0.000 020 830 729 321 671 205 134 999 154 510 63 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 510 63(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
0.000 020 830 729 321 671 205 134 999 154 510 63(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: 0.000 020 830 729 321 671 205 134 999 154 510 63.

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) 0.000 020 830 729 321 671 205 134 999 154 510 63 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 021 26;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 021 26 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 042 52;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 042 52 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 085 04;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 085 04 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 170 08;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 170 08 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 340 16;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 340 16 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 680 32;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 680 32 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 360 64;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 360 64 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 721 28;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 721 28 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 109 442 56;
  • 10) 0.010 665 333 412 695 657 029 119 567 109 442 56 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 218 885 12;
  • 11) 0.021 330 666 825 391 314 058 239 134 218 885 12 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 437 770 24;
  • 12) 0.042 661 333 650 782 628 116 478 268 437 770 24 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 875 540 48;
  • 13) 0.085 322 667 301 565 256 232 956 536 875 540 48 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 751 080 96;
  • 14) 0.170 645 334 603 130 512 465 913 073 751 080 96 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 502 161 92;
  • 15) 0.341 290 669 206 261 024 931 826 147 502 161 92 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 004 323 84;
  • 16) 0.682 581 338 412 522 049 863 652 295 004 323 84 × 2 = 1 + 0.365 162 676 825 044 099 727 304 590 008 647 68;
  • 17) 0.365 162 676 825 044 099 727 304 590 008 647 68 × 2 = 0 + 0.730 325 353 650 088 199 454 609 180 017 295 36;
  • 18) 0.730 325 353 650 088 199 454 609 180 017 295 36 × 2 = 1 + 0.460 650 707 300 176 398 909 218 360 034 590 72;
  • 19) 0.460 650 707 300 176 398 909 218 360 034 590 72 × 2 = 0 + 0.921 301 414 600 352 797 818 436 720 069 181 44;
  • 20) 0.921 301 414 600 352 797 818 436 720 069 181 44 × 2 = 1 + 0.842 602 829 200 705 595 636 873 440 138 362 88;
  • 21) 0.842 602 829 200 705 595 636 873 440 138 362 88 × 2 = 1 + 0.685 205 658 401 411 191 273 746 880 276 725 76;
  • 22) 0.685 205 658 401 411 191 273 746 880 276 725 76 × 2 = 1 + 0.370 411 316 802 822 382 547 493 760 553 451 52;
  • 23) 0.370 411 316 802 822 382 547 493 760 553 451 52 × 2 = 0 + 0.740 822 633 605 644 765 094 987 521 106 903 04;
  • 24) 0.740 822 633 605 644 765 094 987 521 106 903 04 × 2 = 1 + 0.481 645 267 211 289 530 189 975 042 213 806 08;
  • 25) 0.481 645 267 211 289 530 189 975 042 213 806 08 × 2 = 0 + 0.963 290 534 422 579 060 379 950 084 427 612 16;
  • 26) 0.963 290 534 422 579 060 379 950 084 427 612 16 × 2 = 1 + 0.926 581 068 845 158 120 759 900 168 855 224 32;
  • 27) 0.926 581 068 845 158 120 759 900 168 855 224 32 × 2 = 1 + 0.853 162 137 690 316 241 519 800 337 710 448 64;
  • 28) 0.853 162 137 690 316 241 519 800 337 710 448 64 × 2 = 1 + 0.706 324 275 380 632 483 039 600 675 420 897 28;
  • 29) 0.706 324 275 380 632 483 039 600 675 420 897 28 × 2 = 1 + 0.412 648 550 761 264 966 079 201 350 841 794 56;
  • 30) 0.412 648 550 761 264 966 079 201 350 841 794 56 × 2 = 0 + 0.825 297 101 522 529 932 158 402 701 683 589 12;
  • 31) 0.825 297 101 522 529 932 158 402 701 683 589 12 × 2 = 1 + 0.650 594 203 045 059 864 316 805 403 367 178 24;
  • 32) 0.650 594 203 045 059 864 316 805 403 367 178 24 × 2 = 1 + 0.301 188 406 090 119 728 633 610 806 734 356 48;
  • 33) 0.301 188 406 090 119 728 633 610 806 734 356 48 × 2 = 0 + 0.602 376 812 180 239 457 267 221 613 468 712 96;
  • 34) 0.602 376 812 180 239 457 267 221 613 468 712 96 × 2 = 1 + 0.204 753 624 360 478 914 534 443 226 937 425 92;
