64 bit double precision IEEE 754 binary floating point number 0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100
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 1010 1101


The last 52 bits contain the mantissa:
1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100

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

The exponent is allways a positive integer.

100 1010 1101(2) =


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


1,024 + 0 + 0 + 128 + 0 + 32 + 0 + 8 + 4 + 0 + 1 =


1,024 + 128 + 32 + 8 + 4 + 1 =


1,197(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 = 1,197 - 1023 = 174


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)

1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100(2) =

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


0.5 + 0.25 + 0.125 + 0 + 0 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 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 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0 + 0 + 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 + 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 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.015 625 + 0.001 953 125 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 000 931 322 574 615 478 515 625 + 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 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =


0.893 197 336 512 076 667 247 583 827 702 328 562 736 511 230 468 75(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 + 0.893 197 336 512 076 667 247 583 827 702 328 562 736 511 230 468 75) × 2174 =


1.893 197 336 512 076 667 247 583 827 702 328 562 736 511 230 468 75 × 2174 =


45 333 069 940 373 986 391 476 432 776 396 710 987 832 706 377 711 616

Conclusion:

0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

45 333 069 940 373 986 391 476 432 776 396 710 987 832 706 377 711 616(10)

More operations of this kind:

0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1011 = ?

0 - 100 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1101 = ?


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 1010 1101 - 1110 0100 1010 1000 1001 0100 1010 0101 0011 0001 1001 0100 1100 = 45 333 069 940 373 986 391 476 432 776 396 710 987 832 706 377 711 616 Nov 29 10:58 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0110 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 783 617 053 277 285 150 384 315 771 169 034 374 120 556 995 277 983 742 896 572 023 463 073 590 388 307 203 155 305 734 043 509 323 149 886 531 361 970 692 558 414 696 226 256 234 845 090 873 344 Nov 29 10:57 UTC (GMT)
1 - 110 0000 1010 - 1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111 = -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 Nov 29 10:54 UTC (GMT)
1 - 110 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -26 815 615 859 885 194 199 148 049 996 411 692 254 958 731 641 184 786 755 447 122 887 443 528 060 147 093 953 603 748 596 333 806 855 380 063 716 372 972 101 707 507 765 623 893 139 892 867 298 012 168 192 Nov 29 10:54 UTC (GMT)
1 - 011 0101 1111 - 1110 0110 1010 1000 1010 1000 0011 0001 0011 0111 1001 0000 0110 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 300 724 510 356 224 390 628 969 214 893 515 984 853 953 183 562 172 292 101 423 353 397 512 767 979 302 587 117 662 479 517 762 864 888 111 735 241 226 221 899 187 714 958 756 600 935 885 217 040 777 206 420 898 437 5 Nov 29 10:53 UTC (GMT)
0 - 100 1000 0100 - 0011 0100 1011 1101 0100 0111 1010 1110 0001 0100 1110 0100 0010 = 13 132 320 660 840 473 421 625 157 393 889 051 541 504 Nov 29 10:51 UTC (GMT)
0 - 011 1111 1111 - 0101 0010 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 1.323 242 187 500 000 222 044 604 925 031 308 084 726 333 618 164 062 5 Nov 29 10:49 UTC (GMT)
0 - 100 1000 1010 - 0101 0100 0011 0010 0100 1010 0110 1010 0101 0101 0101 0101 0111 = 926 102 820 316 576 866 998 025 703 517 700 935 909 376 Nov 29 10:49 UTC (GMT)
0 - 100 0000 1100 - 1000 0001 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 12 328 Nov 29 10:49 UTC (GMT)
0 - 100 0010 0000 - 0010 1001 0001 1110 1011 1000 0101 0000 0000 0000 0000 0000 0000 = 9 969 692 832 Nov 29 10:48 UTC (GMT)
0 - 000 1000 0100 - 1111 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 119 724 881 240 077 600 750 442 461 668 824 285 003 Nov 29 10:48 UTC (GMT)
0 - 111 1111 1011 - 1001 1001 1001 1001 0110 1010 1100 1010 0010 0000 0010 0010 0101 = 17 976 900 000 000 001 131 263 126 597 574 217 263 396 481 902 578 175 138 403 913 437 777 028 790 527 439 603 364 502 930 460 677 390 792 873 005 190 202 715 684 097 701 052 772 925 098 274 606 265 671 862 282 068 943 709 466 085 560 282 706 513 077 345 426 748 148 655 344 345 059 351 783 647 273 667 856 596 377 154 335 538 285 408 147 908 746 020 334 699 795 216 859 612 853 175 701 809 318 359 744 905 216 Nov 29 10:48 UTC (GMT)
0 - 111 0101 1111 - 0101 1100 1011 1000 1010 0001 1001 0000 0001 1100 1111 0100 0011 = 167 553 945 093 568 456 688 041 326 566 310 003 516 777 753 540 350 940 093 576 762 106 843 966 223 625 769 614 599 281 914 211 527 360 035 871 033 102 812 049 734 349 936 601 880 404 913 022 106 793 252 997 769 142 941 807 906 161 494 242 956 091 399 686 321 844 840 604 386 639 934 034 831 002 804 996 703 557 520 501 691 125 640 321 524 826 112 Nov 29 10:48 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)