64 bit double precision IEEE 754 binary floating point number 1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1101 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1101 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:
000 1111 0000


The last 52 bits contain the mantissa:
0000 0110 0110 1111 0111 0011 1111 1011 1101 0000 0000 0000 0000

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

The exponent is allways a positive integer.

000 1111 0000(2) =


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


0 + 0 + 0 + 128 + 64 + 32 + 16 + 0 + 0 + 0 + 0 =


128 + 64 + 32 + 16 =


240(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 = 240 - 1023 = -783


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)

0000 0110 0110 1111 0111 0011 1111 1011 1101 0000 0000 0000 0000(2) =

0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 + 1 × 2-24 + 1 × 2-25 + 1 × 2-26 + 1 × 2-27 + 1 × 2-28 + 1 × 2-29 + 0 × 2-30 + 1 × 2-31 + 1 × 2-32 + 1 × 2-33 + 1 × 2-34 + 0 × 2-35 + 1 × 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 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 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 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 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 + 0.000 000 001 862 645 149 230 957 031 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 + 0.000 000 000 116 415 321 826 934 814 453 125 + 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.015 625 + 0.007 812 5 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 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 + 0.000 000 001 862 645 149 230 957 031 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 + 0.000 000 000 116 415 321 826 934 814 453 125 + 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 =


0.025 138 138 749 753 125 011 920 928 955 078 125(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 + 0.025 138 138 749 753 125 011 920 928 955 078 125) × 2-783 =


-1.025 138 138 749 753 125 011 920 928 955 078 125 × 2-783 =


-0

Conclusion:

1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1101 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

-0(10)

More operations of this kind:

1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1100 1111 1111 1111 1111 = ?

1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1101 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)

1 - 000 1111 0000 - 0000 0110 0110 1111 0111 0011 1111 1011 1101 0000 0000 0000 0000 = -0 Oct 25 15:58 UTC (GMT)
0 - 111 1111 1110 - 1001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 140 444 776 161 118 430 291 351 968 030 392 557 313 904 451 479 867 700 994 867 250 904 478 652 973 047 627 447 428 497 908 130 887 516 500 088 968 649 526 060 670 929 506 886 262 986 322 537 055 187 089 159 670 131 166 738 176 160 372 111 109 063 473 511 030 822 721 056 316 410 756 904 805 220 580 049 126 151 494 617 610 021 279 033 867 586 472 333 045 499 958 785 889 437 278 363 152 622 132 518 925 107 200 Oct 25 15:58 UTC (GMT)
0 - 100 0001 0101 - 0110 0000 0101 0110 0010 1011 0110 0111 1011 0100 1010 0111 0111 = 5 772 682.851 275 078 020 989 894 866 943 359 375 Oct 25 15:57 UTC (GMT)
1 - 101 1010 0011 - 0101 0010 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3 582 923 350 843 147 406 319 380 969 537 619 035 869 953 381 666 404 208 522 796 470 387 279 388 500 807 494 372 525 520 583 256 843 873 944 369 175 626 602 009 067 520 Oct 25 15:57 UTC (GMT)
0 - 100 0000 0111 - 1011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 447.999 999 999 999 943 156 581 139 191 985 130 310 058 593 75 Oct 25 15:56 UTC (GMT)
0 - 010 0001 0011 - 0100 0010 0111 0000 1001 1001 1001 1001 1111 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 098 503 320 378 584 092 191 770 838 038 179 114 068 916 165 094 408 482 861 891 636 966 243 527 717 201 748 782 023 375 552 367 483 720 698 707 418 617 235 268 506 534 124 373 746 500 575 186 427 550 428 101 825 1 Oct 25 15:55 UTC (GMT)
0 - 100 0001 1000 - 0110 1100 1100 1111 0110 0101 1110 1001 0111 0011 0001 1100 0111 = 47 816 395.823 825 411 498 546 600 341 796 875 Oct 25 15:52 UTC (GMT)
0 - 100 0001 0010 - 1111 0001 1001 0100 0000 1111 1111 1100 1100 1110 1001 1011 0111 = 1 019 040.499 610 236 729 495 227 336 883 544 921 875 Oct 25 15:52 UTC (GMT)
0 - 100 0000 0111 - 1000 0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 = 386.269 999 999 999 981 810 105 964 541 435 241 699 218 75 Oct 25 15:51 UTC (GMT)
0 - 000 1001 0100 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 007 939 328 826 636 875 658 288 403 126 113 181 566 034 423 2 Oct 25 15:51 UTC (GMT)
0 - 011 1111 0100 - 0110 0011 1101 1111 1111 1101 0011 1111 1100 1111 1111 1001 1110 = 0.000 678 777 614 645 287 774 228 221 699 956 975 498 935 207 724 571 228 027 343 75 Oct 25 15:49 UTC (GMT)
0 - 011 1111 1101 - 1110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0110 0111 0000 = 0.475 000 000 000 000 532 907 051 820 075 139 403 343 200 683 593 75 Oct 25 15:49 UTC (GMT)
0 - 110 1101 0110 - 0101 1110 1010 0101 0010 0101 0010 0000 0000 0000 0000 0000 0000 = 967 018 213 427 061 763 520 025 962 175 627 982 708 296 233 366 986 107 150 260 074 410 028 002 682 267 664 306 928 318 295 066 700 004 120 992 870 261 417 921 173 480 526 771 888 571 518 220 012 794 722 224 656 892 813 715 384 535 670 515 442 843 979 888 053 923 473 717 839 591 890 747 392 Oct 25 15:48 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)