64 bit double precision IEEE 754 binary floating point number 1 - 101 0000 1001 - 1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001 converted to decimal base ten (double)

How to convert 64 bit double precision IEEE 754 binary floating point:
1 - 101 0000 1001 - 1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001.

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:
101 0000 1001


The last 52 bits contain the mantissa:
1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001

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

101 0000 1001(2) =


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


1,024 + 0 + 256 + 0 + 0 + 0 + 0 + 8 + 0 + 0 + 1 =


1,024 + 256 + 8 + 1 =


1,289(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 = 1,289 - 1023 = 266

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):

1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001(2) =

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


0.5 + 0 + 0.125 + 0 + 0 + 0 + 0 + 0.003 906 25 + 0 + 0 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 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 + 0 + 0.000 000 001 862 645 149 230 957 031 25 + 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 + 0 + 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.5 + 0.125 + 0.003 906 25 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 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 + 0.000 000 001 862 645 149 230 957 031 25 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 003 637 978 807 091 712 951 660 156 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 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =


0.629 520 404 389 490 417 784 713 827 131 781 727 075 576 782 226 562 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)1 × (1 + 0.629 520 404 389 490 417 784 713 827 131 781 727 075 576 782 226 562 5) × 2266 =


-1.629 520 404 389 490 417 784 713 827 131 781 727 075 576 782 226 562 5 × 2266 =


-193 214 025 808 993 738 604 101 112 860 072 347 767 196 659 945 695 490 958 841 510 140 506 787 897 409 536

1 - 101 0000 1001 - 1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001
converted from
64 bit double precision IEEE 754 binary floating point
to
base ten decimal system (double) =


-193 214 025 808 993 738 604 101 112 860 072 347 767 196 659 945 695 490 958 841 510 140 506 787 897 409 536(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)

1 - 101 0000 1001 - 1010 0001 0010 1000 0011 1111 1100 1101 0000 0100 0111 1110 1001 = -193 214 025 808 993 738 604 101 112 860 072 347 767 196 659 945 695 490 958 841 510 140 506 787 897 409 536 Jun 24 15:37 UTC (GMT)
0 - 100 0000 0010 - 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 10.375 Jun 24 15:33 UTC (GMT)
1 - 100 0000 0000 - 0110 0010 1110 1101 1001 1011 1111 1100 1010 0110 0000 0000 0000 = -2.772 876 261 104 102 013 632 655 143 737 792 968 75 Jun 24 15:32 UTC (GMT)
0 - 100 0001 0010 - 0110 0111 1101 0001 1100 0000 0000 0000 0000 0000 0000 0000 0000 = 736 910 Jun 24 15:28 UTC (GMT)
1 - 110 0110 1010 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = -3 263 311 827 866 217 054 348 829 941 058 367 246 313 020 115 314 884 136 137 384 785 103 279 471 785 076 179 389 414 413 917 568 731 149 656 040 363 498 502 238 289 383 450 538 233 278 932 090 749 541 912 690 342 899 328 012 655 937 730 937 618 432 Jun 24 15:26 UTC (GMT)
1 - 111 1100 0110 - 1100 1100 1100 1101 0100 0000 1100 0110 1111 1100 1001 1110 0000 = -2 245 328 971 236 279 803 757 806 293 868 626 284 142 583 072 539 824 605 203 195 507 396 828 364 822 105 444 825 803 858 528 555 408 079 091 660 699 319 927 705 388 681 351 349 252 342 020 131 169 186 922 703 498 125 638 356 245 639 596 424 468 203 794 654 578 821 226 463 947 761 939 855 555 462 077 010 927 238 560 338 434 724 803 165 416 998 323 061 121 608 355 544 801 478 973 587 456 Jun 24 15:25 UTC (GMT)
1 - 100 0100 0010 - 1000 0010 0101 0011 1010 1010 0111 1101 0101 0111 1101 1010 1000 = -222 702 249 416 229 912 576 Jun 24 15:21 UTC (GMT)
0 - 100 0000 1000 - 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 768 Jun 24 15:17 UTC (GMT)
0 - 111 0010 0110 - 0101 0111 0011 0111 0011 0110 0001 0100 1110 0110 1000 0100 0001 = 1 144 283 517 142 482 188 112 020 266 720 057 419 136 298 954 335 226 472 190 306 906 372 871 084 264 114 568 379 010 961 296 102 317 094 940 241 499 474 946 985 053 109 511 587 646 001 110 865 029 627 005 804 607 780 259 462 420 924 572 488 588 619 024 838 453 814 155 236 534 054 031 332 999 004 074 641 232 794 316 963 840 Jun 24 15:16 UTC (GMT)
0 - 101 1111 0000 - 0011 0010 0111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 = 489 789 860 550 377 917 134 847 078 965 628 620 706 732 054 191 563 313 685 198 290 337 382 324 333 865 946 734 395 546 346 249 636 307 092 206 185 224 213 505 804 205 592 177 614 811 415 703 453 696 Jun 24 15:16 UTC (GMT)
1 - 100 0011 0101 - 0110 1010 1010 1111 1010 1010 0010 0101 0111 0100 0101 0101 0110 = -25 521 771 719 234 904 Jun 24 15:15 UTC (GMT)
1 - 100 0001 0011 - 1010 0001 0000 0010 0000 0000 0100 0000 0000 0000 0000 0000 0000 = -1 708 064.015 625 Jun 24 15:10 UTC (GMT)
1 - 100 0011 0000 - 1000 0100 0101 0000 1010 1010 0111 1101 0101 0100 0101 1010 1011 = -853 913 938 602 165.375 Jun 24 15:09 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)