64 bit double precision IEEE 754 binary floating point number 0 - 111 1111 1111 - 0000 0001 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 - 0000 0001 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:
0000 0001 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 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 - 0000 0001 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) =


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 - 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = SNaN, Signalling Not a Number Oct 16 20:02 UTC (GMT)
0 - 100 0011 0110 - 1011 1010 1111 1110 0010 0001 0001 1000 0110 0000 1001 0111 0001 = 62 345 678 901 234 568 Oct 16 20:00 UTC (GMT)
0 - 100 0000 0101 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 96.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 Oct 16 20:00 UTC (GMT)
0 - 111 1111 0100 - 0011 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 107 150 860 718 626 732 094 842 504 906 000 181 056 140 481 170 553 360 744 375 038 837 035 105 112 493 612 249 319 837 881 569 585 812 759 467 291 755 314 682 518 714 528 569 231 404 359 845 775 746 985 748 039 345 677 748 242 309 854 210 746 050 623 711 418 779 541 821 530 464 749 835 819 412 673 987 675 591 655 439 460 770 629 145 711 964 776 865 421 676 604 298 316 526 243 868 372 056 680 693 760 000 Oct 16 19:54 UTC (GMT)
1 - 100 0010 1111 - 0000 0000 0101 1100 1001 0110 0000 0000 0000 0000 0000 0000 0000 = -281 872 630 284 288 Oct 16 19:53 UTC (GMT)
1 - 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 Oct 16 19:51 UTC (GMT)
1 - 100 0000 0000 - 1110 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3.812 5 Oct 16 19:49 UTC (GMT)
1 - 111 1110 1100 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -514 324 131 449 408 314 055 244 023 548 800 869 069 474 309 618 656 131 573 000 186 417 768 504 539 969 338 796 735 221 831 534 011 901 245 443 000 425 510 476 089 829 737 132 310 740 927 259 723 585 531 590 588 859 253 191 563 087 300 211 581 042 993 814 810 141 800 743 346 230 799 211 933 180 835 140 842 839 946 109 411 699 019 899 417 430 928 954 024 047 700 631 919 325 970 568 185 872 067 330 048 Oct 16 19:49 UTC (GMT)
0 - 100 0000 0010 - 1000 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 = 12.339 999 999 999 999 857 891 452 847 979 962 825 775 146 484 375 Oct 16 19:49 UTC (GMT)
0 - 100 0000 0101 - 1110 1101 1101 0010 1111 0001 1010 1001 1111 1011 1110 0111 0110 = 123.455 999 999 999 988 858 689 903 281 629 085 540 771 484 375 Oct 16 19:48 UTC (GMT)
1 - 100 0001 0100 - 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3 211 264 Oct 16 19:47 UTC (GMT)
1 - 011 1111 1001 - 1100 1101 0111 1111 1110 1000 1110 1110 0110 1011 1000 0011 1001 = -0.028 167 703 125 000 029 956 703 784 250 748 867 634 683 847 427 368 164 062 5 Oct 16 19:46 UTC (GMT)
1 - 000 0000 0000 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0 Oct 16 19:44 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)