Перевод десятичных чисел в другие системы счисления

Ɇɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ, ɱɬɨ ɭɦɧɨɠɟɧɢɟ ɢɥɢ ɞɟɥɟɧɢɟ ɱɢɫɥɚ ɧɚ ɨɫɧɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɩɪɢɜɟɞɟɬ ɤ ɩɟɪɟɦɟɳɟɧɢɸ ɡɚɩɹɬɨɣ, ɨɬɞɟɥɹɸɳɟɣ ɰɟɥɭɸ ɱɚɫɬɶ ɨɬ ɞɪɨɛɧɨɣ, ɜɩɪɚɜɨ ɢɥɢ ɜɥɟɜɨ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɧɚɩɪɢɦɟɪ: 536,1510 ·10 = 5361,510; 1001,112 / 102 = 100,1112 1.1. ɉɟɪɟɜɨɞ ɞɟɫɹɬɢɱɧɵɯ ɱɢɫɟɥ ɜ ɞɪɭɝɢɟ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ ɉɟɪɟɜɨɞ ɞɟɫɹɬɢɱɧɨɝɨ ɱɢɫɥɚ ɜ ɞɪɭɝɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ ɦɨɠɟɬ ɜɵɩɨɥɧɹɬɶɫɹ ɪɚɡɧɵɦɢ ɫɩɨɫɨɛɚɦɢ. ɉɪɢ ɷɬɨɦ ɧɚɞɨ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɚɥɝɨɪɢɬɦɵ ɩɟɪɟɜɨɞɚ ɰɟɥɵɯ ɱɢɫɟɥ ɢ ɩɪɚɜɢɥɶɧɵɯ ɞɪɨɛɟɣ ɛɭɞɭɬ ɨɬɥɢɱɚɬɶɫɹ. Ⱦɥɹ ɫɦɟɲɚɧɧɨɝɨ ɱɢɫɥɚ ɰɟɥɚɹ ɢ ɞɪɨɛɧɚɹ ɱɚɫɬɢ ɩɟɪɟɜɨɞɹɬɫɹ ɨɬɞɟɥɶɧɨ ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɚɥɝɨɪɢɬɦɚɦ. ȼ ɢɬɨɝɨɜɨɣ ɡɚɩɢɫɢ ɢɫɤɨɦɨɝɨ ɱɢɫɥɚ ɨɧɢ ɨɛɴɟɞɢɧɹɸɬɫɹ ɢ ɪɚɡɞɟɥɹɸɬɫɹ ɡɚɩɹɬɨɣ. Ɍɚɤ ɧɚɡɵɜɚɟɦɵɣ ɦɟɬɨɞ ɩɨɷɬɚɩɧɨɝɨ ɞɟɥɟɧɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɦ ɞɟɥɟɧɢɢ ɢɫɯɨɞɧɨɝɨ ɱɢɫɥɚ ɢ ɩɨɥɭɱɚɟɦɵɯ ɧɟɩɨɥɧɵɯ ɱɚɫɬɧɵɯ ɧɚ ɨɫɧɨɜɚɧɢɟ ɬɨɣ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ, ɜ ɤɨɬɨɪɭɸ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɜɨɞ. Ɉɫɬɚɬɤɢ ɞɟɥɟɧɢɹ ɫɨɫɬɚɜɥɹɸɬ ɢɫɤɨɦɨɟ ɱɢɫɥɨ. 