64 bit double precision IEEE 754 binary floating point number 1 - 111 0001 0101 - 1111 0000 0000 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:
1 - 111 0001 0101 - 1111 0000 0000 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:
111 0001 0101


The last 52 bits contain the mantissa:
1111 0000 0000 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):

111 0001 0101(2) =


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


1,024 + 512 + 256 + 0 + 0 + 0 + 16 + 0 + 4 + 0 + 1 =


1,024 + 512 + 256 + 16 + 4 + 1 =


1,813(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 = 1,813 - 1023 = 790

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

1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 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 =


0.5 + 0.25 + 0.125 + 0.062 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 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =


0.5 + 0.25 + 0.125 + 0.062 5 =


0.937 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 + 0.937 5) × 2790 =


-1.937 5 × 2790 =


-12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184

1 - 111 0001 0101 - 1111 0000 0000 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) =


-12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184(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 - 111 0001 0101 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -12 616 482 386 430 388 335 966 516 449 728 942 365 203 094 970 202 218 809 343 505 381 145 334 129 171 545 057 199 975 477 723 277 771 791 432 325 951 930 569 691 317 923 051 617 451 018 753 908 546 280 257 045 223 992 002 254 300 006 857 631 034 134 902 024 328 486 053 479 765 909 511 255 687 815 049 456 788 701 184 Apr 20 19:15 UTC (GMT)
0 - 100 1000 0000 - 1111 1011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 1 349 166 415 721 689 610 997 364 166 184 549 744 640 Apr 20 19:14 UTC (GMT)
0 - 100 0000 0001 - 0010 0010 1000 1111 0101 1100 0010 1000 1111 0101 1100 0010 1000 = 4.539 999 999 999 999 147 348 717 087 879 776 954 650 878 906 25 Apr 20 19:14 UTC (GMT)
0 - 100 0010 1010 - 1101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 15 942 918 602 752 Apr 20 19:12 UTC (GMT)
0 - 100 0001 1000 - 0100 0000 0110 0100 0001 0010 1011 1010 0101 0011 1110 0001 1100 = 41 994 277.455 684 870 481 491 088 867 187 5 Apr 20 19:10 UTC (GMT)
0 - 011 1111 1100 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0100 0000 0000 = 0.166 666 666 666 657 192 763 523 198 664 188 385 009 765 625 Apr 20 19:09 UTC (GMT)
0 - 000 0000 0010 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Apr 20 19:08 UTC (GMT)
0 - 100 0000 0000 - 0010 1111 0010 1011 0000 0010 0000 1100 0100 1001 1011 1010 0100 = 2.368 499 999 999 999 161 559 571 803 081 780 672 073 364 257 812 5 Apr 20 19:08 UTC (GMT)
0 - 000 0000 0000 - 0000 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0 Apr 20 19:06 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0010 0001 1001 0000 0101 0000 = 0 Apr 20 19:04 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 0000 = 0 Apr 20 19:01 UTC (GMT)
0 - 010 1101 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 490 909 346 529 772 655 309 577 195 498 627 564 297 521 551 249 944 956 511 154 911 718 710 525 472 171 585 646 009 788 403 733 195 227 718 357 156 513 187 851 316 791 861 042 471 890 280 751 482 410 896 345 225 310 546 445 986 192 853 894 181 098 439 730 703 830 718 994 140 625 Apr 20 19:01 UTC (GMT)
0 - 100 0001 1000 - 1110 1001 1101 1100 1011 0100 0000 1010 0100 0110 1001 1111 0101 = 64 207 208.080 280 221 998 691 558 837 890 625 Apr 20 19:01 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)