64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 0111 - 1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 0111 - 1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000.

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:
100 0001 0111


The last 52 bits contain the mantissa:
1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000

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

100 0001 0111(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 0 + 4 + 2 + 1 =


1,024 + 16 + 4 + 2 + 1 =


1,047(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 = 1,047 - 1023 = 24

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):

1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0 + 0 + 0.003 906 25 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 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 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 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 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.003 906 25 + 0.001 953 125 + 0.000 488 281 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 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 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 =


0.881 407 803 679 724 821 677 154 977 805 912 494 659 423 828 125(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 + 0.881 407 803 679 724 821 677 154 977 805 912 494 659 423 828 125) × 224 =


1.881 407 803 679 724 821 677 154 977 805 912 494 659 423 828 125 × 224 =


31 564 785.106 420 338 153 839 111 328 125

0 - 100 0001 0111 - 1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


31 564 785.106 420 338 153 839 111 328 125(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 - 100 0001 0111 - 1110 0001 1010 0011 1111 0001 0001 1011 0011 1110 0101 1101 0000 = 31 564 785.106 420 338 153 839 111 328 125 Dec 13 09:05 UTC (GMT)
1 - 111 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -179 769 313 486 231 570 814 527 423 731 704 356 798 070 567 525 844 996 598 917 476 803 157 260 780 028 538 760 589 558 632 766 878 171 540 458 953 514 382 464 234 321 326 889 464 182 768 467 546 703 537 516 986 049 910 576 551 282 076 245 490 090 389 328 944 075 868 508 455 133 942 304 583 236 903 222 948 165 808 559 332 123 348 274 797 826 204 144 723 168 738 177 180 919 299 881 250 404 026 184 124 858 368 Dec 13 09:03 UTC (GMT)
1 - 100 0110 0010 - 1000 0000 0101 0011 1010 1010 0111 1101 0101 0111 1101 1010 1001 = -951 547 117 831 201 185 768 750 972 928 Dec 13 09:03 UTC (GMT)
0 - 001 0000 0011 - 0111 0101 1110 0110 0110 0111 0011 0101 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 015 052 160 026 692 079 701 002 824 353 330 632 500 433 781 380 350 910 208 622 926 515 813 302 761 017 3 Dec 13 09:02 UTC (GMT)
0 - 100 0000 1001 - 0000 1011 1110 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 = 1 071.729 999 999 999 790 816 218 592 226 505 279 541 015 625 Dec 13 09:02 UTC (GMT)
0 - 100 0001 1101 - 0010 1001 0000 0100 0100 1110 0001 0000 0000 0000 0000 0000 0000 = 1 245 778 820 Dec 13 08:59 UTC (GMT)
0 - 100 0001 1011 - 0111 1001 0100 0011 1100 1000 0001 1011 1000 1111 0110 1000 0000 = 395 590 785.722 511 291 503 906 25 Dec 13 08:58 UTC (GMT)
0 - 111 1110 1000 - 1000 1011 1011 1110 1001 0000 0000 0000 0000 0000 0000 0000 0000 = 33 128 399 572 031 336 841 019 584 719 652 978 417 661 853 718 290 464 518 060 467 210 508 338 413 065 654 402 574 105 890 474 313 725 533 085 047 998 298 775 788 818 010 456 149 761 255 984 277 203 906 200 602 262 520 582 941 167 595 304 515 612 589 543 700 177 699 113 255 389 388 099 025 413 044 498 658 850 885 415 276 963 406 745 089 641 157 606 270 925 465 315 406 387 563 514 150 845 402 841 088 Dec 13 08:57 UTC (GMT)
0 - 011 1111 1110 - 0101 0110 0000 0100 0001 1000 1001 0011 0111 0100 0000 0000 0000 = 0.667 999 999 999 665 305 949 747 562 408 447 265 625 Dec 13 08:56 UTC (GMT)
0 - 010 0100 0000 - 0110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 783 507 823 781 784 385 008 671 819 696 767 919 374 061 782 681 257 739 146 242 645 026 470 426 212 709 840 062 720 496 782 514 395 838 304 552 162 787 354 679 547 675 607 846 089 238 139 475 213 279 437 274 496 323 759 185 386 6 Dec 13 08:56 UTC (GMT)
0 - 111 1111 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 = 89 884 656 743 197 525 047 140 706 315 738 565 143 497 707 486 395 790 765 829 872 565 190 867 212 328 285 393 326 272 739 697 662 039 246 980 371 895 305 252 027 664 567 087 448 982 582 048 432 980 144 331 274 023 710 559 094 846 870 458 456 759 076 440 229 419 688 140 367 632 539 377 764 414 858 758 337 425 093 754 912 838 617 204 918 202 750 543 065 721 110 185 179 145 112 732 797 360 752 018 658 951 168 Dec 13 08:56 UTC (GMT)
0 - 000 0000 0111 - 1111 1011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 002 7 Dec 13 08:54 UTC (GMT)
0 - 100 0000 1000 - 1010 1110 0011 1001 1100 1011 1010 1010 0000 0000 0000 0000 0000 = 860.451 527 833 938 598 632 812 5 Dec 13 08:53 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)