  • 35) 0.204 753 624 360 478 914 534 443 226 937 425 92 × 2 = 0 + 0.409 507 248 720 957 829 068 886 453 874 851 84;
  • 36) 0.409 507 248 720 957 829 068 886 453 874 851 84 × 2 = 0 + 0.819 014 497 441 915 658 137 772 907 749 703 68;
  • 37) 0.819 014 497 441 915 658 137 772 907 749 703 68 × 2 = 1 + 0.638 028 994 883 831 316 275 545 815 499 407 36;
  • 38) 0.638 028 994 883 831 316 275 545 815 499 407 36 × 2 = 1 + 0.276 057 989 767 662 632 551 091 630 998 814 72;
  • 39) 0.276 057 989 767 662 632 551 091 630 998 814 72 × 2 = 0 + 0.552 115 979 535 325 265 102 183 261 997 629 44;
  • 40) 0.552 115 979 535 325 265 102 183 261 997 629 44 × 2 = 1 + 0.104 231 959 070 650 530 204 366 523 995 258 88;
  • 41) 0.104 231 959 070 650 530 204 366 523 995 258 88 × 2 = 0 + 0.208 463 918 141 301 060 408 733 047 990 517 76;
  • 42) 0.208 463 918 141 301 060 408 733 047 990 517 76 × 2 = 0 + 0.416 927 836 282 602 120 817 466 095 981 035 52;
  • 43) 0.416 927 836 282 602 120 817 466 095 981 035 52 × 2 = 0 + 0.833 855 672 565 204 241 634 932 191 962 071 04;
  • 44) 0.833 855 672 565 204 241 634 932 191 962 071 04 × 2 = 1 + 0.667 711 345 130 408 483 269 864 383 924 142 08;
  • 45) 0.667 711 345 130 408 483 269 864 383 924 142 08 × 2 = 1 + 0.335 422 690 260 816 966 539 728 767 848 284 16;
  • 46) 0.335 422 690 260 816 966 539 728 767 848 284 16 × 2 = 0 + 0.670 845 380 521 633 933 079 457 535 696 568 32;
  • 47) 0.670 845 380 521 633 933 079 457 535 696 568 32 × 2 = 1 + 0.341 690 761 043 267 866 158 915 071 393 136 64;
  • 48) 0.341 690 761 043 267 866 158 915 071 393 136 64 × 2 = 0 + 0.683 381 522 086 535 732 317 830 142 786 273 28;
  • 49) 0.683 381 522 086 535 732 317 830 142 786 273 28 × 2 = 1 + 0.366 763 044 173 071 464 635 660 285 572 546 56;
  • 50) 0.366 763 044 173 071 464 635 660 285 572 546 56 × 2 = 0 + 0.733 526 088 346 142 929 271 320 571 145 093 12;
  • 51) 0.733 526 088 346 142 929 271 320 571 145 093 12 × 2 = 1 + 0.467 052 176 692 285 858 542 641 142 290 186 24;
  • 52) 0.467 052 176 692 285 858 542 641 142 290 186 24 × 2 = 0 + 0.934 104 353 384 571 717 085 282 284 580 372 48;
  • 53) 0.934 104 353 384 571 717 085 282 284 580 372 48 × 2 = 1 + 0.868 208 706 769 143 434 170 564 569 160 744 96;
  • 54) 0.868 208 706 769 143 434 170 564 569 160 744 96 × 2 = 1 + 0.736 417 413 538 286 868 341 129 138 321 489 92;
  • 55) 0.736 417 413 538 286 868 341 129 138 321 489 92 × 2 = 1 + 0.472 834 827 076 573 736 682 258 276 642 979 84;
  • 56) 0.472 834 827 076 573 736 682 258 276 642 979 84 × 2 = 0 + 0.945 669 654 153 147 473 364 516 553 285 959 68;
  • 57) 0.945 669 654 153 147 473 364 516 553 285 959 68 × 2 = 1 + 0.891 339 308 306 294 946 729 033 106 571 919 36;
  • 58) 0.891 339 308 306 294 946 729 033 106 571 919 36 × 2 = 1 + 0.782 678 616 612 589 893 458 066 213 143 838 72;
  • 59) 0.782 678 616 612 589 893 458 066 213 143 838 72 × 2 = 1 + 0.565 357 233 225 179 786 916 132 426 287 677 44;
  • 60) 0.565 357 233 225 179 786 916 132 426 287 677 44 × 2 = 1 + 0.130 714 466 450 359 573 832 264 852 575 354 88;
  • 61) 0.130 714 466 450 359 573 832 264 852 575 354 88 × 2 = 0 + 0.261 428 932 900 719 147 664 529 705 150 709 76;
  • 62) 0.261 428 932 900 719 147 664 529 705 150 709 76 × 2 = 0 + 0.522 857 865 801 438 295 329 059 410 301 419 52;
  • 63) 0.522 857 865 801 438 295 329 059 410 301 419 52 × 2 = 1 + 0.045 715 731 602 876 590 658 118 820 602 839 04;
  • 64) 0.045 715 731 602 876 590 658 118 820 602 839 04 × 2 = 0 + 0.091 431 463 205 753 181 316 237 641 205 678 08;
  • 65) 0.091 431 463 205 753 181 316 237 641 205 678 08 × 2 = 0 + 0.182 862 926 411 506 362 632 475 282 411 356 16;
  • 66) 0.182 862 926 411 506 362 632 475 282 411 356 16 × 2 = 0 + 0.365 725 852 823 012 725 264 950 564 822 712 32;
  • 67) 0.365 725 852 823 012 725 264 950 564 822 712 32 × 2 = 0 + 0.731 451 705 646 025 450 529 901 129 645 424 64;
  • 68) 0.731 451 705 646 025 450 529 901 129 645 424 64 × 2 = 1 + 0.462 903 411 292 050 901 059 802 259 290 849 28;

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:


0.000 020 830 729 321 671 205 134 999 154 510 63(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:

0.000 020 830 729 321 671 205 134 999 154 510 63(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:


0.000 020 830 729 321 671 205 134 999 154 510 63(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 0.000 020 830 729 321 671 205 134 999 154 510 63 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