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БОГ СОЗДАЛ

ЦЕЛЫЕ ЧИСЛА
МАТЕМАТИЧЕСКИЕ ОТКРЫТИЯ,
ИЗМЕНИВШИЕ ИСТОРИЮ

ИЗДАТЕЛЬСТВО АСТ
МОСКВА

ɍȾɄ
ȻȻɄ

51(091)
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ɉɟɪɟɜɨɞɨɪɢɝɢɧɚɥɶɧɨɝɨɢɡɞɚɧɢɹ
Stephen Hawking
GOD CREATED THE INTEGERS
The Mathematical Breakthroughs that Changed History

ɉɟɱɚɬɚɟɬɫɹɫɪɚɡɪɟɲɟɧɢɹɢɡɞɚɬɟɥɶɫɬɜɚ5XQQLQJ3UHVV$GXOW
DQLPSULQWRI3HUVHXV%RRNV//&DVXEVLGLDU\RI+DFKHWWH%RRNV1HZɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ ɨɤɪɭɠɧɨɫɬɶɸ@ ɧɚ ɤɨɬɨɪɭɸ ɜɫɟ
ɢɡ ɨɞɧɨɣ ɬɨɱɤɢ ɜɧɭɬɪɢ ɮɢɝɭɪɵ ɩɚɞɚɸɳɢɟ >ɧɚ ɨɤɪɭɠɧɨɫɬɶ
ɤɪɭɝɚ@ɩɪɹɦɵɟɪɚɜɧɵɦɟɠɞɭɫɨɛɨɣ
 ɐɟɧɬɪɨɦɠɟɤɪɭɝɚɧɚɡɵɜɚɟɬɫɹɷɬɚɬɨɱɤɚ
 Ⱦɢɚɦɟɬɪ ɠɟ ɤɪɭɝɚ ɟɫɬɶ ɤɚɤɚɹ ɭɝɨɞɧɨ ɩɪɹɦɚɹ ɩɪɨɜɟɞɟɧɧɚɹ
ɱɟɪɟɡɰɟɧɬɪɢɨɝɪɚɧɢɱɢɜɚɟɦɚɹɫɨɛɟɢɯɫɬɨɪɨɧɨɤɪɭɠɧɨɫɬɶɸ
ɤɪɭɝɚɨɧɚɠɟɢɪɚɫɫɟɤɚɟɬɤɪɭɝɩɨɩɨɥɚɦ
 ɉɨɥɭɤɪɭɝɠɟɟɫɬɶɮɢɝɭɪɚɫɨɞɟɪɠɚɳɚɹɫɹɦɟɠɞɭɞɢɚɦɟɬɪɨɦ
ɢɨɬɫɟɤɚɟɦɨɣɢɦɱɚɫɬɶɸ!ɨɤɪɭɠɧɨɫɬɢɐɟɧɬɪɠɟɩɨɥɭɤɪɭɝɚ ± ɬɨɠɟɫɚɦɨɟɱɬɨɢɭɤɪɭɝɚ
1.
2.



