0.000 020 830 729 321 671 205 134 999 154 509 568 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 568(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
0.000 020 830 729 321 671 205 134 999 154 509 568(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: 0.000 020 830 729 321 671 205 134 999 154 509 568.

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) 0.000 020 830 729 321 671 205 134 999 154 509 568 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 136;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 136 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 272;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 272 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 076 544;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 076 544 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 153 088;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 153 088 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 306 176;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 306 176 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 612 352;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 612 352 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 224 704;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 224 704 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 449 408;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 449 408 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 898 816;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 898 816 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 797 632;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 797 632 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 595 264;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 595 264 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 190 528;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 190 528 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 742 381 056;
  • 14) 0.170 645 334 603 130 512 465 913 073 742 381 056 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 484 762 112;
  • 15) 0.341 290 669 206 261 024 931 826 147 484 762 112 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 969 524 224;
  • 16) 0.682 581 338 412 522 049 863 652 294 969 524 224 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 939 048 448;
  • 17) 0.365 162 676 825 044 099 727 304 589 939 048 448 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 878 096 896;
  • 18) 0.730 325 353 650 088 199 454 609 179 878 096 896 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 756 193 792;
  • 19) 0.460 650 707 300 176 398 909 218 359 756 193 792 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 512 387 584;
  • 20) 0.921 301 414 600 352 797 818 436 719 512 387 584 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 024 775 168;
  • 21) 0.842 602 829 200 705 595 636 873 439 024 775 168 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 049 550 336;
  • 22) 0.685 205 658 401 411 191 273 746 878 049 550 336 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 099 100 672;
  • 23) 0.370 411 316 802 822 382 547 493 756 099 100 672 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 198 201 344;
  • 24) 0.740 822 633 605 644 765 094 987 512 198 201 344 × 2 = 1 + 0.481 645 267 211 289 530 189 975 024 396 402 688;
  • 25) 0.481 645 267 211 289 530 189 975 024 396 402 688 × 2 = 0 + 0.963 290 534 422 579 060 379 950 048 792 805 376;
  • 26) 0.963 290 534 422 579 060 379 950 048 792 805 376 × 2 = 1 + 0.926 581 068 845 158 120 759 900 097 585 610 752;
  • 27) 0.926 581 068 845 158 120 759 900 097 585 610 752 × 2 = 1 + 0.853 162 137 690 316 241 519 800 195 171 221 504;
  • 28) 0.853 162 137 690 316 241 519 800 195 171 221 504 × 2 = 1 + 0.706 324 275 380 632 483 039 600 390 342 443 008;
  • 29) 0.706 324 275 380 632 483 039 600 390 342 443 008 × 2 = 1 + 0.412 648 550 761 264 966 079 200 780 684 886 016;
  • 30) 0.412 648 550 761 264 966 079 200 780 684 886 016 × 2 = 0 + 0.825 297 101 522 529 932 158 401 561 369 772 032;
  • 31) 0.825 297 101 522 529 932 158 401 561 369 772 032 × 2 = 1 + 0.650 594 203 045 059 864 316 803 122 739 544 064;
  • 32) 0.650 594 203 045 059 864 316 803 122 739 544 064 × 2 = 1 + 0.301 188 406 090 119 728 633 606 245 479 088 128;
  • 33) 0.301 188 406 090 119 728 633 606 245 479 088 128 × 2 = 0 + 0.602 376 812 180 239 457 267 212 490 958 176 256;
  • 34) 0.602 376 812 180 239 457 267 212 490 958 176 256 × 2 = 1 + 0.204 753 624 360 478 914 534 424 981 916 352 512;
