0.000 020 830 729 321 671 205 134 999 154 722 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 722(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
0.000 020 830 729 321 671 205 134 999 154 722(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: 0.000 020 830 729 321 671 205 134 999 154 722.

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) 0.000 020 830 729 321 671 205 134 999 154 722 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 444;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 444 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 888;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 888 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 237 776;
  • 4) 0.000 166 645 834 573 369 641 079 993 237 776 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 475 552;
  • 5) 0.000 333 291 669 146 739 282 159 986 475 552 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 951 104;
  • 6) 0.000 666 583 338 293 478 564 319 972 951 104 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 902 208;
  • 7) 0.001 333 166 676 586 957 128 639 945 902 208 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 804 416;
  • 8) 0.002 666 333 353 173 914 257 279 891 804 416 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 608 832;
  • 9) 0.005 332 666 706 347 828 514 559 783 608 832 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 217 664;
  • 10) 0.010 665 333 412 695 657 029 119 567 217 664 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 435 328;
  • 11) 0.021 330 666 825 391 314 058 239 134 435 328 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 870 656;
  • 12) 0.042 661 333 650 782 628 116 478 268 870 656 × 2 = 0 + 0.085 322 667 301 565 256 232 956 537 741 312;
  • 13) 0.085 322 667 301 565 256 232 956 537 741 312 × 2 = 0 + 0.170 645 334 603 130 512 465 913 075 482 624;
  • 14) 0.170 645 334 603 130 512 465 913 075 482 624 × 2 = 0 + 0.341 290 669 206 261 024 931 826 150 965 248;
  • 15) 0.341 290 669 206 261 024 931 826 150 965 248 × 2 = 0 + 0.682 581 338 412 522 049 863 652 301 930 496;
  • 16) 0.682 581 338 412 522 049 863 652 301 930 496 × 2 = 1 + 0.365 162 676 825 044 099 727 304 603 860 992;
  • 17) 0.365 162 676 825 044 099 727 304 603 860 992 × 2 = 0 + 0.730 325 353 650 088 199 454 609 207 721 984;
  • 18) 0.730 325 353 650 088 199 454 609 207 721 984 × 2 = 1 + 0.460 650 707 300 176 398 909 218 415 443 968;
  • 19) 0.460 650 707 300 176 398 909 218 415 443 968 × 2 = 0 + 0.921 301 414 600 352 797 818 436 830 887 936;
  • 20) 0.921 301 414 600 352 797 818 436 830 887 936 × 2 = 1 + 0.842 602 829 200 705 595 636 873 661 775 872;
  • 21) 0.842 602 829 200 705 595 636 873 661 775 872 × 2 = 1 + 0.685 205 658 401 411 191 273 747 323 551 744;
  • 22) 0.685 205 658 401 411 191 273 747 323 551 744 × 2 = 1 + 0.370 411 316 802 822 382 547 494 647 103 488;
  • 23) 0.370 411 316 802 822 382 547 494 647 103 488 × 2 = 0 + 0.740 822 633 605 644 765 094 989 294 206 976;
  • 24) 0.740 822 633 605 644 765 094 989 294 206 976 × 2 = 1 + 0.481 645 267 211 289 530 189 978 588 413 952;
  • 25) 0.481 645 267 211 289 530 189 978 588 413 952 × 2 = 0 + 0.963 290 534 422 579 060 379 957 176 827 904;
  • 26) 0.963 290 534 422 579 060 379 957 176 827 904 × 2 = 1 + 0.926 581 068 845 158 120 759 914 353 655 808;
  • 27) 0.926 581 068 845 158 120 759 914 353 655 808 × 2 = 1 + 0.853 162 137 690 316 241 519 828 707 311 616;
  • 28) 0.853 162 137 690 316 241 519 828 707 311 616 × 2 = 1 + 0.706 324 275 380 632 483 039 657 414 623 232;
  • 29) 0.706 324 275 380 632 483 039 657 414 623 232 × 2 = 1 + 0.412 648 550 761 264 966 079 314 829 246 464;
  • 30) 0.412 648 550 761 264 966 079 314 829 246 464 × 2 = 0 + 0.825 297 101 522 529 932 158 629 658 492 928;
  • 31) 0.825 297 101 522 529 932 158 629 658 492 928 × 2 = 1 + 0.650 594 203 045 059 864 317 259 316 985 856;
  • 32) 0.650 594 203 045 059 864 317 259 316 985 856 × 2 = 1 + 0.301 188 406 090 119 728 634 518 633 971 712;
  • 33) 0.301 188 406 090 119 728 634 518 633 971 712 × 2 = 0 + 0.602 376 812 180 239 457 269 037 267 943 424;
  • 34) 0.602 376 812 180 239 457 269 037 267 943 424 × 2 = 1 + 0.204 753 624 360 478 914 538 074 535 886 848;