22

ȻɈȽɋɈɁȾȺɅɐȿɅɕȿɑɂɋɅȺ

19. ɉɪɹɦɨɥɢɧɟɣɧɵɟɮɢɝɭɪɵɫɭɬɶɬɟɤɨɬɨɪɵɟɫɨɞɟɪɠɚɬɫɹɦɟɠɞɭ ɩɪɹɦɵɦɢ ɬɪɟɯɫɬɨɪɨɧɧɢɟ ± ɦɟɠɞɭ ɬɪɟɦɹ ɱɟɬɵɪɟɯɫɬɨɪɨɧɧɢɟɠɟ ± ɱɟɬɵɪɶɦɹɦɧɨɝɨɫɬɨɪɨɧɧɢɟɠɟ ± ɤɨɬɨɪɵɟɫɨɞɟɪɠɚɬɫɹɦɟɠɞɭɛɨɥɟɟɱɟɦɱɟɬɵɪɶɦɹɩɪɹɦɵɦɢ
 ɂɡ ɬɪɟɯɫɬɨɪɨɧɧɢɯ ɮɢɝɭɪ ɪɚɜɧɨɫɬɨɪɨɧɧɢɣ ɬɪɟɭɝɨɥɶɧɢɤ
ɟɫɬɶ ɮɢɝɭɪɚ ɢɦɟɸɳɚɹ ɬɪɢ ɪɚɜɧɵɟ ɫɬɨɪɨɧɵ ɪɚɜɧɨɛɟɞɪɟɧɧɵɣɠɟ ± ɢɦɟɸɳɚɹɬɨɥɶɤɨɞɜɟɪɚɜɧɵɟɫɬɨɪɨɧɵɪɚɡɧɨɫɬɨɪɨɧɧɢɣɠɟ ± ɢɦɟɸɳɚɹɬɪɢɧɟɪɚɜɧɵɟɫɬɨɪɨɧɵ
 Ʉɪɨɦɟ ɬɨɝɨ ɢɡ ɬɪɟɯɫɬɨɪɨɧɧɢɯ ɮɢɝɭɪ ɩɪɹɦɨɭɝɨɥɶɧɵɣ ɬɪɟɭɝɨɥɶɧɢɤ ɟɫɬɶ ɢɦɟɸɳɢɣ ɩɪɹɦɨɣ ɭɝɨɥ ɬɭɩɨɭɝɨɥɶɧɵɣ
ɠɟ ± ɢɦɟɸɳɢɣɬɭɩɨɣɭɝɨɥɚɨɫɬɪɨɭɝɨɥɶɧɵɣ ± ɢɦɟɸɳɢɣɬɪɢ
ɨɫɬɪɵɯɭɝɥɚ
 ɂɡɱɟɬɵɪɟɯɫɬɨɪɨɧɧɢɯɮɢɝɭɪɤɜɚɞɪɚɬɟɫɬɶɬɚɤɨɬɨɪɚɹɢɪɚɜɧɨɫɬɨɪɨɧɧɹɹ ɢ ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɪɚɡɧɨɫɬɨɪɨɧɧɢɤ ɠɟ ± ɩɪɹɦɨɭɝɨɥɶɧɚɹ ɧɨ ɧɟ ɪɚɜɧɨɫɬɨɪɨɧɧɹɹ ɪɨɦɛ ± ɪɚɜɧɨɫɬɨɪɨɧɧɹɹ
ɧɨɧɟɩɪɹɦɨɭɝɨɥɶɧɚɹɪɨɦɛɨɢɞ ɩɚɪɚɥɥɟɥɨɝɪɚɦɦ ± ɢɦɟɸɳɚɹ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɫɬɨɪɨɧɵ ɢ ɭɝɥɵ ɪɚɜɧɵɟ ɦɟɠɞɭ ɫɨɛɨɣ
ɧɨ ɧɟ ɹɜɥɹɸɳɚɹɫɹ ɧɢ ɪɚɜɧɨɫɬɨɪɨɧɧɟɣ ɧɢ ɩɪɹɦɨɭɝɨɥɶɧɨɣ
Ɉɫɬɚɥɶɧɵɟ ɠɟ ɱɟɬɵɪɟɯɫɬɨɪɨɧɧɢɤɢ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɬɪɚɩɟ
ɰɢɹɦɢ.
 ɉɚɪɚɥɥɟɥɶɧɵɟ ɫɭɬɶ ɩɪɹɦɵɟ ɤɨɬɨɪɵɟ ɧɚɯɨɞɹɫɶ ɜ ɨɞɧɨɣ
ɩɥɨɫɤɨɫɬɢɢɛɭɞɭɱɢɩɪɨɞɨɥɠɟɧɵɜɨɛɟɫɬɨɪɨɧɵɧɟɨɝɪɚɧɢɱɟɧɧɨɧɢɫɬɨɣɧɢɫɞɪɭɝɨɣ©ɫɬɨɪɨɧɵªɦɟɠɞɭɫɨɛɨɣɧɟɜɫɬɪɟɱɚɸɬɫɹ

