0.029 999 999 999 999 998 889 776 975 374 843 459 576 446 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.029 999 999 999 999 998 889 776 975 374 843 459 576 446(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.029 999 999 999 999 998 889 776 975 374 843 459 576 446(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.029 999 999 999 999 998 889 776 975 374 843 459 576 446.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.029 999 999 999 999 998 889 776 975 374 843 459 576 446 × 2 = 0 + 0.059 999 999 999 999 997 779 553 950 749 686 919 152 892;
  • 2) 0.059 999 999 999 999 997 779 553 950 749 686 919 152 892 × 2 = 0 + 0.119 999 999 999 999 995 559 107 901 499 373 838 305 784;
  • 3) 0.119 999 999 999 999 995 559 107 901 499 373 838 305 784 × 2 = 0 + 0.239 999 999 999 999 991 118 215 802 998 747 676 611 568;
  • 4) 0.239 999 999 999 999 991 118 215 802 998 747 676 611 568 × 2 = 0 + 0.479 999 999 999 999 982 236 431 605 997 495 353 223 136;
  • 5) 0.479 999 999 999 999 982 236 431 605 997 495 353 223 136 × 2 = 0 + 0.959 999 999 999 999 964 472 863 211 994 990 706 446 272;
  • 6) 0.959 999 999 999 999 964 472 863 211 994 990 706 446 272 × 2 = 1 + 0.919 999 999 999 999 928 945 726 423 989 981 412 892 544;
  • 7) 0.919 999 999 999 999 928 945 726 423 989 981 412 892 544 × 2 = 1 + 0.839 999 999 999 999 857 891 452 847 979 962 825 785 088;
  • 8) 0.839 999 999 999 999 857 891 452 847 979 962 825 785 088 × 2 = 1 + 0.679 999 999 999 999 715 782 905 695 959 925 651 570 176;
  • 9) 0.679 999 999 999 999 715 782 905 695 959 925 651 570 176 × 2 = 1 + 0.359 999 999 999 999 431 565 811 391 919 851 303 140 352;
  • 10) 0.359 999 999 999 999 431 565 811 391 919 851 303 140 352 × 2 = 0 + 0.719 999 999 999 998 863 131 622 783 839 702 606 280 704;
  • 11) 0.719 999 999 999 998 863 131 622 783 839 702 606 280 704 × 2 = 1 + 0.439 999 999 999 997 726 263 245 567 679 405 212 561 408;
  • 12) 0.439 999 999 999 997 726 263 245 567 679 405 212 561 408 × 2 = 0 + 0.879 999 999 999 995 452 526 491 135 358 810 425 122 816;
  • 13) 0.879 999 999 999 995 452 526 491 135 358 810 425 122 816 × 2 = 1 + 0.759 999 999 999 990 905 052 982 270 717 620 850 245 632;
  • 14) 0.759 999 999 999 990 905 052 982 270 717 620 850 245 632 × 2 = 1 + 0.519 999 999 999 981 810 105 964 541 435 241 700 491 264;
  • 15) 0.519 999 999 999 981 810 105 964 541 435 241 700 491 264 × 2 = 1 + 0.039 999 999 999 963 620 211 929 082 870 483 400 982 528;
  • 16) 0.039 999 999 999 963 620 211 929 082 870 483 400 982 528 × 2 = 0 + 0.079 999 999 999 927 240 423 858 165 740 966 801 965 056;
  • 17) 0.079 999 999 999 927 240 423 858 165 740 966 801 965 056 × 2 = 0 + 0.159 999 999 999 854 480 847 716 331 481 933 603 930 112;
  • 18) 0.159 999 999 999 854 480 847 716 331 481 933 603 930 112 × 2 = 0 + 0.319 999 999 999 708 961 695 432 662 963 867 207 860 224;
  • 19) 0.319 999 999 999 708 961 695 432 662 963 867 207 860 224 × 2 = 0 + 0.639 999 999 999 417 923 390 865 325 927 734 415 720 448;
  • 20) 0.639 999 999 999 417 923 390 865 325 927 734 415 720 448 × 2 = 1 + 0.279 999 999 998 835 846 781 730 651 855 468 831 440 896;
  • 21) 0.279 999 999 998 835 846 781 730 651 855 468 831 440 896 × 2 = 0 + 0.559 999 999 997 671 693 563 461 303 710 937 662 881 792;
  • 22) 0.559 999 999 997 671 693 563 461 303 710 937 662 881 792 × 2 = 1 + 0.119 999 999 995 343 387 126 922 607 421 875 325 763 584;
  • 23) 0.119 999 999 995 343 387 126 922 607 421 875 325 763 584 × 2 = 0 + 0.239 999 999 990 686 774 253 845 214 843 750 651 527 168;
  • 24) 0.239 999 999 990 686 774 253 845 214 843 750 651 527 168 × 2 = 0 + 0.479 999 999 981 373 548 507 690 429 687 501 303 054 336;
  • 25) 0.479 999 999 981 373 548 507 690 429 687 501 303 054 336 × 2 = 0 + 0.959 999 999 962 747 097 015 380 859 375 002 606 108 672;
  • 26) 0.959 999 999 962 747 097 015 380 859 375 002 606 108 672 × 2 = 1 + 0.919 999 999 925 494 194 030 761 718 750 005 212 217 344;
  • 27) 0.919 999 999 925 494 194 030 761 718 750 005 212 217 344 × 2 = 1 + 0.839 999 999 850 988 388 061 523 437 500 010 424 434 688;
  • 28) 0.839 999 999 850 988 388 061 523 437 500 010 424 434 688 × 2 = 1 + 0.679 999 999 701 976 776 123 046 875 000 020 848 869 376;
