64 bit double precision IEEE 754 binary floating point number 0 - 111 1111 1111 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 111 1111 1111 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000.

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:
1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

Notice that all the exponent bits are on 1 (set) and the first mantissa bit (the most significant) is on 1 (set).

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

A QNaN is generated by an operation when the result is not mathematically defined. A QNaN 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 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


QNaN, Quiet 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 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = QNaN, Quiet Not a Number Feb 18 18:55 UTC (GMT)
1 - 001 1100 0010 - 0010 0101 1100 0010 0001 0100 0000 0010 0111 1110 0001 0100 1000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 037 116 105 126 320 520 716 997 705 386 996 021 902 392 624 251 194 940 405 986 526 493 037 202 773 199 688 637 785 653 371 582 238 897 358 503 705 934 546 402 684 459 385 144 137 8 Feb 18 18:52 UTC (GMT)
0 - 011 1111 1110 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.5 Feb 18 18:50 UTC (GMT)
0 - 100 0000 0000 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1010 = 3.200 000 000 000 000 177 635 683 940 025 046 467 781 066 894 531 25 Feb 18 18:47 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 Feb 18 18:46 UTC (GMT)
0 - 100 0000 1000 - 0001 1011 1000 0000 0000 0000 0000 0000 0000 0000 0001 0101 1100 = 567.000 000 000 039 563 019 527 122 378 349 304 199 218 75 Feb 18 18:39 UTC (GMT)
0 - 100 0111 1111 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 425 352 958 651 173 079 329 218 259 289 710 264 320 Feb 18 18:37 UTC (GMT)
0 - 100 0000 1010 - 1001 0010 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 3 223.999 999 999 999 545 252 649 113 535 881 042 480 468 75 Feb 18 18:35 UTC (GMT)
0 - 011 1111 1011 - 0101 1100 0010 1000 1111 0000 0001 0000 0000 0000 0000 0000 1010 = 0.084 999 978 775 158 663 291 122 707 050 817 552 953 958 511 352 539 062 5 Feb 18 18:34 UTC (GMT)
0 - 100 0001 0111 - 0011 0001 1011 1111 0101 0110 1101 0000 1001 0110 1101 0001 0101 = 20 037 462.814 801 294 356 584 548 950 195 312 5 Feb 18 18:34 UTC (GMT)
0 - 111 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 179 769 313 486 231 570 814 527 423 731 704 356 798 070 567 525 844 996 598 917 476 803 157 260 780 028 538 760 589 558 632 766 878 171 540 458 953 514 382 464 234 321 326 889 464 182 768 467 546 703 537 516 986 049 910 576 551 282 076 245 490 090 389 328 944 075 868 508 455 133 942 304 583 236 903 222 948 165 808 559 332 123 348 274 797 826 204 144 723 168 738 177 180 919 299 881 250 404 026 184 124 858 368 Feb 18 18:33 UTC (GMT)
1 - 100 0000 0001 - 1010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -6.625 Feb 18 18:33 UTC (GMT)
0 - 011 1111 1100 - 0001 1101 1000 1110 0111 1001 1010 1001 0010 1111 1000 0000 0000 = 0.139 431 906 200 059 074 762 975 797 057 151 794 433 593 75 Feb 18 18:29 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)