0.000 020 830 729 321 671 205 134 999 154 399 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 399(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
0.000 020 830 729 321 671 205 134 999 154 399(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: 0.000 020 830 729 321 671 205 134 999 154 399.

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) 0.000 020 830 729 321 671 205 134 999 154 399 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 308 798;
  • 2) 0.000 041 661 458 643 342 410 269 998 308 798 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 617 596;
  • 3) 0.000 083 322 917 286 684 820 539 996 617 596 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 235 192;
  • 4) 0.000 166 645 834 573 369 641 079 993 235 192 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 470 384;
  • 5) 0.000 333 291 669 146 739 282 159 986 470 384 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 940 768;
  • 6) 0.000 666 583 338 293 478 564 319 972 940 768 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 881 536;
  • 7) 0.001 333 166 676 586 957 128 639 945 881 536 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 763 072;
  • 8) 0.002 666 333 353 173 914 257 279 891 763 072 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 526 144;
  • 9) 0.005 332 666 706 347 828 514 559 783 526 144 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 052 288;
  • 10) 0.010 665 333 412 695 657 029 119 567 052 288 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 104 576;
  • 11) 0.021 330 666 825 391 314 058 239 134 104 576 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 209 152;
  • 12) 0.042 661 333 650 782 628 116 478 268 209 152 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 418 304;
  • 13) 0.085 322 667 301 565 256 232 956 536 418 304 × 2 = 0 + 0.170 645 334 603 130 512 465 913 072 836 608;
  • 14) 0.170 645 334 603 130 512 465 913 072 836 608 × 2 = 0 + 0.341 290 669 206 261 024 931 826 145 673 216;
  • 15) 0.341 290 669 206 261 024 931 826 145 673 216 × 2 = 0 + 0.682 581 338 412 522 049 863 652 291 346 432;
  • 16) 0.682 581 338 412 522 049 863 652 291 346 432 × 2 = 1 + 0.365 162 676 825 044 099 727 304 582 692 864;
  • 17) 0.365 162 676 825 044 099 727 304 582 692 864 × 2 = 0 + 0.730 325 353 650 088 199 454 609 165 385 728;
  • 18) 0.730 325 353 650 088 199 454 609 165 385 728 × 2 = 1 + 0.460 650 707 300 176 398 909 218 330 771 456;
  • 19) 0.460 650 707 300 176 398 909 218 330 771 456 × 2 = 0 + 0.921 301 414 600 352 797 818 436 661 542 912;
  • 20) 0.921 301 414 600 352 797 818 436 661 542 912 × 2 = 1 + 0.842 602 829 200 705 595 636 873 323 085 824;
  • 21) 0.842 602 829 200 705 595 636 873 323 085 824 × 2 = 1 + 0.685 205 658 401 411 191 273 746 646 171 648;
  • 22) 0.685 205 658 401 411 191 273 746 646 171 648 × 2 = 1 + 0.370 411 316 802 822 382 547 493 292 343 296;
  • 23) 0.370 411 316 802 822 382 547 493 292 343 296 × 2 = 0 + 0.740 822 633 605 644 765 094 986 584 686 592;
  • 24) 0.740 822 633 605 644 765 094 986 584 686 592 × 2 = 1 + 0.481 645 267 211 289 530 189 973 169 373 184;
  • 25) 0.481 645 267 211 289 530 189 973 169 373 184 × 2 = 0 + 0.963 290 534 422 579 060 379 946 338 746 368;
  • 26) 0.963 290 534 422 579 060 379 946 338 746 368 × 2 = 1 + 0.926 581 068 845 158 120 759 892 677 492 736;
  • 27) 0.926 581 068 845 158 120 759 892 677 492 736 × 2 = 1 + 0.853 162 137 690 316 241 519 785 354 985 472;
  • 28) 0.853 162 137 690 316 241 519 785 354 985 472 × 2 = 1 + 0.706 324 275 380 632 483 039 570 709 970 944;
  • 29) 0.706 324 275 380 632 483 039 570 709 970 944 × 2 = 1 + 0.412 648 550 761 264 966 079 141 419 941 888;
  • 30) 0.412 648 550 761 264 966 079 141 419 941 888 × 2 = 0 + 0.825 297 101 522 529 932 158 282 839 883 776;
  • 31) 0.825 297 101 522 529 932 158 282 839 883 776 × 2 = 1 + 0.650 594 203 045 059 864 316 565 679 767 552;
  • 32) 0.650 594 203 045 059 864 316 565 679 767 552 × 2 = 1 + 0.301 188 406 090 119 728 633 131 359 535 104;
  • 33) 0.301 188 406 090 119 728 633 131 359 535 104 × 2 = 0 + 0.602 376 812 180 239 457 266 262 719 070 208;
  • 34) 0.602 376 812 180 239 457 266 262 719 070 208 × 2 = 1 + 0.204 753 624 360 478 914 532 525 438 140 416;
