BAB I
PENDAHULUAN
1.1
LATAR
BELAKANG
Banyak
orang yang beranggapan bahwa Matematika itu rumit, karena alasan itulah banyak
orang yang menghindari Matematika. Padahal Matematika dapat kita jumpai di
dalam kehidupan sehari-hari, dan mau tidak mau kita pasti menggunakan
Matematika. Oleh karena itu kami membuat makalah ini dengan maksud membantu
pemahaman masyarakat agar mereka tidak menilai Matematika adalah sesuatu yang
buruk.
1.2
TUJUAN
Makalah
ini dibuat dengan tujuan utama untuk memenuhi tugas mata kuliah Analisa real ,
yang diberikan oleh dosen kami ibu Nur Afifah,Mpd. Dan tujuan berikutnya adalah
sebagai sumber informasi yang kami harapkan bermanfaat dan dapat menambah
wawasan para pembaca makalah ini.
1.3
METODE
PENULISAN
Penulis
menggunakan metode observasi dan kepusatakaan.
Cara
yang digunakan dalam penulisan adalah Studi pustaka.
Dalam
metode ini penulis membaca buku-buku yang berkaitan dengan penulisan makalah
ini, selain itu penulis juga mencari sumber-sumber dari internet.
BAB II
Sifat-sifat pada bilangan real
2.2.1 Definisi
P
R, P
0
disebut bilangan real positif murni jika memenuhi sifat-sifat berikut :


(i)
Jika a,b
P, maka a + b
P (positif real murni )


(ii)
Jika a,b
P, maka ab
P


(iii)
Jika a
R, maka a
P, a = 0, -a
P (sifat Trikotomi )



2.2.2
Definisi
(i)
Jika a
P, maka a > 0

(ii)
Jika a
P
V a = 0, maka a
0


(iii)
Jika -a
P, maka a < 0

(iv)
Jika -a
P
V a =0, maka a
0


2.2.3
Definisi
(i)
Jika a-b
P, maka a > b V b <a

(ii)
Jika a-b
P V {0}, maka a
b V b
a



2.2.4
Teorema
(a)
Jika a > b
b > c, maka
a > c

(b)
Jika a > b, a = 0, a < b
(c)
Jika a
b
b
c, maka a = b



a.
Jika a > b
b > c, maka
a > c

Bukti :




















b.
Jika a > b, a = 0, a < b
Bukti :



-(a-b)
P

Ø
a - b
P a
> b def
2.2.3


Ø
a -b = 0 (a + (-b)) = 0 def +





Ø
-(a-b)
P -1
(a-b )
P teor
2.1.5 (b)












c. Jika
a
b
b
c, maka a = b



Bukti
:
Pengandaian
a
bb









Ø
a – b > 0 (a- b )
P def
2.2.2



a
> b kontradiksi dengan hipotesis a
b

Ø
a – b < 0 -(a-b )
P def
2.2.2











a
b kontradiksi
dengan hipotesis b
a






2.2.5 teorema
(a) Jika
a
R dan a
0, maka a2 > 0


(b) Jika
1 > 0
(c) Jika n
N, maka n = 0

a.
a
R a
P , -a
P ( karena
a
0, maka a = 0 di abaikan )





Ø
a
P a.a
P sifat
2.2.1







Ø
-a
P (-a)
(-a )
P sifat
2.2.1













![]() |
b.
1 > 0
(dengan menggunakan teorema 2.2.5 )



dimisalkan a = 1



1.1
> 0
1
> 0
c.
Jika n
N, maka n = 0

Menggunakan induksi matematika
Bukti :
Ø
P(1) adalah 1 > 0, P (1) benar
Ø
Diasumsikan bahwa P (n) benar untuk n
N yaitu n > 0, P (n +1 ) benar yaitu n+1
> 0

Ø
Jadi, P (n+1 ) benar , sebagai n > 0
benar, n
R

2.2.6
teorema
(a)
Jika a > b, maka a + c > b + c
(b)
Jika a > b
c > d, maka a + c > b + d

(c)
Jika a > b
c > 0, maka ca > cb

Jika a > b
c < 0, maka ca
< cb

(d)
Jika a > 0, maka 1/a > 0
Jika a < 0, maka 1/a <
0
a.
Jika a > b, maka a + c > b + c
Bukti
:













b.
Jika a > b
c > d, maka a + c > b + d

Bukti
:

















c.
1. Jika a > b
c > 0, maka ca > cb

Bukti
:













2.
Jika a > b
c < 0, maka ca
< cb

Bukti :














d.
1. Jika a > 0, maka 1/a > 0
Bukti
:




