64 bit double precision IEEE 754 binary floating point number 0 - 100 0101 0100 - 0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 0001 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0101 0100 - 0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 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:
100 0101 0100


The last 52 bits contain the mantissa:
0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 0001

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

100 0101 0100(2) =


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


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


1,024 + 64 + 16 + 4 =


1,108(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 = 1,108 - 1023 = 85

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):

0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 0001(2) =

0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 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 + 1 × 2-52 =


0 + 0.25 + 0 + 0.062 5 + 0 + 0.015 625 + 0 + 0 + 0 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 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 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.25 + 0.062 5 + 0.015 625 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.329 411 506 652 832 253 294 604 925 031 308 084 726 333 618 164 062 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 + 0.329 411 506 652 832 253 294 604 925 031 308 084 726 333 618 164 062 5) × 285 =


1.329 411 506 652 832 253 294 604 925 031 308 084 726 333 618 164 062 5 × 285 =


51 429 116 649 132 616 885 207 040

0 - 100 0101 0100 - 0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 0001
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


51 429 116 649 132 616 885 207 040(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 - 100 0101 0100 - 0101 0100 0101 0100 0101 0000 0000 0000 0000 0000 0000 0000 0001 = 51 429 116 649 132 616 885 207 040 Jul 08 03:38 UTC (GMT)
0 - 011 1111 1110 - 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 0.999 999 999 999 999 888 977 697 537 484 345 957 636 833 190 917 968 75 Jul 08 03:32 UTC (GMT)
0 - 100 0001 1011 - 1000 0101 1101 1010 0101 0110 1011 0000 1010 0001 0100 1010 1110 = 408 790 379.039 378 046 989 440 917 968 75 Jul 08 03:10 UTC (GMT)
0 - 011 1101 1001 - 0000 0011 1101 0101 0101 0101 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 003 692 453 677 886 509 133 543 313 510 017 469 525 337 219 238 281 25 Jul 08 03:10 UTC (GMT)
0 - 100 0001 0110 - 0000 1001 0110 0100 1101 0101 1010 1100 0110 0111 1110 1111 0100 = 8 696 426.836 730 457 842 350 006 103 515 625 Jul 08 03:01 UTC (GMT)
1 - 100 0000 0110 - 1110 0111 1000 0111 0010 0111 0000 0000 0000 0000 0000 0000 0000 = -243.763 969 421 386 718 75 Jul 08 03:00 UTC (GMT)
0 - 111 1111 0000 - 0000 0000 0000 0000 0011 1111 0000 0111 1010 1110 0001 0100 0000 = 5 486 144 679 507 641 409 794 658 457 588 129 439 150 701 058 632 804 178 184 486 391 214 724 452 103 028 833 746 334 646 986 852 996 358 520 264 707 948 180 188 214 062 997 583 640 142 575 503 341 412 730 959 843 431 667 192 571 145 468 440 716 699 784 488 125 980 885 784 371 204 217 586 454 332 443 102 722 343 336 736 430 982 210 463 158 645 248 953 055 692 406 538 673 218 051 253 832 742 244 188 160 Jul 08 02:51 UTC (GMT)
1 - 100 1100 1100 - 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100 0000 = -92 559 631 349 317 693 721 053 283 428 559 145 536 795 620 813 586 901 720 104 960 Jul 08 02:49 UTC (GMT)
0 - 100 0000 0000 - 1001 1101 1001 1001 1100 0000 0000 0000 0000 0000 0000 0000 0000 = 3.231 254 577 636 718 75 Jul 08 02:39 UTC (GMT)
1 - 000 1000 1000 - 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 120 587 030 737 169 006 233 390 313 169 074 730 780 8 Jul 08 02:26 UTC (GMT)
1 - 100 0100 1010 - 1000 0100 0000 0101 0100 0001 1000 0001 0101 1000 0100 1000 0000 = -57 261 723 413 273 175 392 256 Jul 08 02:19 UTC (GMT)
0 - 100 0001 1010 - 1111 0011 0101 0111 0001 1100 0111 1100 0000 1000 0010 0010 1000 = 261 798 115.875 993 013 381 958 007 812 5 Jul 08 02:04 UTC (GMT)
0 - 011 1010 0000 - 0100 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 032 448 067 703 036 149 652 287 116 376 670 855 462 729 936 569 132 860 313 402 488 827 705 383 300 781 25 Jul 08 01:50 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)