  • 29) 0.679 999 999 701 976 776 123 046 875 000 020 848 869 376 × 2 = 1 + 0.359 999 999 403 953 552 246 093 750 000 041 697 738 752;
  • 30) 0.359 999 999 403 953 552 246 093 750 000 041 697 738 752 × 2 = 0 + 0.719 999 998 807 907 104 492 187 500 000 083 395 477 504;
  • 31) 0.719 999 998 807 907 104 492 187 500 000 083 395 477 504 × 2 = 1 + 0.439 999 997 615 814 208 984 375 000 000 166 790 955 008;
  • 32) 0.439 999 997 615 814 208 984 375 000 000 166 790 955 008 × 2 = 0 + 0.879 999 995 231 628 417 968 750 000 000 333 581 910 016;
  • 33) 0.879 999 995 231 628 417 968 750 000 000 333 581 910 016 × 2 = 1 + 0.759 999 990 463 256 835 937 500 000 000 667 163 820 032;
  • 34) 0.759 999 990 463 256 835 937 500 000 000 667 163 820 032 × 2 = 1 + 0.519 999 980 926 513 671 875 000 000 001 334 327 640 064;
  • 35) 0.519 999 980 926 513 671 875 000 000 001 334 327 640 064 × 2 = 1 + 0.039 999 961 853 027 343 750 000 000 002 668 655 280 128;
  • 36) 0.039 999 961 853 027 343 750 000 000 002 668 655 280 128 × 2 = 0 + 0.079 999 923 706 054 687 500 000 000 005 337 310 560 256;
  • 37) 0.079 999 923 706 054 687 500 000 000 005 337 310 560 256 × 2 = 0 + 0.159 999 847 412 109 375 000 000 000 010 674 621 120 512;
  • 38) 0.159 999 847 412 109 375 000 000 000 010 674 621 120 512 × 2 = 0 + 0.319 999 694 824 218 750 000 000 000 021 349 242 241 024;
  • 39) 0.319 999 694 824 218 750 000 000 000 021 349 242 241 024 × 2 = 0 + 0.639 999 389 648 437 500 000 000 000 042 698 484 482 048;
  • 40) 0.639 999 389 648 437 500 000 000 000 042 698 484 482 048 × 2 = 1 + 0.279 998 779 296 875 000 000 000 000 085 396 968 964 096;
  • 41) 0.279 998 779 296 875 000 000 000 000 085 396 968 964 096 × 2 = 0 + 0.559 997 558 593 750 000 000 000 000 170 793 937 928 192;
  • 42) 0.559 997 558 593 750 000 000 000 000 170 793 937 928 192 × 2 = 1 + 0.119 995 117 187 500 000 000 000 000 341 587 875 856 384;
  • 43) 0.119 995 117 187 500 000 000 000 000 341 587 875 856 384 × 2 = 0 + 0.239 990 234 375 000 000 000 000 000 683 175 751 712 768;
  • 44) 0.239 990 234 375 000 000 000 000 000 683 175 751 712 768 × 2 = 0 + 0.479 980 468 750 000 000 000 000 001 366 351 503 425 536;
  • 45) 0.479 980 468 750 000 000 000 000 001 366 351 503 425 536 × 2 = 0 + 0.959 960 937 500 000 000 000 000 002 732 703 006 851 072;
  • 46) 0.959 960 937 500 000 000 000 000 002 732 703 006 851 072 × 2 = 1 + 0.919 921 875 000 000 000 000 000 005 465 406 013 702 144;
  • 47) 0.919 921 875 000 000 000 000 000 005 465 406 013 702 144 × 2 = 1 + 0.839 843 750 000 000 000 000 000 010 930 812 027 404 288;
  • 48) 0.839 843 750 000 000 000 000 000 010 930 812 027 404 288 × 2 = 1 + 0.679 687 500 000 000 000 000 000 021 861 624 054 808 576;
  • 49) 0.679 687 500 000 000 000 000 000 021 861 624 054 808 576 × 2 = 1 + 0.359 375 000 000 000 000 000 000 043 723 248 109 617 152;
  • 50) 0.359 375 000 000 000 000 000 000 043 723 248 109 617 152 × 2 = 0 + 0.718 750 000 000 000 000 000 000 087 446 496 219 234 304;
  • 51) 0.718 750 000 000 000 000 000 000 087 446 496 219 234 304 × 2 = 1 + 0.437 500 000 000 000 000 000 000 174 892 992 438 468 608;
  • 52) 0.437 500 000 000 000 000 000 000 174 892 992 438 468 608 × 2 = 0 + 0.875 000 000 000 000 000 000 000 349 785 984 876 937 216;
  • 53) 0.875 000 000 000 000 000 000 000 349 785 984 876 937 216 × 2 = 1 + 0.750 000 000 000 000 000 000 000 699 571 969 753 874 432;
  • 54) 0.750 000 000 000 000 000 000 000 699 571 969 753 874 432 × 2 = 1 + 0.500 000 000 000 000 000 000 001 399 143 939 507 748 864;
  • 55) 0.500 000 000 000 000 000 000 001 399 143 939 507 748 864 × 2 = 1 + 0.000 000 000 000 000 000 000 002 798 287 879 015 497 728;
  • 56) 0.000 000 000 000 000 000 000 002 798 287 879 015 497 728 × 2 = 0 + 0.000 000 000 000 000 000 000 005 596 575 758 030 995 456;
  • 57) 0.000 000 000 000 000 000 000 005 596 575 758 030 995 456 × 2 = 0 + 0.000 000 000 000 000 000 000 011 193 151 516 061 990 912;
  • 58) 0.000 000 000 000 000 000 000 011 193 151 516 061 990 912 × 2 = 0 + 0.000 000 000 000 000 000 000 022 386 303 032 123 981 824;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.029 999 999 999 999 998 889 776 975 374 843 459 576 446(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2)

