64 bit double precision IEEE 754 binary floating point number 0 - 010 0101 1000 - 1100 0011 1000 0111 1100 1100 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 - 010 0101 1000 - 1100 0011 1000 0111 1100 1100 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:
010 0101 1000


The last 52 bits contain the mantissa:
1100 0011 1000 0111 1100 1100 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):

010 0101 1000(2) =


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


0 + 512 + 0 + 0 + 64 + 0 + 16 + 8 + 0 + 0 + 0 =


512 + 64 + 16 + 8 =


600(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 = 600 - 1023 = -423

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):

1100 0011 1000 0111 1100 1100 0000 0000 0000 0000 0000 0000 0000(2) =

1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 1 × 2-21 + 1 × 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 + 0 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 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 =


0.5 + 0.25 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 =


0.763 790 845 870 971 679 687 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 + 0.763 790 845 870 971 679 687 5) × 2-423 =


1.763 790 845 870 971 679 687 5 × 2-423 =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 081 425 215 830 864 489 184 715 717 929 132 296 644 478 217 630 699 955 191 233 483 076 582 399 721 653 384 168 884 358 967 665 290 723 235 773 931 145 543 277 090 789 121 999 692 763 411 203 113 290 006 158 624 187 610 559 952 270 427 597 1

0 - 010 0101 1000 - 1100 0011 1000 0111 1100 1100 0000 0000 0000 0000 0000 0000 0000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 081 425 215 830 864 489 184 715 717 929 132 296 644 478 217 630 699 955 191 233 483 076 582 399 721 653 384 168 884 358 967 665 290 723 235 773 931 145 543 277 090 789 121 999 692 763 411 203 113 290 006 158 624 187 610 559 952 270 427 597 1(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 - 010 0101 1000 - 1100 0011 1000 0111 1100 1100 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 081 425 215 830 864 489 184 715 717 929 132 296 644 478 217 630 699 955 191 233 483 076 582 399 721 653 384 168 884 358 967 665 290 723 235 773 931 145 543 277 090 789 121 999 692 763 411 203 113 290 006 158 624 187 610 559 952 270 427 597 1 Oct 20 19:55 UTC (GMT)
1 - 011 0000 0110 - 1011 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 925 865 587 786 061 151 461 156 465 404 489 104 358 868 081 269 290 748 860 285 873 847 595 972 105 221 233 927 646 173 274 320 274 523 002 046 549 991 374 982 129 907 065 143 524 112 684 619 842 283 950 674 755 033 105 611 801 147 460 937 5 Oct 20 19:54 UTC (GMT)
0 - 100 0000 0101 - 0111 1101 0111 0000 1010 0011 1101 0111 0000 1010 0011 1101 0111 = 95.359 999 999 999 999 431 565 811 391 919 851 303 100 585 937 5 Oct 20 19:48 UTC (GMT)
0 - 100 0000 0100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1000 = 32.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 Oct 20 19:47 UTC (GMT)
0 - 000 0000 0010 - 0000 0000 0000 0000 1000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Oct 20 19:45 UTC (GMT)
0 - 100 0001 0101 - 0000 1011 1100 0101 1111 0101 0110 0000 0101 1001 0110 0000 0010 = 4 387 197.344 090 940 430 760 383 605 957 031 25 Oct 20 19:44 UTC (GMT)
0 - 011 0000 0000 - 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 021 050 660 853 042 708 774 379 232 665 575 973 954 595 250 888 313 435 875 995 277 208 535 942 520 208 415 953 805 549 539 713 702 552 241 334 028 993 739 805 017 675 553 682 902 870 065 779 196 706 579 057 263 297 727 331 519 126 892 089 843 75 Oct 20 19:42 UTC (GMT)
1 - 100 0100 0010 - 1000 0010 0101 0011 1010 1010 0111 1101 0101 0111 1101 1010 1000 = -222 702 249 416 229 912 576 Oct 20 19:42 UTC (GMT)
0 - 110 0001 0010 - 0000 0100 1010 1110 0110 1100 0110 0100 0010 1101 1000 0100 0000 = 7 158 098 650 486 605 062 682 343 968 676 472 237 252 701 633 419 710 363 299 580 437 065 553 747 172 154 642 059 767 957 845 356 450 045 175 467 179 225 693 921 233 514 564 618 935 305 306 183 745 279 048 548 352 Oct 20 19:34 UTC (GMT)
1 - 100 0000 1111 - 1000 0001 1100 1101 0110 1110 1001 1110 0001 1011 0000 1000 1010 = -98 765.432 100 000 005 448 237 061 500 549 316 406 25 Oct 20 19:34 UTC (GMT)
0 - 011 1111 1111 - 0101 0100 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 = 1.329 999 999 999 999 849 009 668 650 978 710 502 386 093 139 648 437 5 Oct 20 19:34 UTC (GMT)
0 - 011 1100 1001 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 055 511 151 231 257 827 021 181 583 404 541 015 625 Oct 20 19:32 UTC (GMT)
0 - 000 0000 0000 - 1000 0000 0010 0001 1101 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Oct 20 19:31 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)