Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 011 1111 1110 - 0100 1001 0010 0100 1001 0010 0100 1001 0010 0100 1001 0110 0111 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 011 1111 1110 - 0100 1001 0010 0100 1001 0010 0100 1001 0010 0100 1001 0110 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.
0
The next 11 bits contain the exponent:
011 1111 1110
The last 52 bits contain the mantissa:
0100 1001 0010 0100 1001 0010 0100 1001 0010 0100 1001 0110 0111
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
011 1111 1110(2) =
0 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =
1,022(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 = 1,022 - 1023 = -1
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).
0100 1001 0010 0100 1001 0010 0100 1001 0010 0100 1001 0110 0111(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 0 × 2-30 + 0 × 2-31 + 1 × 2-32 + 0 × 2-33 + 0 × 2-34 + 1 × 2-35 + 0 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 0 × 2-40 + 1 × 2-41 + 0 × 2-42 + 0 × 2-43 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 1 × 2-51 + 1 × 2-52 =
0 + 0.25 + 0 + 0 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0 + 0 + 0.000 488 281 25 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0.000 000 119 209 289 550 781 25 + 0 + 0 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 0 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 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 =
0.25 + 0.031 25 + 0.003 906 25 + 0.000 488 281 25 + 0.000 061 035 156 25 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 119 209 289 550 781 25 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 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 =
0.285 714 285 714 300 464 391 612 877 079 751 342 535 018 920 898 437 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)0 × (1 + 0.285 714 285 714 300 464 391 612 877 079 751 342 535 018 920 898 437 5) × 2-1 =
1.285 714 285 714 300 464 391 612 877 079 751 342 535 018 920 898 437 5 × 2-1 =
0.642 857 142 857 150 232 195 806 438 539 875 671 267 509 460 449 218 75
0 - 011 1111 1110 - 0100 1001 0010 0100 1001 0010 0100 1001 0010 0100 1001 0110 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) = 0.642 857 142 857 150 232 195 806 438 539 875 671 267 509 460 449 218 75(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: