64 bit double precision IEEE 754 binary floating point number 1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101
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:
101 1101 0100


The last 52 bits contain the mantissa:
1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101

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

The exponent is allways a positive integer.

101 1101 0100(2) =


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


1,024 + 0 + 256 + 128 + 64 + 0 + 16 + 0 + 4 + 0 + 0 =


1,024 + 256 + 128 + 64 + 16 + 4 =


1,492(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 = 1,492 - 1023 = 469


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)

1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101(2) =

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


0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 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 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 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 + 0 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0 + 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 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 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 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.003 906 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 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 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 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 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 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 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.972 861 108 945 749 508 691 847 040 608 990 937 471 389 770 507 812 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 + 0.972 861 108 945 749 508 691 847 040 608 990 937 471 389 770 507 812 5) × 2469 =


-1.972 861 108 945 749 508 691 847 040 608 990 937 471 389 770 507 812 5 × 2469 =


-3 007 214 993 567 477 705 783 069 177 247 056 296 518 426 424 952 941 838 845 452 541 670 846 705 497 336 854 736 269 169 018 418 246 319 194 581 547 434 596 690 062 858 282 996 556 890 112

Conclusion:

1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-3 007 214 993 567 477 705 783 069 177 247 056 296 518 426 424 952 941 838 845 452 541 670 846 705 497 336 854 736 269 169 018 418 246 319 194 581 547 434 596 690 062 858 282 996 556 890 112(10)

More operations of this kind:

1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1100 = ?

1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1110 = ?


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)

1 - 101 1101 0100 - 1111 1001 0000 1101 0110 1100 1111 0110 0111 1000 1110 0111 1101 = -3 007 214 993 567 477 705 783 069 177 247 056 296 518 426 424 952 941 838 845 452 541 670 846 705 497 336 854 736 269 169 018 418 246 319 194 581 547 434 596 690 062 858 282 996 556 890 112 Nov 25 20:00 UTC (GMT)
0 - 111 1111 1100 - 1111 1111 1111 1111 1111 1111 1111 1111 1101 1111 1111 1111 1111 = 44 942 328 370 903 895 751 003 518 945 042 528 988 909 730 620 132 266 720 709 878 473 589 760 514 605 309 097 419 767 513 115 304 516 752 487 871 845 639 883 076 675 334 880 821 157 212 080 073 905 133 367 431 650 504 084 327 361 060 148 760 768 116 065 475 456 078 263 473 771 633 638 093 656 211 671 097 071 518 133 023 999 736 974 334 401 441 504 247 413 419 496 555 504 456 938 595 521 881 372 861 923 328 Nov 25 19:56 UTC (GMT)
0 - 011 1111 1101 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 = 0.399 999 999 999 999 966 693 309 261 245 303 787 291 049 957 275 390 625 Nov 25 19:55 UTC (GMT)
0 - 011 0010 1001 - 1101 1100 0000 0100 0101 0000 0000 0000 0000 0000 0000 0000 0010 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 070 625 784 476 514 276 224 462 381 065 628 500 322 768 383 798 189 439 424 784 400 273 112 451 038 205 554 793 591 355 717 345 977 475 727 935 030 535 650 507 224 942 275 216 192 227 643 738 919 563 135 758 496 619 925 894 677 294 309 076 387 435 197 830 200 195 312 5 Nov 25 19:54 UTC (GMT)
0 - 010 0000 0000 - 0101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 195 781 444 194 005 427 011 387 839 530 902 013 231 132 377 783 260 680 974 627 356 246 664 807 608 090 605 504 574 590 873 011 229 838 239 679 585 465 697 663 152 084 112 499 664 194 837 761 826 776 792 1 Nov 25 19:52 UTC (GMT)
0 - 100 0000 0000 - 1001 0010 0001 1111 1011 0101 0100 0100 0100 0010 1101 0001 1000 = 3.141 592 653 589 793 115 997 963 468 544 185 161 590 576 171 875 Nov 25 19:51 UTC (GMT)
1 - 100 0000 0100 - 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 1111 0100 = -49.919 999 999 999 987 494 447 850 622 236 728 668 212 890 625 Nov 25 19:51 UTC (GMT)
0 - 100 0000 0010 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 10 Nov 25 19:48 UTC (GMT)
1 - 110 0101 1010 - 0010 1000 1101 0111 1100 0011 1011 0101 0101 0101 0101 0100 0011 = -38 492 310 771 490 926 843 302 835 977 141 538 850 783 260 154 795 274 230 932 574 521 784 971 799 388 363 860 040 400 422 816 268 124 464 986 746 653 425 227 788 850 982 341 686 469 046 447 310 425 887 608 923 976 214 035 497 634 477 637 632 Nov 25 19:47 UTC (GMT)
0 - 011 1111 1000 - 1011 0011 0011 0011 0011 0010 0000 0000 0000 0000 0000 0000 0000 = 0.013 281 249 441 206 455 230 712 890 625 Nov 25 19:47 UTC (GMT)
0 - 000 1000 1010 - 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 845 781 754 539 108 355 360 634 351 784 601 868 892 Nov 25 19:46 UTC (GMT)
1 - 000 0000 0000 - 0010 1000 0010 0010 0100 1001 0100 1010 1001 0101 0010 1010 1000 = -0 Nov 25 19:46 UTC (GMT)
0 - 011 0000 0000 - 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 021 050 660 853 042 708 774 379 232 665 575 973 954 595 250 888 313 435 875 995 277 208 535 942 520 208 415 953 805 549 539 713 702 552 241 334 028 993 739 805 017 675 553 682 902 870 065 779 196 706 579 057 263 297 727 331 519 126 892 089 843 75 Nov 25 19:46 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)