Obzornik mat. fiz. 55 (2008) 2

O realnih številih
 
 
 
 
 
 
 
 
 
 
 
 
 
 64–76

Peter Šemrl
 
 
 
 
 1
2
=
2
4
 
 
 
 
 
 0 1 
 
 P P 
 
 x 0 P x 5
3
 5
3
 
 
 [0,1] √
2 √
2 √
2 
 
 
 P R P ≤ R P = R P R 
 
 
 ≤ 
 
 
 −2 −2 −4 Obzornik mat. fiz. 55 (2008) 2

O realnih številih
 −2 ≤ 1 −4 ≤ −2 −2 −4 
 x ≤ y y x −2 1 −4 P 
 x 
 P 
 
 
 c
0
 c
0
≤P ≤c
0
+1.
 
 c
0
 c
0
 c
0
+ 1 c
0
 c
0
+
1
10
 c
0
+
2
10
 c
0
+
9
10
 c
0
+1 P c
1
 c
0
+
1
10
c
1
≤P ≤c
0
+
1
10
c
1
+
1
10
.
 P 
 1
10
 c
2
 c
0
+
1
10
c
1
+
1
100
c
2
≤P ≤c
0
+
1
10
c
1
+
1
100
c
2
+
1
100
.
 n c
0
+
1
10
c
1
+···+
1
10
n
c
n
≤P ≤c
0
+
1
10
c
1
+···+
1
10
n
c
n
+
1
10
n
,
 c
1
,...,c
n
 P 
 1
10
n
 
 P 
 c
0
 c
1
 c
2
 c
3
 
 c
0
 c
1
 c
2
 c
3
 P 
 c
0
 c
1
 c
2
 c
3
 
 c
0
 c
0
+ 1 T c
0
≤T ≤c
0
+1 [c
0
,c
0
+1] 64–76

Peter Šemrl
 I
0
 I
1
=

c
0
+
1
10
c
1
,c
0
+
1
10
c
1
+
1
10

 T c
0
+
1
10
c
1
≤T ≤c
0
+
1
10
c
1
+
1
10
.
 I
1
 I
0
 I
2
=

c
0
+
1
10
c
1
+
1
100
c
2
,c
0
+
1
10
c
1
+
1
100
c
2
+
1
100

 
 I
3
 I
4
 P
 P 
 c
0
≥ 0 c
1
 c
2
 c
3
 
 
 c
0
≥ 0 c
1
 c
2
 c
3
 P 
 
 c
0
,c
1
c
2
c
3
 
 
 
 9
8
 
 9
8
= 1 
 
 9
8
= 1 
 c
0
 c
1
 c
2
 c
3
 
 c
0
+0,c
1
c
2
c
3
...
 
 −1/3 
 −1
 
 −
1
3
=−1+0,666... =−0,333...
 
 
 
 1
5
= 0,2
1
3
= 0,333... = 0,3
 Obzornik mat. fiz. 55 (2008) 2

O realnih številih
1
7
= 0,142857142857142857... = 0,142857
 
 
 
 
 
 
 
 1,010010001000010000010000001...
 
 
 
 
 
 
 
 √
2 
 
 
 
 0,9999... = 0,9 1
10
 1
100
 1
1000
 
 x = 0,99999...
 10x = 9,99999... ,
 9x = 10x−x = 9,99999...− 0,99999... = 9 x = 1 64–76

Peter Šemrl
 
 
 
 1 = 1,000 
 1 = 0,999 c
0
= 1
 c
0
= 0 c
0
≤ 1≤c
0
+1 [0,1] 
 
 
 1 = 0,999 
 
 
 
 
 
 
 
 
 
 + = 
 4·3,14 π 
 Obzornik mat. fiz. 55 (2008) 2

O realnih številih

 ≤ 
 
 a b c a≤b a+c≤b+c 0≤a
 0≤b 0≤ab 
 
 
 
 
 A ={r∈Q: 1≤r≤ 2}, B ={r∈Q: 1<r < 2}
 C ={r∈Q: r≤ 2}.
 A B A B C 
 X 
 Y,≤ y ∈ Y X x≤ y x∈ X X 17 A B C ⊂Q 
 ≥ 2 X 
 (Y,≤) y
0
∈Y X y
0
 X y X
 y
0
≤y X X X supX 64–76

Peter Šemrl
 supA = supB = supC = 2.
 D ={r∈Q: r≤ 0 r
2
≤ 2}.
 
 √
2 √
2 
 √
2 
 
 D 
 √
2 
 √
2 
 
 D √
2 
 
 
 
 D D 
 
 √
2 Obzornik mat. fiz. 55 (2008) 2

O realnih številih
 D 
 
 
 x 
 r r < x 
 
 x (−∞,x) 
 α 
 
 r 
 q r Q Q 
 
 (−∞,x) 
 (−∞,x] 
 
 A⊂Q 
 r∈ A p≤ r
 p ∈ A 
 
 α ⊂ Q α6=∅ α6=Q α r q∈Q r∈α q <r q∈α R 
 
 x 64–76

Peter Šemrl
 x,y x ≤ y (−∞,x) ⊂ (−∞,y) α β α≤β ⇐⇒ α⊂β.
 
 
 
 A ⊂ R A ={−1,−1/2,−1/3,−1/4,...},
B = (−1,0) C = [−1,0] 0 0 
 A B C Q 
 A B C A⊂R 
 [
α∈A
α
 A 
 Obzornik mat. fiz. 55 (2008) 2

O realnih številih
 
 
 √
2 
 √
2 √
3 
 √
3 √
2 +
√
3 
 √
2 +
√
3 
 p <
√
2 q <
√
3 p+q <
√
2+
√
3 
 r <
√
2 +
√
3 r = p +q p <
√
2 q <
√
3 α,β α+β ={p+q: p∈α q∈β}.
 α β ∈ R α + β
 
 
 
 Q 
 
 
 64–76

Peter Šemrl
 
 
 R 
 
 
 
 ≤ N Z Q 
 
 
 
 
 
 Obzornik mat. fiz. 55 (2008) 2