, , - . - () - .
, , .. Ì Û (Î Þ Î).
, . - ( ): = Û Ì Ù Ì.
, .. ÎÈ Û ÎÚ Î.
, : ÎÇ Û Î Ù Î.
, .. Î \ Û Î ÙÏ.
=´ (,), , .
F ´ , : (" Î)($! (.)ÎF). : .
- . ("Î)($! Î): (,)ÎF, Î F : =F(). ( ) .
Q - ( ). , , 1 =4/3=8/6=12/9=. , .
p/q, pÎZ, qÎN, p, q- .
Q:
1. . , , , , ( 0) : ; ; .
2. : ( " , ÎQ, ¹)(x<y Ú x>y).
: 1) a>b, b>c Þ a>c; 2) a<b Þ -a>-b.
3. . , (, = ):
( " , ÎQ, x < y)($cÎQ): ( <c< y).
Q 4 , , 2=, Î N Q.
. ÎQ, 2.
g =p/q, p q 2=2. (p/q)2=2. ,
p2=2q2. (1)
(1) 2, p2 . =2n (n-). q .
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(1), 4n2=2q2. q2=2n2. , q 2, .. q - . .n
. 1, 2, 3 , , . , , . 1/q (q= 2,3,4 ) . , p/q (pÎZ, qÎN). p, q , , . , .
? ?
, . . N , ON=x - , 2= 2, .
, N, . =ON, .. . , . , = , , = . .
, .
R .
. p/q 2 5, .. q=2m×5k, p/q=a0,a1a2an. .
, , . :
a0,a1a2an = a0,a1a2an 000=a0,a1a2(an-1)999 (2)
, =0,(9) 10 =9,(9). 10 , 9 =9 1=1,(0)=0,(9).
, 9 0 (2).
, 9 . 9 , , .. 1,999 2,000
. . , 0,1010010001 27,1234567891011 ( ).
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a0, a1a2an (3)
+ , a0 a1,a2,,an,( : 0, 1, 2,, 9).
(3) () . (3) +, a0, a1a2an (4)
(4) , , a0, a1, a2, , an , . (3) -, .
(QÈJ=R). (3) , , .
=a0,a1a2an, b=b0,b1b2bn. ( =b), an=bn n=0,1,2 b ( a < b), a0<b0, a0=b0 m, ak=bk (k=0,1,2,m-1), am<bm, .. a<b Û (a0<b0) Ú ( $ mÎN: ak=bk (k= ), am<bm). > b.
. =a0, a1a2an , +, .. ½ ½= a0, a1a2an ½ ½³0. , b , >b. (a<0, b<0), , : 1) =b, ½ ½ = ½ b ½; 2) <b, ½ ½ > ½ b ½.
R:
I. :
1. b : a=b, a<b, a>b.
2. a<b, b<c, <c ().
3. a<b, , a< <b.
II. :
4. a+b=b+a ().
5. (a+b)+c=a+(b+c) ().
6. +0=.
7. +(-)= 0.
8. a<b Þ +<b+c (" cÎR).
III. :
9. a×b=b×a.
10. (a×b)×c=a×(b×c).
11. ×1=.
12. ×(1/)=1 (¹0).
13. (+b)×c = ac + bc ().
14. a<b c>0, ×<b×c.
IV. ("cÎR)($nÎN): (n>c).
ÎR, nÎN, n>c.
V. . ÌR BÌR , Î (a £ b) bÎB. , Î bÎB a £c£ b:
(" AÌR, BÌR):(" a ÎA, bÎB a £b)($cÎR): (" a ÎA, bÎB a £c£b).
R , .