ɉɈɋɌɍɅȺɌɕ
Ⱦɨɩɭɫɬɢɦ
 ɑɬɨɨɬɜɫɹɤɨɣɬɨɱɤɢɞɨɜɫɹɤɨɣɬɨɱɤɢɦɨɠɧɨ!ɩɪɨɜɟɫɬɢɩɪɹɦɭɸɥɢɧɢɸ
 ɂɱɬɨɨɝɪɚɧɢɱɟɧɧɭɸɩɪɹɦɭɸɦɨɠɧɨ!ɧɟɩɪɟɪɵɜɧɨɩɪɨɞɨɥɠɚɬɶɩɨɩɪɹɦɨɣ
 ɂɱɬɨɢɡɜɫɹɤɨɝɨɰɟɧɬɪɚɢɜɫɹɤɢɦɪɚɫɬɜɨɪɨɦɦɨɠɟɬɛɵɬɶ!
ɨɩɢɫɚɧɤɪɭɝ
 Ⱥɤɫ ɂɱɬɨɜɫɟɩɪɹɦɵɟɭɝɥɵɪɚɜɧɵɦɟɠɞɭɫɨɛɨɣ
 Ⱥɤɫ ɂɟɫɥɢɩɪɹɦɚɹɩɚɞɚɸɳɚɹɧɚɞɜɟɩɪɹɦɵɟɨɛɪɚɡɭɟɬ
ɜɧɭɬɪɟɧɧɢɟɢɩɨɨɞɧɭɫɬɨɪɨɧɭɭɝɥɵɦɟɧɶɲɢɟɞɜɭɯɩɪɹɦɵɯ
ɬɨɩɪɨɞɨɥɠɟɧɧɵɟɷɬɢɞɜɟɩɪɹɦɵɟɧɟɨɝɪɚɧɢɱɟɧɧɨɜɫɬɪɟɬɹɬɫɹ
ɫɬɨɣɫɬɨɪɨɧɵɝɞɟɭɝɥɵɦɟɧɶɲɢɟɞɜɭɯɩɪɹɦɵɯ



ȿȼɄɅɂȾ

ɈȻɓɂȿɉɈɇəɌɂə
Ⱥɤɫɢɨɦɵ
 Ɋɚɜɧɵɟɨɞɧɨɦɭɢɬɨɦɭɠɟɪɚɜɧɵɢɦɟɠɞɭɫɨɛɨɣ
 ɂɟɫɥɢɤɪɚɜɧɵɦɩɪɢɛɚɜɥɹɸɬɫɹɪɚɜɧɵɟɬɨɢɰɟɥɵɟɛɭɞɭɬɪɚɜɧɵ
 ɂɟɫɥɢɨɬɪɚɜɧɵɯɨɬɧɢɦɚɸɬɫɹɪɚɜɧɵɟɬɨɨɫɬɚɬɤɢɛɭɞɭɬɪɚɜɧɵ
>ɂɟɫɥɢɤɧɟɪɚɜɧɵɦɩɪɢɛɚɜɥɹɸɬɫɹɪɚɜɧɵɟɬɨɰɟɥɵɟɛɭɞɭɬɧɟ
ɪɚɜɧɵ
 ɂɭɞɜɨɟɧɧɵɟɨɞɧɨɝɨɢɬɨɝɨɠɟɪɚɜɧɵɦɟɠɞɭɫɨɛɨɣ
 ɂɩɨɥɨɜɢɧɵɨɞɧɨɝɨɢɬɨɝɨɠɟɪɚɜɧɵɦɟɠɞɭɫɨɛɨɣ@
 ɂɫɨɜɦɟɳɚɸɳɢɟɫɹɞɪɭɝɫɞɪɭɝɨɦɪɚɜɧɵɦɟɠɞɭɫɨɛɨɣ
 ɂɰɟɥɨɟɛɨɥɶɲɟɱɚɫɬɢ

