0.000 020 830 729 321 671 205 134 999 154 973 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 973(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
0.000 020 830 729 321 671 205 134 999 154 973(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: 0.000 020 830 729 321 671 205 134 999 154 973.

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) 0.000 020 830 729 321 671 205 134 999 154 973 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 946;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 946 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 619 892;
  • 3) 0.000 083 322 917 286 684 820 539 996 619 892 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 239 784;
  • 4) 0.000 166 645 834 573 369 641 079 993 239 784 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 479 568;
  • 5) 0.000 333 291 669 146 739 282 159 986 479 568 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 959 136;
  • 6) 0.000 666 583 338 293 478 564 319 972 959 136 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 918 272;
  • 7) 0.001 333 166 676 586 957 128 639 945 918 272 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 836 544;
  • 8) 0.002 666 333 353 173 914 257 279 891 836 544 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 673 088;
  • 9) 0.005 332 666 706 347 828 514 559 783 673 088 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 346 176;
  • 10) 0.010 665 333 412 695 657 029 119 567 346 176 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 692 352;
  • 11) 0.021 330 666 825 391 314 058 239 134 692 352 × 2 = 0 + 0.042 661 333 650 782 628 116 478 269 384 704;
  • 12) 0.042 661 333 650 782 628 116 478 269 384 704 × 2 = 0 + 0.085 322 667 301 565 256 232 956 538 769 408;
  • 13) 0.085 322 667 301 565 256 232 956 538 769 408 × 2 = 0 + 0.170 645 334 603 130 512 465 913 077 538 816;
  • 14) 0.170 645 334 603 130 512 465 913 077 538 816 × 2 = 0 + 0.341 290 669 206 261 024 931 826 155 077 632;
  • 15) 0.341 290 669 206 261 024 931 826 155 077 632 × 2 = 0 + 0.682 581 338 412 522 049 863 652 310 155 264;
  • 16) 0.682 581 338 412 522 049 863 652 310 155 264 × 2 = 1 + 0.365 162 676 825 044 099 727 304 620 310 528;
  • 17) 0.365 162 676 825 044 099 727 304 620 310 528 × 2 = 0 + 0.730 325 353 650 088 199 454 609 240 621 056;
  • 18) 0.730 325 353 650 088 199 454 609 240 621 056 × 2 = 1 + 0.460 650 707 300 176 398 909 218 481 242 112;
  • 19) 0.460 650 707 300 176 398 909 218 481 242 112 × 2 = 0 + 0.921 301 414 600 352 797 818 436 962 484 224;
  • 20) 0.921 301 414 600 352 797 818 436 962 484 224 × 2 = 1 + 0.842 602 829 200 705 595 636 873 924 968 448;
  • 21) 0.842 602 829 200 705 595 636 873 924 968 448 × 2 = 1 + 0.685 205 658 401 411 191 273 747 849 936 896;
  • 22) 0.685 205 658 401 411 191 273 747 849 936 896 × 2 = 1 + 0.370 411 316 802 822 382 547 495 699 873 792;
  • 23) 0.370 411 316 802 822 382 547 495 699 873 792 × 2 = 0 + 0.740 822 633 605 644 765 094 991 399 747 584;
  • 24) 0.740 822 633 605 644 765 094 991 399 747 584 × 2 = 1 + 0.481 645 267 211 289 530 189 982 799 495 168;
  • 25) 0.481 645 267 211 289 530 189 982 799 495 168 × 2 = 0 + 0.963 290 534 422 579 060 379 965 598 990 336;
