64 bit double precision IEEE 754 binary floating point number 0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100.

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:
000 0000 0011


The last 52 bits contain the mantissa:
0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100

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

000 0000 0011(2) =


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


0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =


2 + 1 =


3(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 = 3 - 1023 = -1020

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):

0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100(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 + 1 × 2-49 + 1 × 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =


0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =


0.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75(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 + 0.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75) × 2-1020 =


1.000 000 000 000 002 664 535 259 100 375 697 016 716 003 417 968 75 × 2-1020 =


0

0 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


0(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 - 000 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1100 = 0 May 20 16:51 UTC (GMT)
1 - 100 0000 0011 - 1101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -29 May 20 16:51 UTC (GMT)
0 - 011 1111 1111 - 0000 0000 0000 0000 0000 0001 0000 0000 0000 0000 0000 0000 0000 = 1.000 000 059 604 644 775 390 625 May 20 16:50 UTC (GMT)
0 - 011 1111 1110 - 1111 1111 1111 1010 0101 1111 1110 0011 0110 0100 0100 0110 0001 = 0.999 957 081 325 316 576 858 597 272 803 308 442 234 992 980 957 031 25 May 20 16:47 UTC (GMT)
0 - 100 0001 1000 - 0110 1100 1011 0110 0100 1100 0011 0010 1110 0100 0000 1111 1100 = 47 803 544.397 584 885 358 810 424 804 687 5 May 20 16:47 UTC (GMT)
1 - 011 1111 0000 - 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 = -0.000 053 405 761 718 750 020 328 790 734 103 208 137 639 740 016 311 407 089 233 398 437 5 May 20 16:36 UTC (GMT)
1 - 110 0000 1010 - 1000 0000 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0111 = -41 190 988 142 913 239 044 198 239 122 595 952 661 378 719 922 849 970 769 254 134 179 287 431 982 982 205 047 315 736 699 622 780 248 558 126 710 111 878 490 060 461 495 498 601 092 991 093 046 588 001 484 800 May 20 16:32 UTC (GMT)
0 - 100 0011 0111 - 1011 0110 1001 1011 0100 1011 1010 1100 1101 0000 0101 1111 0001 = 123 456 789 123 456 784 May 20 16:26 UTC (GMT)
0 - 100 0000 1001 - 0101 0110 0101 0110 1010 1100 1000 0000 0000 0000 0000 0000 0000 = 1 369.354 278 564 453 125 May 20 16:26 UTC (GMT)
1 - 100 0000 0011 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -16 May 20 16:26 UTC (GMT)
1 - 011 1000 0001 - 1110 0000 1001 0100 0000 0000 0000 0000 0000 0100 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 022 067 065 276 068 824 344 366 864 336 481 199 213 765 376 181 764 592 893 150 941 173 457 722 816 921 101 812 457 153 834 579 060 003 306 949 511 170 387 268 066 406 25 May 20 16:25 UTC (GMT)
0 - 100 0001 0100 - 1101 1000 1000 0000 0000 1000 1000 0000 0000 1000 1000 0000 0000 = 3 870 721.062 516 212 463 378 906 25 May 20 16:23 UTC (GMT)
0 - 010 1011 1001 - 0011 0111 0101 0111 0011 0110 0100 0110 1111 0110 1111 0100 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 008 896 461 560 931 691 470 448 519 261 433 399 650 176 248 902 394 983 948 918 504 339 967 287 726 371 518 872 071 675 329 128 948 212 724 788 182 720 361 222 358 209 819 255 577 572 048 489 928 592 926 954 938 166 606 555 841 393 536 959 903 875 103 772 975 860 198 528 661 250 7 May 20 16:22 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)