64 bit double precision IEEE 754 binary floating point number 0 - 110 0101 0110 - 1100 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 110 0101 0110 - 1100 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001.

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:
110 0101 0110


The last 52 bits contain the mantissa:
1100 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001

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

110 0101 0110(2) =


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


1,024 + 512 + 0 + 0 + 64 + 0 + 16 + 0 + 4 + 2 + 0 =


1,024 + 512 + 64 + 16 + 4 + 2 =


1,622(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 = 1,622 - 1023 = 599

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 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001(2) =

1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 1 × 2-9 + 0 × 2-10 + 0 × 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 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 + 1 × 2-24 + 0 × 2-25 + 0 × 2-26 + 1 × 2-27 + 1 × 2-28 + 0 × 2-29 + 0 × 2-30 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 1 × 2-39 + 1 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =


0.5 + 0.25 + 0 + 0 + 0 + 0.015 625 + 0.007 812 5 + 0 + 0.001 953 125 + 0 + 0 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.25 + 0.015 625 + 0.007 812 5 + 0.001 953 125 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 001 818 989 403 545 856 475 830 078 125 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.775 727 700 910 476 380 968 816 556 560 341 268 777 847 290 039 062 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 + 0.775 727 700 910 476 380 968 816 556 560 341 268 777 847 290 039 062 5) × 2599 =


1.775 727 700 910 476 380 968 816 556 560 341 268 777 847 290 039 062 5 × 2599 =


3 684 204 870 510 636 558 931 979 970 881 447 794 220 096 953 796 727 533 268 637 489 160 945 120 033 697 995 116 093 422 234 431 107 393 634 820 732 689 918 921 340 846 997 331 637 005 419 926 392 810 872 188 403 887 425 705 830 716 014 592

0 - 110 0101 0110 - 1100 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


3 684 204 870 510 636 558 931 979 970 881 447 794 220 096 953 796 727 533 268 637 489 160 945 120 033 697 995 116 093 422 234 431 107 393 634 820 732 689 918 921 340 846 997 331 637 005 419 926 392 810 872 188 403 887 425 705 830 716 014 592(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 - 110 0101 0110 - 1100 0110 1001 0110 0001 0111 0011 0010 0000 0011 0000 0011 0001 = 3 684 204 870 510 636 558 931 979 970 881 447 794 220 096 953 796 727 533 268 637 489 160 945 120 033 697 995 116 093 422 234 431 107 393 634 820 732 689 918 921 340 846 997 331 637 005 419 926 392 810 872 188 403 887 425 705 830 716 014 592 Oct 20 18:45 UTC (GMT)
0 - 100 0001 0111 - 0001 1110 1110 0000 0000 0011 1101 1110 0000 0010 0001 0000 1001 = 18 800 643.867 219 004 780 054 092 407 226 562 5 Oct 20 18:44 UTC (GMT)
0 - 011 1100 1011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 Oct 20 18:43 UTC (GMT)
0 - 011 1111 0100 - 0110 0011 0100 0100 0001 0110 1110 1101 1010 0000 1111 1100 1010 = 0.000 677 616 071 428 571 416 571 107 199 899 870 465 742 424 130 439 758 300 781 25 Oct 20 18:42 UTC (GMT)
1 - 100 0001 1011 - 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -411 041 792 Oct 20 18:42 UTC (GMT)
0 - 100 0000 0100 - 0011 0111 0010 1110 0111 0000 1110 1001 0111 0111 0010 0100 0000 = 38.897 676 299 999 602 633 761 242 032 051 086 425 781 25 Oct 20 18:41 UTC (GMT)
0 - 100 0000 1001 - 1110 1010 1011 0000 1111 1010 1010 1100 1101 1001 1110 1000 0100 = 1 962.765 300 000 000 024 738 255 888 223 648 071 289 062 5 Oct 20 18:39 UTC (GMT)
0 - 011 1111 1100 - 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1011 = 0.200 000 000 000 000 038 857 805 861 880 478 914 827 108 383 178 710 937 5 Oct 20 18:36 UTC (GMT)
1 - 011 1111 1110 - 0111 1111 1110 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.749 755 859 375 Oct 20 18:35 UTC (GMT)
0 - 100 0001 1000 - 0000 0000 0000 0000 0000 0000 0010 0000 0000 0000 0000 0000 0000 = 33 554 432.25 Oct 20 18:35 UTC (GMT)
0 - 100 0001 1000 - 0011 0000 1110 0010 0001 0110 0000 1111 0101 1000 0001 0001 1111 = 39 961 644.119 875 185 191 631 317 138 671 875 Oct 20 18:34 UTC (GMT)
0 - 110 1111 1101 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 388 129 523 075 177 233 787 244 872 115 625 638 814 221 504 279 174 152 784 763 009 506 512 738 171 594 221 582 719 602 207 161 619 487 621 932 674 282 768 301 542 895 011 028 703 597 861 071 818 760 295 284 801 113 744 005 212 476 387 566 321 407 899 611 206 315 749 798 429 117 187 723 211 713 454 014 464 Oct 20 18:32 UTC (GMT)
1 - 100 0000 0100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -32 Oct 20 18:30 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)