Для перевода двоичного числа 1110111.010000111001010001010011000100100101 в десятичное необходимо записать его в виде многочлена, состоящего из произведений цифр вида:
A2-10 = an-1 ∙ 2-10n-1 + an-2 ∙ 2-10n-2 + ∙∙∙ + a0 ∙ 2-100 + a-1 ∙ 2-10-1 + ∙∙∙ + a-m ∙ 2-10-m
В результате преобразований получим:
1110111.0100001110010100010100110001001001012-10=1 ∙ 2-106 + 1 ∙ 2-105 + 1 ∙ 2-104 + 0 ∙ 2-103 + 1 ∙ 2-102 + 1 ∙ 2-101 + 1 ∙ 2-100 + 0 ∙ 2-10-1 + 1 ∙ 2-10-2 + 0 ∙ 2-10-3 + 0 ∙ 2-10-4 + 0 ∙ 2-10-5 + 0 ∙ 2-10-6 + 1 ∙ 2-10-7 + 1 ∙ 2-10-8 + 1 ∙ 2-10-9 + 0 ∙ 2-10-10 + 0 ∙ 2-10-11 + 1 ∙ 2-10-12 + 0 ∙ 2-10-13 + 1 ∙ 2-10-14 + 0 ∙ 2-10-15 + 0 ∙ 2-10-16 + 0 ∙ 2-10-17 + 1 ∙ 2-10-18 + 0 ∙ 2-10-19 + 1 ∙ 2-10-20 + 0 ∙ 2-10-21 + 0 ∙ 2-10-22 + 1 ∙ 2-10-23 + 1 ∙ 2-10-24 + 0 ∙ 2-10-25 + 0 ∙ 2-10-26 + 0 ∙ 2-10-27 + 1 ∙ 2-10-28 + 0 ∙ 2-10-29 + 0 ∙ 2-10-30 + 1 ∙ 2-10-31 + 0 ∙ 2-10-32 + 0 ∙ 2-10-33 + 1 ∙ 2-10-34 + 0 ∙ 2-10-35 + 1 ∙ 2-10-36 = 1 ∙ 64 + 1 ∙ 32 + 1 ∙ 16 + 0 ∙ 8 + 1 ∙ 4 + 1 ∙ 2 + 1 ∙ 1 + 0 ∙ 0.5 + 1 ∙ 0.25 + 0 ∙ 0.125 + 0 ∙ 0.0625 + 0 ∙ 0.03125 + 0 ∙ 0.015625 + 1 ∙ 0.0078125 + 1 ∙ 0.00390625 + 1 ∙ 0.001953125 + 0 ∙ 0.0009765625 + 0 ∙ 0.00048828125 + 1 ∙ 0.000244140625 + 0 ∙ 0.0001220703125 + 1 ∙ 6.103515625E-5 + 0 ∙ 3.0517578125E-5 + 0 ∙ 1.52587890625E-5 + 0 ∙ 7.62939453125E-6 + 1 ∙ 3.814697265625E-6 + 0 ∙ 1.9073486328125E-6 + 1 ∙ 9.5367431640625E-7 + 0 ∙ 4.7683715820312E-7 + 0 ∙ 2.3841857910156E-7 + 1 ∙ 1.1920928955078E-7 + 1 ∙ 5.9604644775391E-8 + 0 ∙ 2.9802322387695E-8 + 0 ∙ 1.4901161193848E-8 + 0 ∙ 7.4505805969238E-9 + 1 ∙ 3.7252902984619E-9 + 0 ∙ 1.862645149231E-9 + 0 ∙ 9.3132257461548E-10 + 1 ∙ 4.6566128730774E-10 + 0 ∙ 2.3283064365387E-10 + 0 ∙ 1.1641532182693E-10 + 1 ∙ 5.8207660913467E-11 + 0 ∙ 2.9103830456734E-11 + 1 ∙ 1.4551915228367E-11 = 64 + 32 + 16 + 0 + 4 + 2 + 1 + 0 + 0.25 + 0 + 0 + 0 + 0 + 0.0078125 + 0.00390625 + 0.001953125 + 0 + 0 + 0.000244140625 + 0 + 6.103515625E-5 + 0 + 0 + 0 + 3.814697265625E-6 + 0 + 9.5367431640625E-7 + 0 + 0 + 1.1920928955078E-7 + 5.9604644775391E-8 + 0 + 0 + 0 + 3.7252902984619E-9 + 0 + 0 + 4.6566128730774E-10 + 0 + 0 + 5.8207660913467E-11 + 0 + 1.4551915228367E-11 = 119.2639820022310
Таким образом:
1110111.0100001110010100010100110001001001012-10 = 119.2639820022310.