Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 110 0000 1010 - 1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 110 0000 1010 - 1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111: 64 bit double precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
1
The next 11 bits contain the exponent:
110 0000 1010
The last 52 bits contain the mantissa:
1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
110 0000 1010(2) =
1 × 210 + 1 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
1,024 + 512 + 0 + 0 + 0 + 0 + 0 + 8 + 0 + 2 + 0 =
1,024 + 512 + 8 + 2 =
1,546(10)
3. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 1,546 - 1023 = 523
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 1 × 2-24 + 1 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 1 × 2-32 + 0 × 2-33 + 1 × 2-34 + 0 × 2-35 + 1 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 1 × 2-51 + 1 × 2-52 =
0.5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.500 080 198 362 668 815 192 932 925 128 843 635 320 663 452 148 437 5(10)
5. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)1 × (1 + 0.500 080 198 362 668 815 192 932 925 128 843 635 320 663 452 148 437 5) × 2523 =
-1.500 080 198 362 668 815 192 932 925 128 843 635 320 663 452 148 437 5 × 2523 =
-41 190 988 142 913 239 044 198 239 122 595 952 661 378 719 922 849 970 769 254 134 179 287 431 982 982 205 047 315 736 699 622 780 248 558 126 710 111 878 490 060 461 495 498 601 092 991 093 046 588 001 484 800
1 - 110 0000 1010 - 1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = -41 190 988 142 913 239 044 198 239 122 595 952 661 378 719 922 849 970 769 254 134 179 287 431 982 982 205 047 315 736 699 622 780 248 558 126 710 111 878 490 060 461 495 498 601 092 991 093 046 588 001 484 800(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 64 bit double precision IEEE 754 binary floating point standard representation numbers: