64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 1000 - 0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 0001 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1000 - 0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 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 0001 1000


The last 52 bits contain the mantissa:
0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 0001

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

100 0001 1000(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 0 + 0 =


1,024 + 16 + 8 =


1,048(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 = 1,048 - 1023 = 25

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):

0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 0001(2) =

0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 0 × 2-10 + 1 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 + 0 × 2-24 + 1 × 2-25 + 0 × 2-26 + 1 × 2-27 + 1 × 2-28 + 1 × 2-29 + 1 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 1 × 2-34 + 1 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 1 × 2-40 + 1 × 2-41 + 1 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 0 × 2-48 + 0 × 2-49 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =


0 + 0.25 + 0 + 0 + 0.031 25 + 0 + 0 + 0 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0.000 244 140 625 + 0 + 0 + 0.000 030 517 578 125 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 + 0 + 0.000 000 029 802 322 387 695 312 5 + 0 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 0 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0 + 0 + 0 + 0 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0 + 0 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 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 + 0.031 25 + 0.001 953 125 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 030 517 578 125 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 454 747 350 886 464 118 957 519 531 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.283 978 148 452 441 070 404 006 495 664 361 864 328 384 399 414 062 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 + 0.283 978 148 452 441 070 404 006 495 664 361 864 328 384 399 414 062 5) × 225 =


1.283 978 148 452 441 070 404 006 495 664 361 864 328 384 399 414 062 5 × 225 =


43 083 157.471 733 339 130 878 448 486 328 125

0 - 100 0001 1000 - 0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 0001
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


43 083 157.471 733 339 130 878 448 486 328 125(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 0001 1000 - 0100 1000 1011 0010 1100 1010 1011 1100 0110 0001 1100 0010 0001 = 43 083 157.471 733 339 130 878 448 486 328 125 Jul 17 23:28 UTC (GMT)
0 - 101 0000 0100 - 0000 0001 0100 0111 0110 1111 1111 1111 1111 1111 1111 1111 1110 = 3 723 859 879 138 864 519 055 087 990 284 755 196 544 297 328 221 455 801 969 975 244 450 500 869 881 856 Jul 17 23:27 UTC (GMT)
0 - 110 0000 0111 - 0101 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 4 612 285 927 900 253 402 253 464 599 382 811 067 852 901 842 283 783 321 936 905 136 640 286 826 345 300 160 019 844 758 569 414 779 125 370 959 216 151 201 493 691 335 687 309 620 061 573 175 258 092 929 024 Jul 17 23:25 UTC (GMT)
0 - 111 1111 1111 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 = SNaN, Signalling Not a Number Jul 17 23:24 UTC (GMT)
0 - 000 0000 0000 - 1101 1100 1000 0100 0110 1011 0101 1011 0000 0000 0000 0000 0000 = 0 Jul 17 23:24 UTC (GMT)
0 - 100 0010 1100 - 1101 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 63 771 674 411 008 Jul 17 23:20 UTC (GMT)
0 - 010 0101 1000 - 1100 0011 1000 0111 1100 1100 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 081 425 215 830 864 489 184 715 717 929 132 296 644 478 217 630 699 955 191 233 483 076 582 399 721 653 384 168 884 358 967 665 290 723 235 773 931 145 543 277 090 789 121 999 692 763 411 203 113 290 006 158 624 187 610 559 952 270 427 597 1 Jul 17 23:20 UTC (GMT)
0 - 001 1111 1110 - 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 044 750 044 387 201 238 803 661 470 837 158 569 268 365 260 311 427 938 313 003 631 593 800 950 741 161 883 626 959 521 900 223 812 583 195 812 433 801 139 322 557 812 633 893 415 892 426 281 624 721 476 3 Jul 17 23:17 UTC (GMT)
0 - 011 1110 1110 - 0100 1111 1000 1001 0101 1000 0000 0000 0000 0000 0000 0000 0000 = 0.000 009 999 766 916 735 097 765 922 546 386 718 75 Jul 17 23:16 UTC (GMT)
0 - 111 1111 0100 - 0011 1000 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 107 150 860 718 626 732 094 842 504 906 000 181 056 140 481 170 553 360 744 375 038 837 035 105 112 493 612 249 319 837 881 569 585 812 759 467 291 755 314 682 518 714 528 569 231 404 359 845 775 746 985 748 039 345 677 748 242 309 854 210 746 050 623 711 418 779 541 821 530 464 749 835 819 412 673 987 675 591 655 439 460 770 629 145 711 964 776 865 421 676 604 298 316 526 243 868 372 056 680 693 760 000 Jul 17 23:16 UTC (GMT)
0 - 000 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1100 = 0 Jul 17 23:15 UTC (GMT)
0 - 000 1100 0100 - 0100 0101 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 422 616 857 683 481 690 287 229 092 469 781 029 414 825 797 336 916 325 319 6 Jul 17 23:15 UTC (GMT)
0 - 100 0000 1001 - 0000 0110 1000 1111 1011 0111 1110 1001 0000 1111 1111 1001 0111 = 1 050.245 599 999 999 967 621 988 616 883 754 730 224 609 375 Jul 17 23:11 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)