64 bit double precision IEEE 754 binary floating point number 0 - 100 0000 0000 - 0000 0010 0100 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 - 100 0000 0000 - 0000 0010 0100 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:
100 0000 0000


The last 52 bits contain the mantissa:
0000 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000

2. Convert the exponent, that is allways a positive integer, from binary (base 2) to decimal (base 10):

100 0000 0000(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


1,024 =


1,024(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 = 1,024 - 1023 = 1

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):

0000 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 + 0 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 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 + 0.007 812 5 + 0 + 0 + 0.000 976 562 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.007 812 5 + 0.000 976 562 5 =


0.008 789 062 5(10)

Conclusion:

5. Put all the numbers into expression to calculate the double precision floating point decimal value:

(-1)Sign × (1 + Mantissa) × 2(Exponent adjusted) =


(-1)0 × (1 + 0.008 789 062 5) × 21 =


1.008 789 062 5 × 21 =


2.017 578 125

0 - 100 0000 0000 - 0000 0010 0100 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) =


2.017 578 125(10)

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 - 100 0000 0000 - 0000 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 2.017 578 125 Aug 25 12:10 UTC (GMT)
0 - 011 1111 1110 - 1111 0111 1111 1111 1111 0101 1110 1101 1011 0111 1111 1111 1111 = 0.984 374 699 848 558 409 875 920 460 763 154 551 386 833 190 917 968 75 Aug 25 12:10 UTC (GMT)
0 - 011 1000 0001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 0.000 000 000 000 000 000 000 000 000 000 000 000 011 754 943 508 222 877 689 809 152 571 632 267 462 304 299 270 339 897 495 553 220 290 370 791 698 001 853 916 404 922 838 994 490 889 158 450 741 547 312 645 707 279 443 740 844 726 562 5 Aug 25 12:09 UTC (GMT)
0 - 000 0011 1000 - 0000 0000 0000 0011 1111 1000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 801 715 891 630 412 7 Aug 25 12:07 UTC (GMT)
0 - 011 1111 1100 - 1000 0101 1110 0110 1111 1001 0101 1000 1000 0111 1111 1111 1110 = 0.190 381 954 206 713 943 644 757 591 755 478 642 880 916 595 458 984 375 Aug 25 12:06 UTC (GMT)
0 - 110 1011 1110 - 1011 1110 1011 1110 1011 1110 1011 1110 1011 1110 1011 1110 1011 = 73 435 622 782 400 813 686 143 151 113 110 365 575 435 840 418 095 255 275 297 506 064 069 659 137 818 951 812 398 040 764 533 185 524 715 064 856 092 106 547 425 013 440 122 109 769 664 608 675 607 646 651 085 489 240 444 353 012 735 852 259 223 372 053 716 078 873 457 780 064 256 Aug 25 12:05 UTC (GMT)
0 - 111 1110 1111 - 0001 0011 1110 0010 0001 0000 0000 0000 0000 0000 0000 0000 0000 = 2 956 110 697 672 861 912 148 868 234 042 080 239 172 770 949 973 271 412 311 436 725 134 519 003 269 157 574 051 261 114 341 250 455 355 884 154 068 053 355 263 568 120 982 183 036 227 062 984 471 979 531 104 545 343 865 528 211 187 854 630 483 883 975 940 095 581 091 565 927 519 170 293 533 819 619 666 289 283 586 541 336 465 918 831 787 043 487 771 806 804 964 828 387 354 407 163 236 972 025 085 952 Aug 25 12:05 UTC (GMT)
0 - 100 0001 1000 - 1010 0110 1011 1001 0011 1010 0001 0011 0100 1001 0111 1110 0001 = 55 407 220.150 680 311 024 188 995 361 328 125 Aug 25 12:05 UTC (GMT)
1 - 100 0000 0101 - 0110 1101 1000 1110 0000 1100 0001 1000 0100 0110 1000 0001 0010 = -91.388 718 013 098 952 042 128 075 845 539 569 854 736 328 125 Aug 25 12:04 UTC (GMT)
1 - 111 1111 1111 - 1111 1111 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = QNaN, Quiet Not a Number Aug 25 12:02 UTC (GMT)
1 - 100 0000 0100 - 1000 0111 0000 0101 1001 0101 0100 0110 1000 0001 0101 1101 0001 = -48.877 726 126 501 904 957 422 084 407 880 902 290 344 238 281 25 Aug 25 12:01 UTC (GMT)
1 - 111 1000 0000 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = -2 113 178 124 542 660 516 189 557 339 372 302 052 878 033 334 815 932 256 526 868 239 262 724 567 858 187 515 233 196 359 848 647 004 865 560 168 702 317 211 680 903 191 193 919 328 450 083 989 269 961 554 447 542 240 071 001 947 834 191 993 408 774 294 680 672 825 063 561 256 483 617 856 618 302 092 112 309 707 397 302 279 575 951 919 786 371 116 097 863 680 Aug 25 12:01 UTC (GMT)
0 - 011 0010 1001 - 1111 1110 0000 0101 0101 0101 0101 0010 1001 0000 0001 1000 0001 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 075 670 896 454 648 527 203 394 053 232 346 692 781 048 251 906 715 810 909 955 934 785 019 259 960 720 967 446 946 423 227 758 977 291 867 986 905 720 675 578 864 154 954 189 387 977 330 775 135 011 000 794 672 342 620 317 273 542 696 057 120 338 082 313 537 597 656 25 Aug 25 12:00 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)