  • 35) 0.204 753 624 360 478 914 538 074 535 886 848 × 2 = 0 + 0.409 507 248 720 957 829 076 149 071 773 696;
  • 36) 0.409 507 248 720 957 829 076 149 071 773 696 × 2 = 0 + 0.819 014 497 441 915 658 152 298 143 547 392;
  • 37) 0.819 014 497 441 915 658 152 298 143 547 392 × 2 = 1 + 0.638 028 994 883 831 316 304 596 287 094 784;
  • 38) 0.638 028 994 883 831 316 304 596 287 094 784 × 2 = 1 + 0.276 057 989 767 662 632 609 192 574 189 568;
  • 39) 0.276 057 989 767 662 632 609 192 574 189 568 × 2 = 0 + 0.552 115 979 535 325 265 218 385 148 379 136;
  • 40) 0.552 115 979 535 325 265 218 385 148 379 136 × 2 = 1 + 0.104 231 959 070 650 530 436 770 296 758 272;
  • 41) 0.104 231 959 070 650 530 436 770 296 758 272 × 2 = 0 + 0.208 463 918 141 301 060 873 540 593 516 544;
  • 42) 0.208 463 918 141 301 060 873 540 593 516 544 × 2 = 0 + 0.416 927 836 282 602 121 747 081 187 033 088;
  • 43) 0.416 927 836 282 602 121 747 081 187 033 088 × 2 = 0 + 0.833 855 672 565 204 243 494 162 374 066 176;
  • 44) 0.833 855 672 565 204 243 494 162 374 066 176 × 2 = 1 + 0.667 711 345 130 408 486 988 324 748 132 352;
  • 45) 0.667 711 345 130 408 486 988 324 748 132 352 × 2 = 1 + 0.335 422 690 260 816 973 976 649 496 264 704;
  • 46) 0.335 422 690 260 816 973 976 649 496 264 704 × 2 = 0 + 0.670 845 380 521 633 947 953 298 992 529 408;
  • 47) 0.670 845 380 521 633 947 953 298 992 529 408 × 2 = 1 + 0.341 690 761 043 267 895 906 597 985 058 816;
  • 48) 0.341 690 761 043 267 895 906 597 985 058 816 × 2 = 0 + 0.683 381 522 086 535 791 813 195 970 117 632;
  • 49) 0.683 381 522 086 535 791 813 195 970 117 632 × 2 = 1 + 0.366 763 044 173 071 583 626 391 940 235 264;
  • 50) 0.366 763 044 173 071 583 626 391 940 235 264 × 2 = 0 + 0.733 526 088 346 143 167 252 783 880 470 528;
  • 51) 0.733 526 088 346 143 167 252 783 880 470 528 × 2 = 1 + 0.467 052 176 692 286 334 505 567 760 941 056;
  • 52) 0.467 052 176 692 286 334 505 567 760 941 056 × 2 = 0 + 0.934 104 353 384 572 669 011 135 521 882 112;
  • 53) 0.934 104 353 384 572 669 011 135 521 882 112 × 2 = 1 + 0.868 208 706 769 145 338 022 271 043 764 224;
  • 54) 0.868 208 706 769 145 338 022 271 043 764 224 × 2 = 1 + 0.736 417 413 538 290 676 044 542 087 528 448;
  • 55) 0.736 417 413 538 290 676 044 542 087 528 448 × 2 = 1 + 0.472 834 827 076 581 352 089 084 175 056 896;
  • 56) 0.472 834 827 076 581 352 089 084 175 056 896 × 2 = 0 + 0.945 669 654 153 162 704 178 168 350 113 792;
  • 57) 0.945 669 654 153 162 704 178 168 350 113 792 × 2 = 1 + 0.891 339 308 306 325 408 356 336 700 227 584;
  • 58) 0.891 339 308 306 325 408 356 336 700 227 584 × 2 = 1 + 0.782 678 616 612 650 816 712 673 400 455 168;
  • 59) 0.782 678 616 612 650 816 712 673 400 455 168 × 2 = 1 + 0.565 357 233 225 301 633 425 346 800 910 336;
  • 60) 0.565 357 233 225 301 633 425 346 800 910 336 × 2 = 1 + 0.130 714 466 450 603 266 850 693 601 820 672;
  • 61) 0.130 714 466 450 603 266 850 693 601 820 672 × 2 = 0 + 0.261 428 932 901 206 533 701 387 203 641 344;
  • 62) 0.261 428 932 901 206 533 701 387 203 641 344 × 2 = 0 + 0.522 857 865 802 413 067 402 774 407 282 688;
  • 63) 0.522 857 865 802 413 067 402 774 407 282 688 × 2 = 1 + 0.045 715 731 604 826 134 805 548 814 565 376;
  • 64) 0.045 715 731 604 826 134 805 548 814 565 376 × 2 = 0 + 0.091 431 463 209 652 269 611 097 629 130 752;
  • 65) 0.091 431 463 209 652 269 611 097 629 130 752 × 2 = 0 + 0.182 862 926 419 304 539 222 195 258 261 504;
  • 66) 0.182 862 926 419 304 539 222 195 258 261 504 × 2 = 0 + 0.365 725 852 838 609 078 444 390 516 523 008;
  • 67) 0.365 725 852 838 609 078 444 390 516 523 008 × 2 = 0 + 0.731 451 705 677 218 156 888 781 033 046 016;
  • 68) 0.731 451 705 677 218 156 888 781 033 046 016 × 2 = 1 + 0.462 903 411 354 436 313 777 562 066 092 032;

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:


0.000 020 830 729 321 671 205 134 999 154 722(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:

0.000 020 830 729 321 671 205 134 999 154 722(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:


0.000 020 830 729 321 671 205 134 999 154 722(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 0.000 020 830 729 321 671 205 134 999 154 722 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