  • 35) 0.204 753 624 360 478 914 534 424 981 916 352 512 × 2 = 0 + 0.409 507 248 720 957 829 068 849 963 832 705 024;
  • 36) 0.409 507 248 720 957 829 068 849 963 832 705 024 × 2 = 0 + 0.819 014 497 441 915 658 137 699 927 665 410 048;
  • 37) 0.819 014 497 441 915 658 137 699 927 665 410 048 × 2 = 1 + 0.638 028 994 883 831 316 275 399 855 330 820 096;
  • 38) 0.638 028 994 883 831 316 275 399 855 330 820 096 × 2 = 1 + 0.276 057 989 767 662 632 550 799 710 661 640 192;
  • 39) 0.276 057 989 767 662 632 550 799 710 661 640 192 × 2 = 0 + 0.552 115 979 535 325 265 101 599 421 323 280 384;
  • 40) 0.552 115 979 535 325 265 101 599 421 323 280 384 × 2 = 1 + 0.104 231 959 070 650 530 203 198 842 646 560 768;
  • 41) 0.104 231 959 070 650 530 203 198 842 646 560 768 × 2 = 0 + 0.208 463 918 141 301 060 406 397 685 293 121 536;
  • 42) 0.208 463 918 141 301 060 406 397 685 293 121 536 × 2 = 0 + 0.416 927 836 282 602 120 812 795 370 586 243 072;
  • 43) 0.416 927 836 282 602 120 812 795 370 586 243 072 × 2 = 0 + 0.833 855 672 565 204 241 625 590 741 172 486 144;
  • 44) 0.833 855 672 565 204 241 625 590 741 172 486 144 × 2 = 1 + 0.667 711 345 130 408 483 251 181 482 344 972 288;
  • 45) 0.667 711 345 130 408 483 251 181 482 344 972 288 × 2 = 1 + 0.335 422 690 260 816 966 502 362 964 689 944 576;
  • 46) 0.335 422 690 260 816 966 502 362 964 689 944 576 × 2 = 0 + 0.670 845 380 521 633 933 004 725 929 379 889 152;
  • 47) 0.670 845 380 521 633 933 004 725 929 379 889 152 × 2 = 1 + 0.341 690 761 043 267 866 009 451 858 759 778 304;
  • 48) 0.341 690 761 043 267 866 009 451 858 759 778 304 × 2 = 0 + 0.683 381 522 086 535 732 018 903 717 519 556 608;
  • 49) 0.683 381 522 086 535 732 018 903 717 519 556 608 × 2 = 1 + 0.366 763 044 173 071 464 037 807 435 039 113 216;
  • 50) 0.366 763 044 173 071 464 037 807 435 039 113 216 × 2 = 0 + 0.733 526 088 346 142 928 075 614 870 078 226 432;
  • 51) 0.733 526 088 346 142 928 075 614 870 078 226 432 × 2 = 1 + 0.467 052 176 692 285 856 151 229 740 156 452 864;
  • 52) 0.467 052 176 692 285 856 151 229 740 156 452 864 × 2 = 0 + 0.934 104 353 384 571 712 302 459 480 312 905 728;
  • 53) 0.934 104 353 384 571 712 302 459 480 312 905 728 × 2 = 1 + 0.868 208 706 769 143 424 604 918 960 625 811 456;
  • 54) 0.868 208 706 769 143 424 604 918 960 625 811 456 × 2 = 1 + 0.736 417 413 538 286 849 209 837 921 251 622 912;
  • 55) 0.736 417 413 538 286 849 209 837 921 251 622 912 × 2 = 1 + 0.472 834 827 076 573 698 419 675 842 503 245 824;
  • 56) 0.472 834 827 076 573 698 419 675 842 503 245 824 × 2 = 0 + 0.945 669 654 153 147 396 839 351 685 006 491 648;
  • 57) 0.945 669 654 153 147 396 839 351 685 006 491 648 × 2 = 1 + 0.891 339 308 306 294 793 678 703 370 012 983 296;
  • 58) 0.891 339 308 306 294 793 678 703 370 012 983 296 × 2 = 1 + 0.782 678 616 612 589 587 357 406 740 025 966 592;
  • 59) 0.782 678 616 612 589 587 357 406 740 025 966 592 × 2 = 1 + 0.565 357 233 225 179 174 714 813 480 051 933 184;
  • 60) 0.565 357 233 225 179 174 714 813 480 051 933 184 × 2 = 1 + 0.130 714 466 450 358 349 429 626 960 103 866 368;
  • 61) 0.130 714 466 450 358 349 429 626 960 103 866 368 × 2 = 0 + 0.261 428 932 900 716 698 859 253 920 207 732 736;
  • 62) 0.261 428 932 900 716 698 859 253 920 207 732 736 × 2 = 0 + 0.522 857 865 801 433 397 718 507 840 415 465 472;
  • 63) 0.522 857 865 801 433 397 718 507 840 415 465 472 × 2 = 1 + 0.045 715 731 602 866 795 437 015 680 830 930 944;
  • 64) 0.045 715 731 602 866 795 437 015 680 830 930 944 × 2 = 0 + 0.091 431 463 205 733 590 874 031 361 661 861 888;
  • 65) 0.091 431 463 205 733 590 874 031 361 661 861 888 × 2 = 0 + 0.182 862 926 411 467 181 748 062 723 323 723 776;
  • 66) 0.182 862 926 411 467 181 748 062 723 323 723 776 × 2 = 0 + 0.365 725 852 822 934 363 496 125 446 647 447 552;
  • 67) 0.365 725 852 822 934 363 496 125 446 647 447 552 × 2 = 0 + 0.731 451 705 645 868 726 992 250 893 294 895 104;
  • 68) 0.731 451 705 645 868 726 992 250 893 294 895 104 × 2 = 1 + 0.462 903 411 291 737 453 984 501 786 589 790 208;

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:


0.000 020 830 729 321 671 205 134 999 154 509 568(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:

0.000 020 830 729 321 671 205 134 999 154 509 568(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:


0.000 020 830 729 321 671 205 134 999 154 509 568(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 0.000 020 830 729 321 671 205 134 999 154 509 568 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