0.000 020 830 729 321 671 205 134 999 154 509 631 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 631 5(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
0.000 020 830 729 321 671 205 134 999 154 509 631 5(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: 0.000 020 830 729 321 671 205 134 999 154 509 631 5.

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) 0.000 020 830 729 321 671 205 134 999 154 509 631 5 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 263;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 263 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 526;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 526 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 052;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 052 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 104;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 104 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 308 208;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 308 208 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 616 416;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 616 416 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 232 832;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 232 832 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 465 664;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 465 664 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 931 328;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 931 328 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 862 656;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 862 656 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 725 312;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 725 312 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 450 624;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 450 624 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 742 901 248;
  • 14) 0.170 645 334 603 130 512 465 913 073 742 901 248 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 485 802 496;
  • 15) 0.341 290 669 206 261 024 931 826 147 485 802 496 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 971 604 992;
  • 16) 0.682 581 338 412 522 049 863 652 294 971 604 992 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 943 209 984;
  • 17) 0.365 162 676 825 044 099 727 304 589 943 209 984 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 886 419 968;
  • 18) 0.730 325 353 650 088 199 454 609 179 886 419 968 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 772 839 936;
  • 19) 0.460 650 707 300 176 398 909 218 359 772 839 936 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 545 679 872;
  • 20) 0.921 301 414 600 352 797 818 436 719 545 679 872 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 091 359 744;
  • 21) 0.842 602 829 200 705 595 636 873 439 091 359 744 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 182 719 488;
  • 22) 0.685 205 658 401 411 191 273 746 878 182 719 488 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 365 438 976;
  • 23) 0.370 411 316 802 822 382 547 493 756 365 438 976 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 730 877 952;
  • 24) 0.740 822 633 605 644 765 094 987 512 730 877 952 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 461 755 904;
  • 25) 0.481 645 267 211 289 530 189 975 025 461 755 904 × 2 = 0 + 0.963 290 534 422 579 060 379 950 050 923 511 808;
  • 26) 0.963 290 534 422 579 060 379 950 050 923 511 808 × 2 = 1 + 0.926 581 068 845 158 120 759 900 101 847 023 616;
  • 27) 0.926 581 068 845 158 120 759 900 101 847 023 616 × 2 = 1 + 0.853 162 137 690 316 241 519 800 203 694 047 232;
  • 28) 0.853 162 137 690 316 241 519 800 203 694 047 232 × 2 = 1 + 0.706 324 275 380 632 483 039 600 407 388 094 464;
  • 29) 0.706 324 275 380 632 483 039 600 407 388 094 464 × 2 = 1 + 0.412 648 550 761 264 966 079 200 814 776 188 928;
  • 30) 0.412 648 550 761 264 966 079 200 814 776 188 928 × 2 = 0 + 0.825 297 101 522 529 932 158 401 629 552 377 856;
  • 31) 0.825 297 101 522 529 932 158 401 629 552 377 856 × 2 = 1 + 0.650 594 203 045 059 864 316 803 259 104 755 712;
  • 32) 0.650 594 203 045 059 864 316 803 259 104 755 712 × 2 = 1 + 0.301 188 406 090 119 728 633 606 518 209 511 424;
  • 33) 0.301 188 406 090 119 728 633 606 518 209 511 424 × 2 = 0 + 0.602 376 812 180 239 457 267 213 036 419 022 848;
  • 34) 0.602 376 812 180 239 457 267 213 036 419 022 848 × 2 = 1 + 0.204 753 624 360 478 914 534 426 072 838 045 696;