  • 26) 0.963 290 534 422 579 060 379 965 598 990 336 × 2 = 1 + 0.926 581 068 845 158 120 759 931 197 980 672;
  • 27) 0.926 581 068 845 158 120 759 931 197 980 672 × 2 = 1 + 0.853 162 137 690 316 241 519 862 395 961 344;
  • 28) 0.853 162 137 690 316 241 519 862 395 961 344 × 2 = 1 + 0.706 324 275 380 632 483 039 724 791 922 688;
  • 29) 0.706 324 275 380 632 483 039 724 791 922 688 × 2 = 1 + 0.412 648 550 761 264 966 079 449 583 845 376;
  • 30) 0.412 648 550 761 264 966 079 449 583 845 376 × 2 = 0 + 0.825 297 101 522 529 932 158 899 167 690 752;
  • 31) 0.825 297 101 522 529 932 158 899 167 690 752 × 2 = 1 + 0.650 594 203 045 059 864 317 798 335 381 504;
  • 32) 0.650 594 203 045 059 864 317 798 335 381 504 × 2 = 1 + 0.301 188 406 090 119 728 635 596 670 763 008;
  • 33) 0.301 188 406 090 119 728 635 596 670 763 008 × 2 = 0 + 0.602 376 812 180 239 457 271 193 341 526 016;
  • 34) 0.602 376 812 180 239 457 271 193 341 526 016 × 2 = 1 + 0.204 753 624 360 478 914 542 386 683 052 032;
  • 35) 0.204 753 624 360 478 914 542 386 683 052 032 × 2 = 0 + 0.409 507 248 720 957 829 084 773 366 104 064;
  • 36) 0.409 507 248 720 957 829 084 773 366 104 064 × 2 = 0 + 0.819 014 497 441 915 658 169 546 732 208 128;
  • 37) 0.819 014 497 441 915 658 169 546 732 208 128 × 2 = 1 + 0.638 028 994 883 831 316 339 093 464 416 256;
  • 38) 0.638 028 994 883 831 316 339 093 464 416 256 × 2 = 1 + 0.276 057 989 767 662 632 678 186 928 832 512;
  • 39) 0.276 057 989 767 662 632 678 186 928 832 512 × 2 = 0 + 0.552 115 979 535 325 265 356 373 857 665 024;
  • 40) 0.552 115 979 535 325 265 356 373 857 665 024 × 2 = 1 + 0.104 231 959 070 650 530 712 747 715 330 048;
  • 41) 0.104 231 959 070 650 530 712 747 715 330 048 × 2 = 0 + 0.208 463 918 141 301 061 425 495 430 660 096;
  • 42) 0.208 463 918 141 301 061 425 495 430 660 096 × 2 = 0 + 0.416 927 836 282 602 122 850 990 861 320 192;
  • 43) 0.416 927 836 282 602 122 850 990 861 320 192 × 2 = 0 + 0.833 855 672 565 204 245 701 981 722 640 384;
  • 44) 0.833 855 672 565 204 245 701 981 722 640 384 × 2 = 1 + 0.667 711 345 130 408 491 403 963 445 280 768;
  • 45) 0.667 711 345 130 408 491 403 963 445 280 768 × 2 = 1 + 0.335 422 690 260 816 982 807 926 890 561 536;
  • 46) 0.335 422 690 260 816 982 807 926 890 561 536 × 2 = 0 + 0.670 845 380 521 633 965 615 853 781 123 072;
  • 47) 0.670 845 380 521 633 965 615 853 781 123 072 × 2 = 1 + 0.341 690 761 043 267 931 231 707 562 246 144;
  • 48) 0.341 690 761 043 267 931 231 707 562 246 144 × 2 = 0 + 0.683 381 522 086 535 862 463 415 124 492 288;
  • 49) 0.683 381 522 086 535 862 463 415 124 492 288 × 2 = 1 + 0.366 763 044 173 071 724 926 830 248 984 576;
  • 50) 0.366 763 044 173 071 724 926 830 248 984 576 × 2 = 0 + 0.733 526 088 346 143 449 853 660 497 969 152;
  • 51) 0.733 526 088 346 143 449 853 660 497 969 152 × 2 = 1 + 0.467 052 176 692 286 899 707 320 995 938 304;
  • 52) 0.467 052 176 692 286 899 707 320 995 938 304 × 2 = 0 + 0.934 104 353 384 573 799 414 641 991 876 608;
  • 53) 0.934 104 353 384 573 799 414 641 991 876 608 × 2 = 1 + 0.868 208 706 769 147 598 829 283 983 753 216;
  • 54) 0.868 208 706 769 147 598 829 283 983 753 216 × 2 = 1 + 0.736 417 413 538 295 197 658 567 967 506 432;