5. Positive number before normalization:

0.029 999 999 999 999 998 889 776 975 374 843 459 576 446(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 6 positions to the right, so that only one non zero digit remains to the left of it:


0.029 999 999 999 999 998 889 776 975 374 843 459 576 446(10) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2) =

0.0000 0111 1010 1110 0001 0100 0111 1010 1110 0001 0100 0111 1010 1110 00(2) × 20 =

1.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000(2) × 2-6


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -6

Mantissa (not normalized):
1.1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-6 + 2(11-1) - 1 =

(-6 + 1 023)(10) =

1 017(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 017 ÷ 2 = 508 + 1;
  • 508 ÷ 2 = 254 + 0;
  • 254 ÷ 2 = 127 + 0;
  • 127 ÷ 2 = 63 + 1;
  • 63 ÷ 2 = 31 + 1;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

1017(10) =

011 1111 1001(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000 =

1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1111 1001

Mantissa (52 bits) =
1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000


Decimal number 0.029 999 999 999 999 998 889 776 975 374 843 459 576 446 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1111 1001 - 1110 1011 1000 0101 0001 1110 1011 1000 0101 0001 1110 1011 1000

» Convert decimal 0.029 999 999 999 999 998 889 776 975 374 843 459 576 417 to 64 bit double precision IEEE 754 binary floating point representation standard

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100