1.1.1. Ⱥɥɝɨɪɢɬɦ ɩɟɪɟɜɨɞɚ ɰɟɥɨɝɨ ɞɟɫɹɬɢɱɧɨɝɨ ɱɢɫɥɚ N ɜ ɩɨɡɢɰɢɨɧɧɭɸ ɫɢɫɬɟɦɭ ɫ ɨɫɧɨɜɚɧɢɟɦ p 1. Ɋɚɡɞɟɥɢɬɶ ɱɢɫɥɨ N ɧɚ p 2. ɉɨɥɭɱɟɧɧɵɣ ɨɫɬɚɬɨɤ ɞɚɟɬ ɰɢɮɪɭ, ɫɬɨɹɳɭɸ ɜ ɧɭɥɟɜɨɦ ɪɚɡɪɹɞɟ p –ɢɱɧɨɣ ɡɚɩɢɫɢ ɱɢɫɥɚ N. 3. ɉɨɥɭɱɟɧɧɨɟ ɱɚɫɬɧɨɟ ɫɧɨɜɚ ɪɚɡɞɟɥɢɬɶ ɧɚ p ɢ ɫɧɨɜɚ ɡɚɩɨɦɧɢɬɶ ɩɨɥɭɱɟɧɧɵɣ ɨɫɬɚɬɨɤ – ɷɬɨ ɰɢɮɪɚ ɩɟɪɜɨɝɨ ɪɚɡɪɹɞɚ, ɢ ɬ.ɞ. 4. Ɍɚɤɨɟ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ ɞɟɥɟɧɢɟ ɩɪɨɢɡɜɨɞɢɬɫɹ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɱɚɫɬɧɨɟ ɧɟ ɫɬɚɧɟɬ ɪɚɜɧɵɦ 0. 5. ɐɢɮɪɚɦɢ ɢɫɤɨɦɨɝɨ ɱɢɫɥɚ ɹɜɥɹɸɬɫɹ ɨɫɬɚɬɤɢ ɨɬ ɞɟɥɟɧɢɹ, ɜɵɩɢɫɚɧɧɵɟ ɫɥɟɜɚ ɧɚɩɪɚɜɨ ɧɚɱɢɧɚɹ ɫ ɩɨɫɥɟɞɧɟɝɨ ɩɨɥɭɱɟɧɧɨɝɨ ɨɫɬɚɬɤɚ. ɉɪɢɦɟɪ. ɉɟɪɟɜɟɫɬɢ ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ 26 ɜ ɞɜɨɢɱɧɭɸ, ɬɪɨɢɱɧɭɸ ɢ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɭɸ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ. Ɋɟɲɟɧɢɟ 2610 ĺ X2 2610 ĺX3 2610 ĺX16 26/2| 0 26/3| 2 26/16| 10 ĺ A 13/2| 1 8/3| 2 1| 1 6/2| 0 2| 2 3/2| 1 1| 1 Ɋɟɡɭɥɶɬɚɬ X2 = 11010 X3 = 222 X7 = 1A16
  1.1.2. Ⱥɥɝɨɪɢɬɦ ɩɟɪɟɜɨɞɚ ɩɪɚɜɢɥɶɧɨɣ ɞɟɫɹɬɢɱɧɨɣ ɞɪɨɛɢ N ɜ ɩɨɡɢɰɢɨɧɧɭɸ ɫɢɫɬɟɦɭ ɫ ɨɫɧɨɜɚɧɢɟɦ p 1. ɍɦɧɨɠɢɬɶ ɞɚɧɧɨɟ ɱɢɫɥɨ (ɞɪɨɛɧɭɸ ɱɚɫɬɶ ɫɦɟɲɚɧɧɨɝɨ ɱɢɫɥɚ) ɧɚ ɨɫɧɨɜɚɧɢɟ ɡɚɞɚɧɧɨɣ ɩɨɡɢɰɢɨɧɧɨɣ ɫɢɫɬɟɦɵ p. 2. ɐɟɥɚɹ ɱɚɫɬɶ ɩɨɥɭɱɟɧɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɹɜɥɹɟɬɫɹ ɰɢɮɪɨɣ ɫɬɚɪɲɟɝɨ ɪɚɡɪɹɞɚ ɢɫɤɨɦɨɣ ɞɪɨɛɢ 3. Ɉɫɬɚɜɲɚɹɫɹ ɞɪɨɛɧɚɹ ɱɚɫɬɶ ɩɨɥɭɱɟɧɧɨɝɨ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜɧɨɜɶ ɭɦɧɨɠɚɟɬɫɹ ɧɚ p, ɢ ɰɟɥɚɹ ɱɚɫɬɶ ɪɟɡɭɥɶɬɚɬɚ ɫɱɢɬɚɟɬɫɹ ɫɥɟɞɭɸɳɟɣ ɰɢɮɪɨɣ ɢɫɤɨɦɨɣ ɞɪɨɛɢ. 