  • 55) 0.736 417 413 538 295 197 658 567 967 506 432 × 2 = 1 + 0.472 834 827 076 590 395 317 135 935 012 864;
  • 56) 0.472 834 827 076 590 395 317 135 935 012 864 × 2 = 0 + 0.945 669 654 153 180 790 634 271 870 025 728;
  • 57) 0.945 669 654 153 180 790 634 271 870 025 728 × 2 = 1 + 0.891 339 308 306 361 581 268 543 740 051 456;
  • 58) 0.891 339 308 306 361 581 268 543 740 051 456 × 2 = 1 + 0.782 678 616 612 723 162 537 087 480 102 912;
  • 59) 0.782 678 616 612 723 162 537 087 480 102 912 × 2 = 1 + 0.565 357 233 225 446 325 074 174 960 205 824;
  • 60) 0.565 357 233 225 446 325 074 174 960 205 824 × 2 = 1 + 0.130 714 466 450 892 650 148 349 920 411 648;
  • 61) 0.130 714 466 450 892 650 148 349 920 411 648 × 2 = 0 + 0.261 428 932 901 785 300 296 699 840 823 296;
  • 62) 0.261 428 932 901 785 300 296 699 840 823 296 × 2 = 0 + 0.522 857 865 803 570 600 593 399 681 646 592;
  • 63) 0.522 857 865 803 570 600 593 399 681 646 592 × 2 = 1 + 0.045 715 731 607 141 201 186 799 363 293 184;
  • 64) 0.045 715 731 607 141 201 186 799 363 293 184 × 2 = 0 + 0.091 431 463 214 282 402 373 598 726 586 368;
  • 65) 0.091 431 463 214 282 402 373 598 726 586 368 × 2 = 0 + 0.182 862 926 428 564 804 747 197 453 172 736;
  • 66) 0.182 862 926 428 564 804 747 197 453 172 736 × 2 = 0 + 0.365 725 852 857 129 609 494 394 906 345 472;
  • 67) 0.365 725 852 857 129 609 494 394 906 345 472 × 2 = 0 + 0.731 451 705 714 259 218 988 789 812 690 944;
  • 68) 0.731 451 705 714 259 218 988 789 812 690 944 × 2 = 1 + 0.462 903 411 428 518 437 977 579 625 381 888;

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:


0.000 020 830 729 321 671 205 134 999 154 973(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

5. Positive number before normalization:

0.000 020 830 729 321 671 205 134 999 154 973(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 16 positions to the right, so that only one non zero digit remains to the left of it:


0.000 020 830 729 321 671 205 134 999 154 973(10) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) =

0.0000 0000 0000 0001 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 20 =

1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001(2) × 2-16


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): -16

Mantissa (not normalized):
1.0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-16 + 2(11-1) - 1 =

(-16 + 1 023)(10) =

1 007(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;
  • 1 007 ÷ 2 = 503 + 1;
  • 503 ÷ 2 = 251 + 1;
  • 251 ÷ 2 = 125 + 1;
  • 125 ÷ 2 = 62 + 1;
  • 62 ÷ 2 = 31 + 0;
  • 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) =

1007(10) =

011 1110 1111(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. 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001 =

0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


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 1110 1111

Mantissa (52 bits) =
0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001


Decimal number 0.000 020 830 729 321 671 205 134 999 154 973 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1110 1111 - 0101 1101 0111 1011 0100 1101 0001 1010 1010 1110 1111 0010 0001

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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