64 bit double precision IEEE 754 binary floating point number 0 - 111 1111 1111 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 111 1111 1111 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111.

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:
111 1111 1111


The last 52 bits contain the mantissa:
0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111

Notice that all the exponent bits are on 1 (set) and the first mantissa bit (the most significant) is on 0 (clear) while there is at least one mantissa bit set on 1.

This is one of the reserved bitpatterns of the special values of: SNaN (Signalling Not a Number).

A SNaN signals an exception when used in operations. A SNaN is a category of NaN (Not A Number). Generally, a NaN is used to represent a value that is not a number.

Conclusion:

0 - 111 1111 1111 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


SNaN, Signalling Not a Number

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 - 111 1111 1111 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = SNaN, Signalling Not a Number Jun 18 08:45 UTC (GMT)
0 - 100 0001 0111 - 0011 0111 0111 1100 0100 0011 1100 0001 0010 1101 0000 0111 0011 = 20 413 507.754 593 323 916 196 823 120 117 187 5 Jun 18 08:44 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 = 0 Jun 18 08:42 UTC (GMT)
0 - 100 0000 1001 - 0010 0000 0001 0000 0000 0000 0000 0000 0000 0000 0001 0010 0000 = 1 152.250 000 000 065 483 618 527 650 833 129 882 812 5 Jun 18 08:42 UTC (GMT)
0 - 011 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Jun 18 08:40 UTC (GMT)
0 - 110 1010 1011 - 0110 1100 0110 0010 1000 0100 0010 1110 1000 0111 1111 1001 1000 = 114 245 040 612 424 147 426 239 380 189 178 347 775 031 085 181 273 423 528 289 397 154 598 883 312 230 586 583 755 551 243 846 433 348 423 146 459 396 856 650 059 194 569 848 918 819 398 791 432 257 063 822 850 038 962 223 214 415 712 848 200 442 462 320 102 088 305 016 832 Jun 18 08:39 UTC (GMT)
0 - 010 0010 0010 - 1011 0110 1101 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 393 008 266 508 794 731 525 728 933 775 240 535 873 800 467 031 775 574 124 341 261 111 007 059 075 752 897 280 735 350 028 478 243 882 116 891 573 871 711 258 841 970 246 313 529 322 187 824 570 646 885 368 226 228 3 Jun 18 08:38 UTC (GMT)
0 - 100 0100 0110 - 0100 0011 1101 1100 0100 0110 0111 0010 0100 0110 1010 1100 0100 = 2 987 085 417 696 381 108 224 Jun 18 08:37 UTC (GMT)
0 - 101 0000 0000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 347 376 267 711 948 586 270 712 955 026 063 723 559 809 953 996 921 692 118 372 752 023 739 388 919 808 Jun 18 08:32 UTC (GMT)
0 - 010 0000 0001 - 1001 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 475 469 221 614 013 179 884 799 038 860 762 032 132 750 060 330 775 939 509 809 293 741 900 247 048 220 041 939 681 149 263 027 272 464 296 364 707 559 551 467 655 061 416 070 613 044 605 993 007 886 495 2 Jun 18 08:30 UTC (GMT)
1 - 011 1111 1101 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.25 Jun 18 08:29 UTC (GMT)
0 - 111 0111 1100 - 0111 0110 1100 1000 1011 1111 1111 1110 1011 1100 0110 1010 0110 = 96 677 945 529 917 390 551 288 397 807 319 303 045 099 594 651 618 823 490 992 596 383 648 930 807 140 694 238 927 521 417 598 986 979 122 814 836 963 129 535 710 333 630 931 114 105 582 871 177 693 347 843 611 893 870 305 618 899 397 312 805 106 717 725 690 280 111 518 771 469 449 036 345 263 661 439 554 884 300 472 282 967 536 574 717 552 643 505 389 568 Jun 18 08:28 UTC (GMT)
0 - 100 0000 0010 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 8 Jun 18 08:27 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)