64 bit double precision IEEE 754 binary floating point number 0 - 010 1111 1101 - 0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 010 1111 1101 - 0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010.

1. Identify the elements that make up the binary representation of the number:

First bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.


The next 11 bits contain the exponent:
010 1111 1101


The last 52 bits contain the mantissa:
0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010

2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):

010 1111 1101(2) =


0 × 210 + 1 × 29 + 0 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =


0 + 512 + 0 + 128 + 64 + 32 + 16 + 8 + 4 + 0 + 1 =


512 + 128 + 64 + 32 + 16 + 8 + 4 + 1 =


765(10)

3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1023, that is due to the 11 bit excess/bias notation:

Exponent adjusted = 765 - 1023 = -258

4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10):

0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010(2) =

0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 + 1 × 2-24 + 0 × 2-25 + 0 × 2-26 + 1 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 1 × 2-32 + 0 × 2-33 + 1 × 2-34 + 0 × 2-35 + 1 × 2-36 + 1 × 2-37 + 0 × 2-38 + 1 × 2-39 + 1 × 2-40 + 1 × 2-41 + 0 × 2-42 + 1 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 0 × 2-47 + 1 × 2-48 + 0 × 2-49 + 0 × 2-50 + 1 × 2-51 + 0 × 2-52 =


0 + 0 + 0 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =


0.062 5 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 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 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =


0.077 685 069 143 317 964 091 011 162 963 695 824 146 270 751 953 125(10)

Conclusion:

5. Put all the numbers into expression to calculate the double precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =


(-1)0 × (1 + 0.077 685 069 143 317 964 091 011 162 963 695 824 146 270 751 953 125) × 2-258 =


1.077 685 069 143 317 964 091 011 162 963 695 824 146 270 751 953 125 × 2-258 =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 326 767 476 607 576 238 122 278 783 522 771 951 414 698 381 569 088 872 935 363 506 146 488 903 576 694 298 498 970 084 868 468 328 718 808 730 531 469 766 228 107 927 957 516 105 094 440 727 258 938 706 010 092 769 099 029 333 264 994 653 391 788 859 315 980 619 868 469 148 059 375 584 125 518 798 828 1

0 - 010 1111 1101 - 0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 326 767 476 607 576 238 122 278 783 522 771 951 414 698 381 569 088 872 935 363 506 146 488 903 576 694 298 498 970 084 868 468 328 718 808 730 531 469 766 228 107 927 957 516 105 094 440 727 258 938 706 010 092 769 099 029 333 264 994 653 391 788 859 315 980 619 868 469 148 059 375 584 125 518 798 828 1(10)

Convert 64 bit double precision IEEE 754 floating point standard binary numbers to base ten decimal system (double)

64 bit double precision IEEE 754 binary floating point standard representation of numbers requires three building blocks: sign (it takes 1 bit and it's either 0 for positive or 1 for negative numbers), exponent (11 bits), mantissa (52 bits)

Latest 64 bit double precision IEEE 754 floating point binary standard numbers converted to decimal base ten (double)