  • 35) 0.204 753 624 360 478 914 532 525 438 140 416 × 2 = 0 + 0.409 507 248 720 957 829 065 050 876 280 832;
  • 36) 0.409 507 248 720 957 829 065 050 876 280 832 × 2 = 0 + 0.819 014 497 441 915 658 130 101 752 561 664;
  • 37) 0.819 014 497 441 915 658 130 101 752 561 664 × 2 = 1 + 0.638 028 994 883 831 316 260 203 505 123 328;
  • 38) 0.638 028 994 883 831 316 260 203 505 123 328 × 2 = 1 + 0.276 057 989 767 662 632 520 407 010 246 656;
  • 39) 0.276 057 989 767 662 632 520 407 010 246 656 × 2 = 0 + 0.552 115 979 535 325 265 040 814 020 493 312;
  • 40) 0.552 115 979 535 325 265 040 814 020 493 312 × 2 = 1 + 0.104 231 959 070 650 530 081 628 040 986 624;
  • 41) 0.104 231 959 070 650 530 081 628 040 986 624 × 2 = 0 + 0.208 463 918 141 301 060 163 256 081 973 248;
  • 42) 0.208 463 918 141 301 060 163 256 081 973 248 × 2 = 0 + 0.416 927 836 282 602 120 326 512 163 946 496;
  • 43) 0.416 927 836 282 602 120 326 512 163 946 496 × 2 = 0 + 0.833 855 672 565 204 240 653 024 327 892 992;
  • 44) 0.833 855 672 565 204 240 653 024 327 892 992 × 2 = 1 + 0.667 711 345 130 408 481 306 048 655 785 984;
  • 45) 0.667 711 345 130 408 481 306 048 655 785 984 × 2 = 1 + 0.335 422 690 260 816 962 612 097 311 571 968;
  • 46) 0.335 422 690 260 816 962 612 097 311 571 968 × 2 = 0 + 0.670 845 380 521 633 925 224 194 623 143 936;
  • 47) 0.670 845 380 521 633 925 224 194 623 143 936 × 2 = 1 + 0.341 690 761 043 267 850 448 389 246 287 872;
  • 48) 0.341 690 761 043 267 850 448 389 246 287 872 × 2 = 0 + 0.683 381 522 086 535 700 896 778 492 575 744;
  • 49) 0.683 381 522 086 535 700 896 778 492 575 744 × 2 = 1 + 0.366 763 044 173 071 401 793 556 985 151 488;
  • 50) 0.366 763 044 173 071 401 793 556 985 151 488 × 2 = 0 + 0.733 526 088 346 142 803 587 113 970 302 976;
  • 51) 0.733 526 088 346 142 803 587 113 970 302 976 × 2 = 1 + 0.467 052 176 692 285 607 174 227 940 605 952;
  • 52) 0.467 052 176 692 285 607 174 227 940 605 952 × 2 = 0 + 0.934 104 353 384 571 214 348 455 881 211 904;
  • 53) 0.934 104 353 384 571 214 348 455 881 211 904 × 2 = 1 + 0.868 208 706 769 142 428 696 911 762 423 808;
  • 54) 0.868 208 706 769 142 428 696 911 762 423 808 × 2 = 1 + 0.736 417 413 538 284 857 393 823 524 847 616;
  • 55) 0.736 417 413 538 284 857 393 823 524 847 616 × 2 = 1 + 0.472 834 827 076 569 714 787 647 049 695 232;
  • 56) 0.472 834 827 076 569 714 787 647 049 695 232 × 2 = 0 + 0.945 669 654 153 139 429 575 294 099 390 464;
  • 57) 0.945 669 654 153 139 429 575 294 099 390 464 × 2 = 1 + 0.891 339 308 306 278 859 150 588 198 780 928;
  • 58) 0.891 339 308 306 278 859 150 588 198 780 928 × 2 = 1 + 0.782 678 616 612 557 718 301 176 397 561 856;
  • 59) 0.782 678 616 612 557 718 301 176 397 561 856 × 2 = 1 + 0.565 357 233 225 115 436 602 352 795 123 712;
  • 60) 0.565 357 233 225 115 436 602 352 795 123 712 × 2 = 1 + 0.130 714 466 450 230 873 204 705 590 247 424;
  • 61) 0.130 714 466 450 230 873 204 705 590 247 424 × 2 = 0 + 0.261 428 932 900 461 746 409 411 180 494 848;
  • 62) 0.261 428 932 900 461 746 409 411 180 494 848 × 2 = 0 + 0.522 857 865 800 923 492 818 822 360 989 696;
  • 63) 0.522 857 865 800 923 492 818 822 360 989 696 × 2 = 1 + 0.045 715 731 601 846 985 637 644 721 979 392;
  • 64) 0.045 715 731 601 846 985 637 644 721 979 392 × 2 = 0 + 0.091 431 463 203 693 971 275 289 443 958 784;
  • 65) 0.091 431 463 203 693 971 275 289 443 958 784 × 2 = 0 + 0.182 862 926 407 387 942 550 578 887 917 568;
  • 66) 0.182 862 926 407 387 942 550 578 887 917 568 × 2 = 0 + 0.365 725 852 814 775 885 101 157 775 835 136;
  • 67) 0.365 725 852 814 775 885 101 157 775 835 136 × 2 = 0 + 0.731 451 705 629 551 770 202 315 551 670 272;
  • 68) 0.731 451 705 629 551 770 202 315 551 670 272 × 2 = 1 + 0.462 903 411 259 103 540 404 631 103 340 544;

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:


0.000 020 830 729 321 671 205 134 999 154 399(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:

0.000 020 830 729 321 671 205 134 999 154 399(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:


0.000 020 830 729 321 671 205 134 999 154 399(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 0.000 020 830 729 321 671 205 134 999 154 399 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