>ɂɞɜɟɩɪɹɦɵɟɧɟɫɨɞɟɪɠɚɬɩɪɨɫɬɪɚɧɫɬɜɚ@

ɉɪɟɞɥɨɠɟɧɢɟ
ȼɩɪɹɦɨɭɝɨɥɶɧɵɯɬɪɟɭɝɨɥɶɧɢɤɚɯɤɜɚɞɪɚɬɧɚɫɬɨɪɨɧɟɫɬɹɝɢɜɚɸ
ɳɟɣɩɪɹɦɨɣɭɝɨɥɪɚɜɟɧɜɦɟɫɬɟɜɡɹɬɵɦ!ɤɜɚɞɪɚɬɚɦɧɚɫɬɨɪɨɧɚɯ
ɡɚɤɥɸɱɚɸɳɢɯɩɪɹɦɨɣɭɝɨɥ
ɉɭɫɬɶABC ± ɩɪɹɦɨɭɝɨɥɶɧɵɣɬɪɟɭɝɨɥɶɧɢɤɢɦɟɸɳɢɣɩɪɹɦɨɣɭɝɨɥ
BACɹɭɬɜɟɪɠɞɚɸɱɬɨɤɜɚɞɪɚɬɧɚBCɪɚɜɟɧɜɦɟɫɬɟɜɡɹɬɵɦ!ɤɜɚɞɪɚɬɚɦɧɚBAɢAC ɱɟɪɬɟɠ 
G
H

K
A

F
C

B

D

L

ɑɟɪɬɟɠ

E



ȻɈȽɋɈɁȾȺɅɐȿɅɕȿɑɂɋɅȺ

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɧɚɞɫɬɪɨɢɦ ɧɚ BC ɤɜɚɞɪɚɬ BDEC ɚ ɧɚ BA AC
ɧɚɞɫɬɪɨɢɦɤɜɚɞɪɚɬɵ!HBGC ɩɪɟɞɥɨɠɟɧɢɟ ɢɱɟɪɟɡAɩɪɨɜɟɞɟɦALɩɚɪɚɥɥɟɥɶɧɭɸɤɚɤBDɬɚɤɢCE ɩɪɟɞɥɨɠɟɧɢɟ ɫɨɟɞɢɧɢɦ
AD FC ɂ ɩɨɫɤɨɥɶɤɭ ɤɚɠɞɵɣ ɢɡ ɭɝɥɨɜ BAC BAH ɩɪɹɦɨɣ ɨɩɪɟɞɟɥɟɧɢɟ ɬɨɜɨɬɧɚɧɟɤɨɬɨɪɨɣɩɪɹɦɨɣBAɩɪɢɟɟɬɨɱɤɟAɞɜɟɩɪɹɦɵɟACAHɪɚɫɩɨɥɨɠɟɧɧɵɟɧɟɩɨɨɞɧɭɫɬɨɪɨɧɭɨɛɪɚɡɭɸɬɫɦɟɠɧɵɟ
ɭɝɥɵɜɦɟɫɬɟ!ɪɚɜɧɵɟɞɜɭɦɩɪɹɦɵɦɡɧɚɱɢɬCAɛɭɞɟɬɩɨɨɞɧɨɣ!
ɩɪɹɦɨɣɫAH ɩɪɟɞɥɨɠɟɧɢɟ ȼɫɥɟɞɫɬɜɢɟɬɨɝɨɠɟɜɨɬɢBAɛɭɞɟɬ
ɩɨɨɞɧɨɣɩɪɹɦɨɣɫAGɂɩɨɫɤɨɥɶɤɭɭɝɨɥDBCɪɚɜɟɧɭɝɥɭFBA ɚɤɫɢɨɦɚ ɢɛɨɤɚɠɞɵɣɢɡɧɢɯɩɪɹɦɨɣɬɨɩɪɢɛɚɜɢɦɨɛɳɢɣɭɝɨɥABC
ɡɧɚɱɢɬɜɟɫɶɭɝɨɥDBAɪɚɜɟɧɜɫɟɦɭFBC ɚɤɫɢɨɦɚ ɂɩɨɫɤɨɥɶɤɭDB
ɪɚɜɧɚBCɚFBɪɚɜɧɚBA ɨɩɪɟɞɟɥɟɧɢɟ ɬɨɜɨɬɞɜɟɫɬɨɪɨɧɵDB
BAɪɚɜɧɵɞɜɭɦɫɬɨɪɨɧɚɦBCFBɤɚɠɞɚɹɤɚɠɞɨɣɢɭɝɨɥDBAɪɚɜɟɧ
ɭɝɥɭFBCɡɧɚɱɢɬɢɨɫɧɨɜɚɧɢɟ ɩɪɟɞɥɨɠɟɧɢɟ --">