0 - 010 1111 1101 - 0001 0011 1110 0011 0010 1011 0010 1111 0101 1011 1010 1001 0010 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 326 767 476 607 576 238 122 278 783 522 771 951 414 698 381 569 088 872 935 363 506 146 488 903 576 694 298 498 970 084 868 468 328 718 808 730 531 469 766 228 107 927 957 516 105 094 440 727 258 938 706 010 092 769 099 029 333 264 994 653 391 788 859 315 980 619 868 469 148 059 375 584 125 518 798 828 1 Aug 13 17:14 UTC (GMT)
0 - 100 0000 0010 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 12 Aug 13 17:10 UTC (GMT)
0 - 110 0001 1011 - 0110 0100 0001 1111 0111 1000 0000 1001 1101 0011 0101 1011 1110 = 5 006 769 817 473 996 012 145 722 500 606 509 168 925 913 256 616 727 389 712 761 258 200 488 377 702 273 333 491 563 777 409 703 055 555 856 355 959 251 677 782 113 227 775 725 752 264 211 181 946 982 493 010 788 352 Aug 13 17:08 UTC (GMT)
1 - 100 1010 1010 - 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 1010 = -4 988 592 255 422 814 851 059 245 634 032 955 072 489 896 319 909 888 Aug 13 17:05 UTC (GMT)
0 - 100 0000 0010 - 0000 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 = 8.139 999 999 999 998 792 077 349 207 829 684 019 088 745 117 187 5 Aug 13 17:02 UTC (GMT)
0 - 110 1011 0110 - 0001 1000 0000 0101 0011 1110 1001 1111 0001 0100 0010 1011 1110 = 179 802 956 013 607 253 124 717 795 920 976 247 283 349 719 650 634 676 445 466 346 409 752 801 893 860 644 619 708 121 738 845 530 537 554 741 355 153 734 086 732 400 874 736 840 890 086 579 048 697 033 309 719 865 835 782 794 776 518 839 278 356 895 399 337 418 322 507 464 704 Aug 13 17:01 UTC (GMT)
0 - 110 1011 0110 - 0001 1000 0000 0101 0011 1110 1001 1111 0001 0100 0010 1011 1110 = 179 802 956 013 607 253 124 717 795 920 976 247 283 349 719 650 634 676 445 466 346 409 752 801 893 860 644 619 708 121 738 845 530 537 554 741 355 153 734 086 732 400 874 736 840 890 086 579 048 697 033 309 719 865 835 782 794 776 518 839 278 356 895 399 337 418 322 507 464 704 Aug 13 17:01 UTC (GMT)
1 - 011 1000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 005 877 471 754 111 437 539 843 682 686 111 228 389 093 327 783 860 437 607 543 758 531 392 086 297 273 635 864 257 812 5 Aug 13 16:41 UTC (GMT)
0 - 100 0000 1001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 1 024 Aug 13 16:40 UTC (GMT)
0 - 101 0111 1000 - 1010 1100 0000 1000 0011 0001 0010 0110 1110 1001 1000 0000 0000 = 514 688 705 940 410 723 816 992 583 622 862 906 376 836 083 775 428 322 859 314 903 866 331 890 602 035 686 883 147 983 517 743 043 629 263 028 748 288 Aug 13 16:36 UTC (GMT)
1 - 100 0000 1000 - 0111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -752 Aug 13 16:32 UTC (GMT)
0 - 100 0000 0000 - 1001 0010 0001 1111 1010 0000 0000 0000 0000 0000 0000 0000 0000 = 3.141 590 118 408 203 125 Aug 13 16:32 UTC (GMT)
1 - 100 1111 1111 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -173 688 133 855 974 293 135 356 477 513 031 861 779 904 976 998 460 846 059 186 376 011 869 694 459 904 Aug 13 16:27 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

How to convert numbers from 64 bit double precision IEEE 754 binary floating point standard to decimal system in base 10

Follow the steps below to convert a number from 64 bit double precision IEEE 754 binary floating point representation to base 10 decimal system:

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent.
    The last 52 bits contain the mantissa.
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10).
  • 3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1,023, that is due to the 11 bit excess/bias notation.
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10).
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted)

Example: convert the number 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 from 64 bit double precision IEEE 754 binary floating point system to base ten decimal (double):

  • 1. Identify the elements that make up the binary representation of the number:
    First bit (leftmost) indicates the sign, 1 = negative, 0 = pozitive.
    The next 11 bits contain the exponent: 100 0011 1101
    The last 52 bits contain the mantissa:
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000
  • 2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):
    100 0011 1101(2) =
    1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
    1,024 + 0 + 0 + 0 + 0 + 32 + 16 + 8 + 4 + 0 + 1 =
    1,024 + 32 + 16 + 8 + 4 + 1 =
    1,085(10)
  • 3. Adjust the exponent, subtract the excess bits, 2(11 - 1) - 1 = 1,023, that is due to the 11 bit excess/bias notation:
    Exponent adjusted = 1,085 - 1,023 = 62
  • 4. Convert the mantissa, that represents the number's fractional part (the excess beyond the number's integer part, comma delimited), from binary (base 2) to decimal (base 10):
    1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000(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 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 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 + 0 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 1 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 1 × 2-45 + 0 × 2-46 + 1 × 2-47 + 0 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =
    0.5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 + 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 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0 =
    0.5 + 0.000 488 281 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 =
    0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5(10)
  • 5. Put all the numbers into expression to calculate the double precision floating point decimal value:
    (-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =
    (-1)1 × (1 + 0.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5) × 262 =
    -1.500 507 372 900 793 612 302 550 172 898 918 390 274 047 851 562 5 × 262 =
    -6 919 868 872 153 800 704(10)
  • 1 - 100 0011 1101 - 1000 0000 0010 0001 0100 0000 0100 1110 0000 0100 0000 1010 1000 converted from 64 bit double precision IEEE 754 binary floating point representation to a decimal number (float) in decimal system (in base 10) = -6 919 868 872 153 800 704(10)