64 bit double precision IEEE 754 binary floating point number 0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011
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:
100 0001 1010


The last 52 bits contain the mantissa:
0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011

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

The exponent is allways a positive integer.

100 0001 1010(2) =


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


1,024 + 0 + 0 + 0 + 0 + 0 + 16 + 8 + 0 + 2 + 0 =


1,024 + 16 + 8 + 2 =


1,050(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 = 1,050 - 1023 = 27


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)

0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011(2) =

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


0 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 029 802 322 387 695 312 5 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 007 450 580 596 923 828 125 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 029 103 830 456 733 703 613 281 25 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 113 686 837 721 616 029 739 379 882 812 5 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.495 833 394 398 224 269 750 130 588 363 390 415 906 906 127 929 687 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)0 × (1 + 0.495 833 394 398 224 269 750 130 588 363 390 415 906 906 127 929 687 5) × 227 =


1.495 833 394 398 224 269 750 130 588 363 390 415 906 906 127 929 687 5 × 227 =


200 767 359.662 657 588 720 321 655 273 437 5

Conclusion:

0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =

200 767 359.662 657 588 720 321 655 273 437 5(10)

More operations of this kind:

0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1010 = ?

0 - 100 0001 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1100 = ?


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 1010 - 0111 1110 1110 1110 1110 1111 1111 0101 0011 0100 0111 1101 1011 = 200 767 359.662 657 588 720 321 655 273 437 5 Nov 26 23:29 UTC (GMT)
1 - 000 0000 0100 - 0010 0010 0110 0001 1101 1001 0110 1110 1001 1011 1011 1111 0001 = -0 Nov 26 23:27 UTC (GMT)
0 - 100 0000 0101 - 1000 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 0000 = 98.500 000 000 000 227 373 675 443 232 059 478 759 765 625 Nov 26 23:26 UTC (GMT)
0 - 100 0000 0101 - 0100 0100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 81 Nov 26 23:23 UTC (GMT)
0 - 100 0000 0000 - 1010 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 = 3.320 312 499 999 999 555 910 790 149 937 383 830 547 332 763 671 875 Nov 26 23:22 UTC (GMT)
0 - 000 0000 0100 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 1 Nov 26 23:18 UTC (GMT)
0 - 100 0000 1010 - 0000 1000 0000 0000 0010 0000 0000 0000 0000 0000 0000 1000 0000 = 2 112.003 906 250 058 207 660 913 467 407 226 562 5 Nov 26 23:17 UTC (GMT)
0 - 111 1111 1000 - 1111 1111 1111 1111 1111 1111 1111 1110 0000 0000 0000 0000 0000 = 2 808 895 522 568 371 653 198 702 372 724 290 935 670 177 768 268 371 590 877 854 290 890 018 379 059 126 956 220 947 813 086 202 724 197 374 912 840 034 788 231 513 593 296 180 371 246 413 928 332 990 771 378 540 649 774 953 063 748 529 610 426 788 203 460 053 565 557 486 316 363 200 043 951 397 466 342 553 095 885 543 169 325 486 312 296 619 914 727 531 234 127 978 063 377 535 545 973 317 477 209 210 880 Nov 26 23:16 UTC (GMT)
1 - 110 0100 0001 - 1011 0111 0000 0110 0001 1111 1011 1000 1100 0000 0000 0000 0000 = -1 696 624 418 271 947 018 590 808 993 713 532 180 696 594 741 154 370 079 635 629 614 890 016 592 369 112 139 728 677 422 710 370 441 458 332 446 530 106 227 503 979 088 502 057 550 167 847 429 397 288 145 169 934 926 244 361 535 488 Nov 26 23:16 UTC (GMT)
0 - 100 0000 0000 - 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0000 0000 0010 = 2.399 999 999 999 637 090 297 710 528 830 066 323 280 334 472 656 25 Nov 26 23:13 UTC (GMT)
0 - 011 1111 1101 - 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0011 0100 = 0.300 000 000 000 000 044 408 920 985 006 261 616 945 266 723 632 812 5 Nov 26 23:13 UTC (GMT)
0 - 010 0000 0000 - 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 1000 1011 0110 = 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 149 166 814 624 213 662 654 119 263 229 409 472 401 866 174 690 134 356 060 958 222 517 885 470 328 093 526 329 095 482 263 994 617 923 729 371 115 270 967 329 283 966 014 357 551 908 316 008 975 338 556 3 Nov 26 23:12 UTC (GMT)
0 - 100 0000 0010 - 0101 1001 0100 0010 1011 1000 1000 1010 1001 1001 1110 1111 1000 = 10.789 394 636 827 168 483 250 716 235 488 653 182 983 398 437 5 Nov 26 23:12 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)