64 bit double precision IEEE 754 binary floating point number 0 - 100 0000 0101 - 0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0000 0101 - 0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000
to decimal system (base ten)

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 0000 0101


The last 52 bits contain the mantissa:
0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000

2. Convert the exponent from binary (base 2) to decimal (base 10):

The exponent is allways a positive integer.

100 0000 0101(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 4 + 0 + 1 =


1,024 + 4 + 1 =


1,029(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 = 1,029 - 1023 = 6


4. Convert the mantissa from binary (base 2) to decimal (base 10):

Mantissa represents the number's fractional part (the excess beyond the number's integer part, comma delimited)

0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

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


0 + 0 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.125 + 0.062 5 + 0.031 25 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 007 629 394 531 25 =


0.225 776 672 363 281 25(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)0 × (1 + 0.225 776 672 363 281 25) × 26 =


1.225 776 672 363 281 25 × 26 =


78.449 707 031 25

Conclusion:

0 - 100 0000 0101 - 0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

78.449 707 031 25(10)

More operations of this kind:

0 - 100 0000 0101 - 0011 1001 1100 1100 0111 1111 1111 1111 1111 1111 1111 1111 1111 = ?

0 - 100 0000 0101 - 0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0001 = ?


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 0000 0101 - 0011 1001 1100 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000 = 78.449 707 031 25 Oct 30 05:24 UTC (GMT)
0 - 011 1111 1111 - 1100 1100 1100 1100 1011 1111 1111 1111 1111 1111 1111 1111 1111 = 1.799 999 237 060 546 652 955 395 074 968 691 915 273 666 381 835 937 5 Oct 30 05:24 UTC (GMT)
1 - 101 1111 0011 - 1011 1110 0101 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -5 707 655 205 467 344 301 392 281 762 679 904 652 124 253 023 333 250 979 974 280 658 126 907 387 924 396 930 438 913 790 750 418 346 262 961 175 406 120 481 021 616 490 963 000 161 481 724 627 255 296 Oct 30 05:23 UTC (GMT)
1 - 100 0000 1000 - 1111 1111 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1 023 Oct 30 05:23 UTC (GMT)
0 - 100 1011 1000 - 0010 0110 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 56 510 773 069 429 640 762 928 305 156 136 820 317 953 567 313 937 563 648 Oct 30 05:22 UTC (GMT)
0 - 111 1111 0000 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 10 972 248 137 587 376 148 347 621 077 374 533 495 975 986 787 466 125 280 695 646 777 536 453 905 031 038 742 711 765 053 269 462 779 024 686 215 424 461 820 326 801 838 799 405 772 874 051 974 286 104 584 776 980 585 362 338 335 087 661 467 901 024 739 315 434 318 146 268 013 546 283 238 722 955 518 978 769 885 776 326 423 544 206 225 453 259 530 317 576 216 929 759 577 593 986 892 724 855 113 777 152 Oct 30 05:22 UTC (GMT)
0 - 111 1111 1010 - 0011 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 6 857 655 085 992 110 854 069 920 313 984 011 587 592 990 794 915 415 087 640 002 485 570 246 727 199 591 183 956 469 624 420 453 492 016 605 906 672 340 139 681 197 729 828 430 809 879 030 129 647 807 087 874 518 123 375 887 507 830 669 487 747 239 917 530 801 890 676 577 949 743 989 492 442 411 135 211 237 865 948 125 489 320 265 325 565 745 719 386 987 302 675 092 257 679 607 575 811 627 564 400 640 000 Oct 30 05:21 UTC (GMT)
1 - 011 1111 1111 - 1000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1.558 593 75 Oct 30 05:21 UTC (GMT)
0 - 100 0000 0000 - 0000 1111 0000 1111 0000 1111 0000 1111 0000 0000 0000 0000 0000 = 2.117 647 058 796 137 571 334 838 867 187 5 Oct 30 05:20 UTC (GMT)
1 - 111 0111 0111 - 1101 1001 0000 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -3 813 294 632 466 379 025 314 516 738 496 847 484 050 210 176 795 440 336 850 536 946 561 252 358 046 198 150 003 982 160 443 187 744 372 905 843 700 415 151 959 260 502 118 908 256 084 797 682 046 792 798 204 414 228 143 275 775 421 390 117 291 064 918 209 243 643 541 258 542 102 580 251 924 891 467 389 138 881 127 994 277 488 420 958 429 671 130 660 864 Oct 30 05:20 UTC (GMT)
0 - 100 0000 1000 - 0000 1010 0011 0000 1000 0011 0001 0010 0110 1110 1001 0111 0110 = 532.378 999 999 999 678 038 875 572 383 403 778 076 171 875 Oct 30 05:19 UTC (GMT)
0 - 000 0001 0000 - 0110 0000 0000 0000 0000 0000 0010 0101 1110 0101 1101 1100 0101 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 002 5 Oct 30 05:18 UTC (GMT)
0 - 110 1111 1101 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 388 129 523 075 177 233 787 244 872 115 625 638 814 221 504 279 174 152 784 763 009 506 512 738 171 594 221 582 719 602 207 161 619 487 621 932 674 282 768 301 542 895 011 028 703 597 861 071 818 760 295 284 801 113 744 005 212 476 387 566 321 407 899 611 206 315 749 798 429 117 187 723 211 713 454 014 464 Oct 30 05:18 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)