Converter: 64 bit double precision IEEE 754 binary floating point numbers converted to decimal base ten (double)

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Exponent: empty
Mantissa: empty

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 - 100 0000 1000 - 1000 0000 1000 1011 1000 0011 1101 0011 0000 0011 1111 1100 0000 = -769.089 960 457 749 839 406 460 523 605 346 679 687 5 Apr 18 22:18 UTC (GMT)
1 - 000 0000 0000 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0 Apr 18 22:15 UTC (GMT)
1 - 011 1111 1111 - 1000 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -1.558 593 75 Apr 18 22:15 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0110 1110 1101 0010 0000 0000 0000 0000 0000 0000 0000 = 489 782 495 174 892 015 340 676 163 009 664 577 454 979 428 592 820 463 518 889 232 013 930 708 222 419 668 849 885 574 999 037 720 880 722 750 968 558 598 937 053 311 644 609 996 490 617 558 925 312 Apr 18 22:13 UTC (GMT)
0 - 100 0000 0101 - 0000 1001 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 = 66.299 999 999 999 997 157 829 056 959 599 256 515 502 929 687 5 Apr 18 22:10 UTC (GMT)
1 - 100 0000 0101 - 0010 0100 0100 0100 0100 0100 0000 0000 0000 0000 0000 0000 0000 = -73.066 665 649 414 062 5 Apr 18 22:08 UTC (GMT)
0 - 010 0001 1010 - 1001 1001 1001 1001 1001 0001 1000 1001 1010 0001 0000 1001 1101 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 016 016 659 950 902 757 451 227 393 948 138 226 018 957 485 595 364 372 853 156 079 619 270 555 272 551 907 850 297 847 680 394 084 513 963 940 788 647 634 088 451 188 696 228 266 764 886 478 950 387 397 349 871 519 1 Apr 18 22:07 UTC (GMT)
0 - 010 0000 0000 - 0101 1110 0101 0110 0000 0100 0001 1000 1001 0011 0111 0100 1100 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 204 134 785 812 949 665 983 131 479 013 393 770 355 397 214 556 874 910 371 973 600 036 212 343 069 648 679 394 008 829 691 220 527 959 644 764 773 200 004 338 788 639 899 323 469 471 261 632 763 253 156 1 Apr 18 22:06 UTC (GMT)
0 - 100 0000 1001 - 1110 1010 1011 0000 1111 1010 1010 1100 1101 1001 1110 1000 0010 = 1 962.765 299 999 999 569 990 905 001 759 529 113 769 531 25 Apr 18 22:02 UTC (GMT)
0 - 100 0000 0000 - 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 3.25 Apr 18 21:59 UTC (GMT)
0 - 100 0000 1000 - 1000 1010 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 789.5 Apr 18 21:47 UTC (GMT)
0 - 111 1111 1111 - 1111 1111 1111 1110 1111 1111 1111 1111 1111 1111 1111 1111 1111 = QNaN, Quiet Not a Number Apr 18 21:46 UTC (GMT)
1 - 000 0000 0000 - 1011 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0 Apr 18 21:45 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)