(i)
1/a < 0 -1/a
P def
2.2.3








Terjadi
kontradiksi dengan hipotesis a > 0
(ii)
1/a = 0 a.
1/a = 0 kanselasi
1 = 0 M4
dan teor. 2.1.5
Kontradiksi
1
0

Kesimpulan
: dari per (i) dan (ii) terjadi kontradiksi dengan hpotesis. Jadi pengandaian
salh, maka harus 1/a > 0
2.
Jika a < 0,
maka 1/a < 0
Bukti :



(i)
1/a >0 1/a
P def
2.2.3







(ii)

1/a = 0 a.1/a = 0 kanselasi



Kontradiksi
1
0

Kesimpulan
: dari per (i) dan (ii) terjadi kontradiksi dengan hipotesis berarti
pengandaian salah maka haruslah 1/a > 0
2.2.7 teorema
Jika a, b
R, jika a > b, maka a > ½ (b+a) > b

Bukti :




a
+ b > 2b


Karena 2
N teor.
2.2.5 ©

2
> 0
Maka
1/a > 0 teor
2.2.6 (d)
1/2
> 0
(1/2)(2a) >
(1/2) (a+b) > (1/2)(2b) setiap
ruas dikali 1/2
(1/2.2) (a) >
(1/2)(a+b) > (1/2.2).b M2
1.a >
(1/2) (a+b ) > 1.b M4
a >
½ (a+b) > b
M3
2.2.8 teorema
Jika a
R dan a > 0, makaa > ½ a > 0

Bukti :




Karena
2
N

2
> 0
Maka 1/a
> 0
½
> 0
(1/2 ) (2a) > ( ½ ) (a) > (1/2 )
(0) setiap
ruas dikali ½
( ½ 2) (a) > ( ½ ) (a) > ½ (0) M2
1.a
> ½ (a) > 0 M4
a
> ½ a > 0 M3

2.2.9
teorema
Jika a
R , 0
a <
,
adalah bilangan
real, maka a = 0




Bukti :







dari pers (i) (a >
0 > 0) terjadi kontradiksi dengan
hipotesis a <
berarti
pengandaian salh, maka haruslah a = 0


2.2.10
teorema
(a)
Jika ab > 0, maka a > 0 dan b > 0
(b)
Jika ab > 0, maka a < 0 dan b <
0
a.
ab > 0 a > 0
b > 0


Bukti :







Ø a
> 0






(ab). (1/a)
P (ab)
(1/a)
P













b.
ab > 0 a <0
b < 0


bukti :
Ø a
< 0




1/a < 0 -1/a
P def 2.2.2





a (
b.(-1/a))
P M2
















2.2.11 Corollary
(a)
jika ab < 0, maka a <0 dan b > 0
(b)
jika ab < 0, maka a >0 dan b < 0
a.
ab < 0 a < 0
b > 0


bukti :






































b
> 0 def
2.2.2
b.
Ab < 0 a > 0
b < 0


Bukti :



a
0
b
0
















-1(ab)
(1/a)
P teor
2.1.5














2.2.12 KETAKSAMAAN BERNOULLI
Jika x > -1, maka (1 + x )n
1 + nx , n
N


Bukti :
Dengan menggunakan induksi matematika
Misal :
S
= { n
N : ( 1 + x )n
1 + nx }


P
(n) = ( 1 + X)n
1 + nx dan
S
n


P
(1) = ( 1 + x )1
1 + 1 x




Bukti :
P(
n) =
( 1 + n)n
(1 + nX )

P
( n + 1) = ( 1 + n ) n+1
( 1 +(n+1) x)

=
( 1+ X )n (1 + x )
( 1 +( n+1) x)

=
(1 + nx) (1 + x)
1 + (n + 1) x

=
1 + x + nx + nx2
1 + (n +1) x

=
1 + (n + 1) + nx2
1 + (n + 1) x

Karena P (1) benar a
b maka P (n +
1) benar

BAB III
PENUTUP
Saran
Alangkah baiknya kita mengenal Matematika dulu
sebelum kita menganggap Matematika itu sulit, karena bila kita telah mengenal
Matematika dengan baik dan menikmati bagaimana Matematika itu bekerja akan
terasa bahwa Matematika itu tidaklah seburuk apa yang kita pikirkan.
Merit Casino - xn--o80b910a26eepc81il5g.online
BalasHapusMerit หาเงินออนไลน์ Casino - Merit Casino - Merit Casino - Merit Casino - Merit Casino - Merit Casino - Merit Casino 제왕 카지노 - Merit Casino - Merit Casino - Merit Casino - Merit Casino 메리트 카지노 주소 - Merit Casino -