  • 35) 0.204 753 624 360 478 914 534 426 072 838 045 696 × 2 = 0 + 0.409 507 248 720 957 829 068 852 145 676 091 392;
  • 36) 0.409 507 248 720 957 829 068 852 145 676 091 392 × 2 = 0 + 0.819 014 497 441 915 658 137 704 291 352 182 784;
  • 37) 0.819 014 497 441 915 658 137 704 291 352 182 784 × 2 = 1 + 0.638 028 994 883 831 316 275 408 582 704 365 568;
  • 38) 0.638 028 994 883 831 316 275 408 582 704 365 568 × 2 = 1 + 0.276 057 989 767 662 632 550 817 165 408 731 136;
  • 39) 0.276 057 989 767 662 632 550 817 165 408 731 136 × 2 = 0 + 0.552 115 979 535 325 265 101 634 330 817 462 272;
  • 40) 0.552 115 979 535 325 265 101 634 330 817 462 272 × 2 = 1 + 0.104 231 959 070 650 530 203 268 661 634 924 544;
  • 41) 0.104 231 959 070 650 530 203 268 661 634 924 544 × 2 = 0 + 0.208 463 918 141 301 060 406 537 323 269 849 088;
  • 42) 0.208 463 918 141 301 060 406 537 323 269 849 088 × 2 = 0 + 0.416 927 836 282 602 120 813 074 646 539 698 176;
  • 43) 0.416 927 836 282 602 120 813 074 646 539 698 176 × 2 = 0 + 0.833 855 672 565 204 241 626 149 293 079 396 352;
  • 44) 0.833 855 672 565 204 241 626 149 293 079 396 352 × 2 = 1 + 0.667 711 345 130 408 483 252 298 586 158 792 704;
  • 45) 0.667 711 345 130 408 483 252 298 586 158 792 704 × 2 = 1 + 0.335 422 690 260 816 966 504 597 172 317 585 408;
  • 46) 0.335 422 690 260 816 966 504 597 172 317 585 408 × 2 = 0 + 0.670 845 380 521 633 933 009 194 344 635 170 816;
  • 47) 0.670 845 380 521 633 933 009 194 344 635 170 816 × 2 = 1 + 0.341 690 761 043 267 866 018 388 689 270 341 632;
  • 48) 0.341 690 761 043 267 866 018 388 689 270 341 632 × 2 = 0 + 0.683 381 522 086 535 732 036 777 378 540 683 264;
  • 49) 0.683 381 522 086 535 732 036 777 378 540 683 264 × 2 = 1 + 0.366 763 044 173 071 464 073 554 757 081 366 528;
  • 50) 0.366 763 044 173 071 464 073 554 757 081 366 528 × 2 = 0 + 0.733 526 088 346 142 928 147 109 514 162 733 056;
  • 51) 0.733 526 088 346 142 928 147 109 514 162 733 056 × 2 = 1 + 0.467 052 176 692 285 856 294 219 028 325 466 112;
  • 52) 0.467 052 176 692 285 856 294 219 028 325 466 112 × 2 = 0 + 0.934 104 353 384 571 712 588 438 056 650 932 224;
  • 53) 0.934 104 353 384 571 712 588 438 056 650 932 224 × 2 = 1 + 0.868 208 706 769 143 425 176 876 113 301 864 448;
  • 54) 0.868 208 706 769 143 425 176 876 113 301 864 448 × 2 = 1 + 0.736 417 413 538 286 850 353 752 226 603 728 896;
  • 55) 0.736 417 413 538 286 850 353 752 226 603 728 896 × 2 = 1 + 0.472 834 827 076 573 700 707 504 453 207 457 792;
  • 56) 0.472 834 827 076 573 700 707 504 453 207 457 792 × 2 = 0 + 0.945 669 654 153 147 401 415 008 906 414 915 584;
  • 57) 0.945 669 654 153 147 401 415 008 906 414 915 584 × 2 = 1 + 0.891 339 308 306 294 802 830 017 812 829 831 168;
  • 58) 0.891 339 308 306 294 802 830 017 812 829 831 168 × 2 = 1 + 0.782 678 616 612 589 605 660 035 625 659 662 336;
  • 59) 0.782 678 616 612 589 605 660 035 625 659 662 336 × 2 = 1 + 0.565 357 233 225 179 211 320 071 251 319 324 672;
  • 60) 0.565 357 233 225 179 211 320 071 251 319 324 672 × 2 = 1 + 0.130 714 466 450 358 422 640 142 502 638 649 344;
  • 61) 0.130 714 466 450 358 422 640 142 502 638 649 344 × 2 = 0 + 0.261 428 932 900 716 845 280 285 005 277 298 688;
  • 62) 0.261 428 932 900 716 845 280 285 005 277 298 688 × 2 = 0 + 0.522 857 865 801 433 690 560 570 010 554 597 376;
  • 63) 0.522 857 865 801 433 690 560 570 010 554 597 376 × 2 = 1 + 0.045 715 731 602 867 381 121 140 021 109 194 752;
  • 64) 0.045 715 731 602 867 381 121 140 021 109 194 752 × 2 = 0 + 0.091 431 463 205 734 762 242 280 042 218 389 504;
  • 65) 0.091 431 463 205 734 762 242 280 042 218 389 504 × 2 = 0 + 0.182 862 926 411 469 524 484 560 084 436 779 008;
  • 66) 0.182 862 926 411 469 524 484 560 084 436 779 008 × 2 = 0 + 0.365 725 852 822 939 048 969 120 168 873 558 016;
  • 67) 0.365 725 852 822 939 048 969 120 168 873 558 016 × 2 = 0 + 0.731 451 705 645 878 097 938 240 337 747 116 032;
  • 68) 0.731 451 705 645 878 097 938 240 337 747 116 032 × 2 = 1 + 0.462 903 411 291 756 195 876 480 675 494 232 064;

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:


0.000 020 830 729 321 671 205 134 999 154 509 631 5(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:

0.000 020 830 729 321 671 205 134 999 154 509 631 5(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:


0.000 020 830 729 321 671 205 134 999 154 509 631 5(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 0.000 020 830 729 321 671 205 134 999 154 509 631 5 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