chont f Number Systems:
R-SetofealNumbers.
Q-Set raienal Numbers
-Set e Rationa Nanbers
Z Set Tateye Ys
W Set a whoe rumbes
N Set of Natwra Nawbera
Nambers wih maqinary
Set d all real numbers is a set Compex
Past Zeyos R -C-{a,b)} aeR ad bro
, Paopertes oReal Number ystem:
) Field Propexties:
w.rt, Addition .T.t. MuHipliealian
A: Closur Propeat M,: Closure Propesty
Y a,b e R Va,beR
a + b eR a.be R
A: Commutative fropert M:ommuthtive Popety
Y a,beR a,be
a b = bta a.b = b a
A Associahive Popes M:Amociathe Pmpen:
a,b,ceR
a+(btc) =(a+b)+tc ab-c)= ab).c
AIelentiExistance: My: Identi Enistance:
VaeR 1 oeR YaeR 1e R
such that
Sueh thot
a+oe o+a = Qa
a.L=1 a 2 a
"'o Called Additie iletity
i
Ts called
A: Inerse Existance: muthpicahie idenl.
M: Invere Eristance:
Y aeR -atR VaeR 3 LeR
Such hat Such That
a+-a) =-a +a=0
-a" is called Additie
a.
a
=1
inverse a" 6 Caled mulhiplèahive
inverse a"
Disthibutie Propest:
V a,b,cE R
a.Cbte) = a-bta.c
a+b)C = a.c+b.c Left Distaibutive La
A
Set that satisfres Riht Distibuhe Lud
all te
above 11
Field ad is
denoted as
propesties is clue
Set of rationa) (F+,
(F+,)-)
numbers
numbe rs, Set of rea) numbers avd Set
are 3
fields. ofamplex