4. Ɉɩɟɪɚɰɢɹ ɩɪɨɞɨɥɠɚɟɬɫɹ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɞɪɨɛɧɚɹ ɱɚɫɬɶ ɧɟ ɨɤɚɠɟɬɫɹ ɪɚɜɧɨɣ ɧɭɥɸ ɢɥɢ ɧɟ ɛɭɞɟɬ ɞɨɫɬɢɝɧɭɬɚ ɬɪɟɛɭɟɦɚɹ ɬɨɱɧɨɫɬɶ ɉɪɢɦɟɪ 1. ɉɟɪɟɜɟɫɬɢ ɞɟɫɹɬɢɱɧɭɸ ɞɪɨɛɶ 0,375 ɜ ɞɜɨɢɱɧɭɸ, ɬɪɨɢɱɧɭɸ ɢ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ. ɉɟɪɟɜɨɞ ɜɵɩɨɥɧɢɬɶ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɬɪɟɬɶɟɝɨ ɡɧɚɤɚ. Ɋɟɲɟɧɢɟ 0,37510 ĺ 0,X2 0,37510 ĺ 0,X3 0,37510 ĺ0,X16 p=2 p=3 p = 16 0,375·2 = 0,75 0,375·3 = 1,125 0,375·16 = 6,0 0,75·2 = 1,5 0,125·3 = 0,375 0,5·2 = 1,0 0,375·3 = 1,125 0,125·3 = 0,375 Ɋɟɡɭɥɶɬɚɬ: 0,37510 = 0,0112 0,37510 = 0,1013 0,37510 = 0,616 ɉɪɢɦɟɪ 2. ɉɟɪɟɜɟɫɬɢ ɞɟɫɹɬɢɱɧɨɟ ɱɢɫɥɨ 26,375 ɜ ɞɜɨɢɱɧɭɸ, ɬɪɨɢɱɧɭɸ ɢ ɲɟɫɬɧɚɞɰɚɬɟɪɢɱɧɭɸ ɫɢɫɬɟɦɭ ɫɱɢɫɥɟɧɢɹ. Ɋɟɲɟɧɢɟ ɂɡ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜɵɲɟ ɩɪɢɦɟɪɨɜ ɫɥɟɞɭɟɬ: 26,37510 = 11010,0112 = 222,1013 = 1A,616 2. ɉɟɪɟɜɨɞ ɱɢɫɟɥ ɥɸɛɵɯ ɩɨɡɢɰɢɨɧɧɵɯ ɫɢɫɬɟɦ ɫɱɢɫɥɟɧɢɹ ɜ ɞɟɫɹɬɢɱɧɭɸ ɫɢɫɬɟɦɭ ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɱɢɫɟɥ ɜ ɪɚɡɜɟɪɧɭɬɨɣ ɮɨɪɦɟ ɨɞɧɨɜɪɟɦɟɧɧɨ ɹɜɥɹɟɬɫɹ ɫɩɨɫɨɛɨɦ ɩɟɪɟɜɨɞɚ ɱɢɫɟɥ ɜ ɞɟɫɹɬɢɱɧɭɸ ɫɢɫɬɟɦɭ ɢɡ ɥɸɛɨɣ ɞɪɭɝɨɣ ɩɨɡɢɰɢɨɧɧɨɣ ɫɢɫɬɟɦɵ ɫɱɢɫɥɟɧɢɹ. Ⱦɨɫɬɚɬɨɱɧɨ ɩɨɞɫɱɢɬɚɬɶ ɪɟɡɭɥɶɬɚɬ ɩɨ ɩɪɚɜɢɥɚɦ ɞɟɫɹɬɢɱɧɨɣ ɚɪɢɮɦɟɬɢɤɢ. ɉɪɢɦɟɪ. ɉɨɥɭɱɢɬɶ ɞɟɫɹɬɢɱɧɵɟ ɷɤɜɢɜɚɥɟɧɬɵ ɱɢɫɟɥ: 101,012; 673,28; 15AC16. Ɋɟɲɟɧɢɟ 101,012 = 1·21 + 0·20 + 0·2-1 + 1·2-2 = 4 +0 + 1 + 0 + 1/4 = 5 + 0,25 = 5,2510 673,28 = 6·82 + 7·81 + 3·80 + 2·8-1 = 384 + 56 + 3 + 2·0,25 = 443,2510 15AC16 = 1·163 + 5·162 + 10·161 + 12·160 = 4096 + 1280 + 160 + 12 = 554810