64 bit double precision IEEE 754 binary floating point number 0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010
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:
011 1111 1111


The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010

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

The exponent is allways a positive integer.

011 1111 1111(2) =


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


0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =


512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 =


1,023(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 = 1,023 - 1023 = 0


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 0010(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 + 1 × 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 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =


0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =


0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125(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)0 × (1 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125) × 20 =


1.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 × 20 =


1.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125

Conclusion:

0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

1.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125(10)

More operations of this kind:

0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = ?

0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 = ?


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 - 011 1111 1111 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0010 = 1.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 Dec 03 00:22 UTC (GMT)
0 - 011 1110 1110 - 1110 1110 1100 1001 1101 0100 0010 1000 0011 0001 0010 1001 1011 = 0.000 014 745 843 217 042 495 522 351 407 133 410 106 098 381 220 363 080 501 556 396 484 375 Dec 03 00:19 UTC (GMT)
1 - 110 0000 0000 - 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -40 223 423 789 827 785 344 459 245 565 005 891 002 377 463 243 244 034 708 140 657 580 655 742 264 252 929 242 608 574 669 544 922 394 913 008 127 408 329 078 794 684 649 903 310 078 323 844 405 406 990 336 Dec 03 00:19 UTC (GMT)
1 - 100 0011 0101 - 0011 0011 0011 0011 0010 0011 0001 0011 0100 0010 0001 0000 0001 = -21 617 260 897 534 980 Dec 03 00:16 UTC (GMT)
0 - 100 0000 0010 - 1000 0010 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 = 12.066 406 250 000 001 776 356 839 400 250 464 677 810 668 945 312 5 Dec 03 00:15 UTC (GMT)
0 - 011 1111 1011 - 0011 0110 0111 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.075 799 942 016 601 548 622 212 192 185 543 244 704 604 148 864 746 093 75 Dec 03 00:15 UTC (GMT)
0 - 100 0001 1010 - 0011 1100 0101 0001 0100 1101 1011 1011 0110 1001 0110 1010 0011 = 165 841 517.856 618 016 958 236 694 335 937 5 Dec 03 00:15 UTC (GMT)
0 - 100 1001 1000 - 0101 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 = 15 119 905 244 603 772 631 166 018 814 690 801 074 680 365 056 Dec 03 00:13 UTC (GMT)
0 - 100 0001 1000 - 0010 0100 1010 1101 0110 0001 0001 0010 1011 0101 1101 0001 1100 = 38 361 794.146 173 685 789 108 276 367 187 5 Dec 03 00:13 UTC (GMT)
0 - 100 0000 0011 - 0001 0000 1000 1010 0001 0011 0111 1111 0011 1000 1100 0101 0010 = 17.033 709 999 999 992 135 144 566 418 603 062 629 699 707 031 25 Dec 03 00:13 UTC (GMT)
1 - 111 0100 0100 - 1100 1010 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = -1 643 157 112 322 500 695 611 876 334 156 072 310 558 120 273 246 156 266 246 139 141 150 439 082 089 294 147 991 885 214 242 896 807 879 364 699 853 088 951 012 940 681 951 573 057 708 866 693 826 917 815 840 341 304 449 222 613 016 561 235 105 372 517 587 002 687 652 167 425 023 437 059 685 194 381 576 742 903 987 586 855 627 390 976 Dec 03 00:11 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0011 1110 0010 0000 1101 1011 0110 0111 0011 = 0 Dec 03 00:10 UTC (GMT)
0 - 100 0001 0000 - 0111 1101 1101 0101 1001 0101 1100 0010 1000 1010 0000 1000 0001 = 195 499.169 999 364 792 602 136 731 147 766 113 281 25 Dec 03 00:06 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)