64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 0111 - 1111 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 0111 - 1111 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000.

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 0001 0111


The last 52 bits contain the mantissa:
1111 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000

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

100 0001 0111(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 0 + 4 + 2 + 1 =


1,024 + 16 + 4 + 2 + 1 =


1,047(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 = 1,047 - 1023 = 24

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 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000(2) =

1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 1 × 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 + 1 × 2-24 + 1 × 2-25 + 1 × 2-26 + 0 × 2-27 + 0 × 2-28 + 1 × 2-29 + 0 × 2-30 + 0 × 2-31 + 1 × 2-32 + 0 × 2-33 + 1 × 2-34 + 0 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 1 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 1 × 2-46 + 1 × 2-47 + 0 × 2-48 + 1 × 2-49 + 0 × 2-50 + 0 × 2-51 + 0 × 2-52 =


0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0 + 0 + 0.003 906 25 + 0 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 0 + 0 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0 + 0 + 0 + 0 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 + 0.25 + 0.125 + 0.062 5 + 0.003 906 25 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 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 =


0.942 230 331 078 940 608 335 869 910 661 131 143 569 946 289 062 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 + 0.942 230 331 078 940 608 335 869 910 661 131 143 569 946 289 062 5) × 224 =


1.942 230 331 078 940 608 335 869 910 661 131 143 569 946 289 062 5 × 224 =


32 585 217.786 262 899 637 222 290 039 062 5

0 - 100 0001 0111 - 1111 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


32 585 217.786 262 899 637 222 290 039 062 5(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 0001 0111 - 1111 0001 0011 0110 0000 0001 1100 1001 0100 1000 1000 0110 1000 = 32 585 217.786 262 899 637 222 290 039 062 5 Jan 29 19:50 UTC (GMT)
0 - 011 1111 1111 - 0100 0000 1000 0011 1100 1010 1111 1011 0011 1011 0111 0101 0010 = 1.252 010 999 999 999 985 021 759 130 177 088 081 836 700 439 453 125 Jan 29 19:50 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 1000 1010 1000 1000 0000 0000 0000 0000 = 0 Jan 29 19:49 UTC (GMT)
0 - 111 1100 0000 - 0010 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 22 574 106 284 219 529 119 724 827 171 914 010 423 240 634 869 350 935 404 033 861 441 903 431 928 220 666 713 914 491 148 703 250 327 033 616 466 114 161 485 284 340 967 695 568 104 568 831 040 885 599 232 335 086 123 734 664 798 854 349 133 850 412 507 083 879 727 183 332 720 697 937 873 279 569 371 722 042 295 287 593 644 908 626 589 254 336 878 267 911 795 017 007 285 957 296 128 Jan 29 19:48 UTC (GMT)
0 - 100 0000 0111 - 1000 0010 0100 0101 0001 1110 1011 1000 0101 0001 1110 1011 1001 = 386.270 000 000 000 038 653 524 825 349 450 111 389 160 156 25 Jan 29 19:48 UTC (GMT)
0 - 000 1000 1010 - 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 004 845 781 754 539 109 216 144 390 543 433 836 673 647 2 Jan 29 19:47 UTC (GMT)
1 - 100 0000 0100 - 1101 1111 0011 1101 1011 0011 1101 0110 0101 1010 1111 0000 1101 = -59.905 128 168 726 243 131 914 088 735 356 926 918 029 785 156 25 Jan 29 19:46 UTC (GMT)
0 - 000 0000 0001 - 0000 0100 0000 0000 0000 0000 0000 0000 0000 1110 0000 0011 1111 = 0 Jan 29 19:44 UTC (GMT)
0 - 011 1111 1111 - 0100 0000 1000 0011 1100 1010 1111 1011 0011 1011 0111 0101 0010 = 1.252 010 999 999 999 985 021 759 130 177 088 081 836 700 439 453 125 Jan 29 19:42 UTC (GMT)
0 - 100 0001 0111 - 1001 1010 1010 1111 0100 0010 1100 0100 0100 0010 1000 0010 1111 = 26 914 626.766 639 884 561 300 277 709 960 937 5 Jan 29 19:42 UTC (GMT)
1 - 110 0111 1111 - 0000 0000 0000 0000 0000 0100 0000 0000 0000 0000 0000 0000 0000 = -4 562 441 705 392 804 507 380 189 721 976 938 344 348 977 278 255 265 035 540 571 230 360 870 201 635 167 981 531 305 941 122 176 038 968 219 359 458 941 195 143 522 784 151 169 921 510 997 087 630 419 130 702 222 891 455 326 707 620 968 186 769 225 809 920 Jan 29 19:40 UTC (GMT)
1 - 100 0000 0111 - 1010 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -420 Jan 29 19:34 UTC (GMT)
1 - 100 0000 1011 - 0011 1011 0011 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -5 043.25 Jan 29 19:34 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)