64 bit double precision IEEE 754 binary floating point number 1 - 111 0110 0000 - 0000 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 0110 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000
to decimal system (base ten)

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 0110 0000


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

2. Convert the exponent from binary (base 2) to decimal (base 10):

The exponent is allways a positive integer.

111 0110 0000(2) =


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


1,024 + 512 + 256 + 0 + 64 + 32 + 0 + 0 + 0 + 0 + 0 =


1,024 + 512 + 256 + 64 + 32 =


1,888(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 = 1,888 - 1023 = 865


4. Convert the mantissa from binary (base 2) to decimal (base 10):

Mantissa represents the number's fractional part (the excess beyond the number's integer part, comma delimited)

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

0 × 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 + 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 + 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 + 0 + 0 =


0(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) × 2865 =


-1 × 2865 =


-246 006 311 446 272 417 135 694 895 366 447 328 831 463 738 361 430 131 889 861 407 236 509 911 043 906 984 606 020 737 387 080 298 687 645 418 100 644 428 599 105 378 407 753 391 907 201 399 550 988 776 412 284 181 771 799 458 695 654 166 637 769 167 516 870 901 097 035 133 833 253 825 096 549 816 225 533 764 062 867 857 067 136 321 933 279 232

Conclusion:

1 - 111 0110 0000 - 0000 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) =

-246 006 311 446 272 417 135 694 895 366 447 328 831 463 738 361 430 131 889 861 407 236 509 911 043 906 984 606 020 737 387 080 298 687 645 418 100 644 428 599 105 378 407 753 391 907 201 399 550 988 776 412 284 181 771 799 458 695 654 166 637 769 167 516 870 901 097 035 133 833 253 825 096 549 816 225 533 764 062 867 857 067 136 321 933 279 232(10)

More operations of this kind:

1 - 111 0110 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ?


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 0110 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -246 006 311 446 272 417 135 694 895 366 447 328 831 463 738 361 430 131 889 861 407 236 509 911 043 906 984 606 020 737 387 080 298 687 645 418 100 644 428 599 105 378 407 753 391 907 201 399 550 988 776 412 284 181 771 799 458 695 654 166 637 769 167 516 870 901 097 035 133 833 253 825 096 549 816 225 533 764 062 867 857 067 136 321 933 279 232 Dec 02 22:22 UTC (GMT)
0 - 100 0001 1001 - 1010 1000 0001 0010 0001 0110 0000 0000 0000 0000 0000 0000 0001 = 111 167 576.000 000 014 901 161 193 847 656 25 Dec 02 22:20 UTC (GMT)
1 - 100 0000 1000 - 1000 0000 0100 0000 1000 0000 0000 1111 1000 0000 0001 1100 0000 = -768.503 908 097 794 919 740 408 658 981 323 242 187 5 Dec 02 22:20 UTC (GMT)
0 - 111 1111 1111 - 0000 0000 0000 0000 0011 1110 0010 0000 1101 1011 0110 0111 0010 = SNaN, Signalling Not a Number Dec 02 22:18 UTC (GMT)
0 - 001 1111 1000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 874 024 304 437 524 357 110 997 580 376 049 571 246 740 327 974 618 233 908 809 055 097 354 464 598 750 285 942 327 326 389 015 307 969 962 858 642 002 456 120 529 908 345 372 959 181 825 360 273 473 6 Dec 02 22:16 UTC (GMT)
1 - 100 1111 0011 - 0101 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = -38 208 067 775 832 573 681 442 279 226 927 140 576 348 185 685 268 092 174 090 623 031 021 928 448 Dec 02 22:16 UTC (GMT)
0 - 100 0000 0000 - 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 3.562 500 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 Dec 02 22:15 UTC (GMT)
0 - 100 0001 1000 - 1000 0011 1101 0011 1110 1111 1111 0101 0001 0001 1110 1000 0010 = 50 833 375.914 608 970 284 461 975 097 656 25 Dec 02 22:15 UTC (GMT)
1 - 100 0010 1010 - 1001 0100 0011 0010 1010 1010 1000 0101 0101 0111 1101 0101 0101 = -13 888 134 589 118.666 015 625 Dec 02 22:15 UTC (GMT)
0 - 000 0100 0000 - 0000 0000 0010 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 205 377 152 691 919 394 2 Dec 02 22:14 UTC (GMT)
0 - 100 0010 0011 - 1111 1011 1000 1110 0000 1000 0110 1011 1101 1111 0100 1100 0001 = 136 245 708 477.956 069 946 289 062 5 Dec 02 22:12 UTC (GMT)
0 - 100 0001 1001 - 0000 1001 1110 1000 0011 0101 0101 1001 1010 1101 1111 0110 1100 = 69 705 941.401 242 911 815 643 310 546 875 Dec 02 22:11 UTC (GMT)
0 - 100 0001 0110 - 0111 0011 0010 1011 0100 0011 0110 0001 0001 1010 0000 1101 0001 = 12 162 465.689 651 878 550 648 689 270 019 531 25 Dec 02 22:10 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)