64 bit double precision IEEE 754 binary floating point number 1 - 011 1111 1110 - 1111 1111 1111 1111 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:
1 - 011 1111 1110 - 1111 1111 1111 1111 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:
011 1111 1110


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

011 1111 1110(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =


512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =


1,022(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 = 1,022 - 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):

1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000(2) =

1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 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 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 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 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 =


0.999 984 741 210 937 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)1 × (1 + 0.999 984 741 210 937 5) × 2-1 =


-1.999 984 741 210 937 5 × 2-1 =


-0.999 992 370 605 468 75

1 - 011 1111 1110 - 1111 1111 1111 1111 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) =


-0.999 992 370 605 468 75(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)

1 - 011 1111 1110 - 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.999 992 370 605 468 75 Mar 25 08:37 UTC (GMT)
1 - 100 0000 0101 - 1011 1110 0111 1011 0110 0111 1010 1100 1011 0101 1110 0001 1011 = -111.620 512 674 904 986 738 511 070 143 431 425 094 604 492 187 5 Mar 25 08:35 UTC (GMT)
1 - 011 1110 0000 - 0111 1000 0010 1010 0111 0100 1001 1110 0000 0111 1111 1111 1010 = -0.000 000 000 684 241 680 462 814 526 618 670 319 627 491 666 070 184 805 903 409 142 047 166 824 340 820 312 5 Mar 25 08:32 UTC (GMT)
1 - 111 1111 1111 - 1111 1111 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = QNaN, Quiet Not a Number Mar 25 08:32 UTC (GMT)
0 - 110 0000 0000 - 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 31 843 543 833 613 668 111 488 309 370 738 884 552 763 493 823 906 934 272 093 458 428 839 189 571 424 674 069 904 451 458 146 395 640 763 825 663 192 904 370 777 665 471 678 373 103 622 779 916 389 449 728 Mar 25 08:30 UTC (GMT)
0 - 100 1010 0011 - 1100 0000 0111 0010 0100 0000 0000 0000 0000 0000 0000 0000 0000 = 40 962 811 607 488 195 688 805 902 609 058 325 755 199 043 403 776 Mar 25 08:30 UTC (GMT)
0 - 110 0000 0000 - 1110 1101 1100 1100 1100 1100 0000 0000 0000 0000 0000 0000 0000 = 51 724 807 751 063 469 163 770 000 339 488 765 440 385 572 120 128 293 341 926 993 490 197 177 134 715 841 225 867 482 689 619 627 314 829 139 640 844 942 230 731 094 663 059 971 788 634 994 400 099 827 712 Mar 25 08:28 UTC (GMT)
0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011 = 200 767 359.662 657 588 720 321 655 273 437 5 Mar 25 08:28 UTC (GMT)
0 - 100 0100 1000 - 1111 0111 0001 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 18 560 306 841 913 326 043 136 Mar 25 08:27 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 Mar 25 08:26 UTC (GMT)
0 - 110 0000 0000 - 1011 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 46 298 836 758 083 030 609 466 555 071 929 562 408 952 185 099 233 108 382 451 673 110 351 716 416 347 716 904 268 972 185 857 588 398 742 141 260 300 209 644 354 368 876 585 002 999 346 278 694 224 134 144 Mar 25 08:25 UTC (GMT)
0 - 100 0010 0011 - 0001 1011 0011 0101 0100 0111 0010 0000 1010 1010 0100 0000 0000 = 76 023 099 914.640 625 Mar 25 08:25 UTC (GMT)
0 - 011 1111 0100 - 0110 0011 0100 0111 1010 1111 0100 0111 0100 0011 0110 1100 1100 = 0.000 677 642 857 142 857 126 667 712 108 769 592 305 179 685 354 232 788 085 937 5 Mar 25 08:22 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)