0.000 020 830 729 321 671 205 134 999 154 505 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 505 9(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
0.000 020 830 729 321 671 205 134 999 154 505 9(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: 0.000 020 830 729 321 671 205 134 999 154 505 9.

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) 0.000 020 830 729 321 671 205 134 999 154 505 9 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 011 8;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 011 8 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 023 6;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 023 6 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 047 2;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 047 2 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 094 4;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 094 4 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 188 8;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 188 8 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 377 6;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 377 6 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 776 755 2;
  • 8) 0.002 666 333 353 173 914 257 279 891 776 755 2 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 553 510 4;
  • 9) 0.005 332 666 706 347 828 514 559 783 553 510 4 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 107 020 8;
  • 10) 0.010 665 333 412 695 657 029 119 567 107 020 8 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 214 041 6;
  • 11) 0.021 330 666 825 391 314 058 239 134 214 041 6 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 428 083 2;
  • 12) 0.042 661 333 650 782 628 116 478 268 428 083 2 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 856 166 4;
  • 13) 0.085 322 667 301 565 256 232 956 536 856 166 4 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 712 332 8;
  • 14) 0.170 645 334 603 130 512 465 913 073 712 332 8 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 424 665 6;
  • 15) 0.341 290 669 206 261 024 931 826 147 424 665 6 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 849 331 2;
  • 16) 0.682 581 338 412 522 049 863 652 294 849 331 2 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 698 662 4;
  • 17) 0.365 162 676 825 044 099 727 304 589 698 662 4 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 397 324 8;
  • 18) 0.730 325 353 650 088 199 454 609 179 397 324 8 × 2 = 1 + 0.460 650 707 300 176 398 909 218 358 794 649 6;
  • 19) 0.460 650 707 300 176 398 909 218 358 794 649 6 × 2 = 0 + 0.921 301 414 600 352 797 818 436 717 589 299 2;
  • 20) 0.921 301 414 600 352 797 818 436 717 589 299 2 × 2 = 1 + 0.842 602 829 200 705 595 636 873 435 178 598 4;
  • 21) 0.842 602 829 200 705 595 636 873 435 178 598 4 × 2 = 1 + 0.685 205 658 401 411 191 273 746 870 357 196 8;
  • 22) 0.685 205 658 401 411 191 273 746 870 357 196 8 × 2 = 1 + 0.370 411 316 802 822 382 547 493 740 714 393 6;
  • 23) 0.370 411 316 802 822 382 547 493 740 714 393 6 × 2 = 0 + 0.740 822 633 605 644 765 094 987 481 428 787 2;
  • 24) 0.740 822 633 605 644 765 094 987 481 428 787 2 × 2 = 1 + 0.481 645 267 211 289 530 189 974 962 857 574 4;
  • 25) 0.481 645 267 211 289 530 189 974 962 857 574 4 × 2 = 0 + 0.963 290 534 422 579 060 379 949 925 715 148 8;
  • 26) 0.963 290 534 422 579 060 379 949 925 715 148 8 × 2 = 1 + 0.926 581 068 845 158 120 759 899 851 430 297 6;
  • 27) 0.926 581 068 845 158 120 759 899 851 430 297 6 × 2 = 1 + 0.853 162 137 690 316 241 519 799 702 860 595 2;
  • 28) 0.853 162 137 690 316 241 519 799 702 860 595 2 × 2 = 1 + 0.706 324 275 380 632 483 039 599 405 721 190 4;
  • 29) 0.706 324 275 380 632 483 039 599 405 721 190 4 × 2 = 1 + 0.412 648 550 761 264 966 079 198 811 442 380 8;
  • 30) 0.412 648 550 761 264 966 079 198 811 442 380 8 × 2 = 0 + 0.825 297 101 522 529 932 158 397 622 884 761 6;
  • 31) 0.825 297 101 522 529 932 158 397 622 884 761 6 × 2 = 1 + 0.650 594 203 045 059 864 316 795 245 769 523 2;
  • 32) 0.650 594 203 045 059 864 316 795 245 769 523 2 × 2 = 1 + 0.301 188 406 090 119 728 633 590 491 539 046 4;
  • 33) 0.301 188 406 090 119 728 633 590 491 539 046 4 × 2 = 0 + 0.602 376 812 180 239 457 267 180 983 078 092 8;
  • 34) 0.602 376 812 180 239 457 267 180 983 078 092 8 × 2 = 1 + 0.204 753 624 360 478 914 534 361 966 156 185 6;
  • 35) 0.204 753 624 360 478 914 534 361 966 156 185 6 × 2 = 0 + 0.409 507 248 720 957 829 068 723 932 312 371 2;
  • 36) 0.409 507 248 720 957 829 068 723 932 312 371 2 × 2 = 0 + 0.819 014 497 441 915 658 137 447 864 624 742 4;
  • 37) 0.819 014 497 441 915 658 137 447 864 624 742 4 × 2 = 1 + 0.638 028 994 883 831 316 274 895 729 249 484 8;
  • 38) 0.638 028 994 883 831 316 274 895 729 249 484 8 × 2 = 1 + 0.276 057 989 767 662 632 549 791 458 498 969 6;
  • 39) 0.276 057 989 767 662 632 549 791 458 498 969 6 × 2 = 0 + 0.552 115 979 535 325 265 099 582 916 997 939 2;
  • 40) 0.552 115 979 535 325 265 099 582 916 997 939 2 × 2 = 1 + 0.104 231 959 070 650 530 199 165 833 995 878 4;
  • 41) 0.104 231 959 070 650 530 199 165 833 995 878 4 × 2 = 0 + 0.208 463 918 141 301 060 398 331 667 991 756 8;
  • 42) 0.208 463 918 141 301 060 398 331 667 991 756 8 × 2 = 0 + 0.416 927 836 282 602 120 796 663 335 983 513 6;
  • 43) 0.416 927 836 282 602 120 796 663 335 983 513 6 × 2 = 0 + 0.833 855 672 565 204 241 593 326 671 967 027 2;
  • 44) 0.833 855 672 565 204 241 593 326 671 967 027 2 × 2 = 1 + 0.667 711 345 130 408 483 186 653 343 934 054 4;
  • 45) 0.667 711 345 130 408 483 186 653 343 934 054 4 × 2 = 1 + 0.335 422 690 260 816 966 373 306 687 868 108 8;
  • 46) 0.335 422 690 260 816 966 373 306 687 868 108 8 × 2 = 0 + 0.670 845 380 521 633 932 746 613 375 736 217 6;
  • 47) 0.670 845 380 521 633 932 746 613 375 736 217 6 × 2 = 1 + 0.341 690 761 043 267 865 493 226 751 472 435 2;
  • 48) 0.341 690 761 043 267 865 493 226 751 472 435 2 × 2 = 0 + 0.683 381 522 086 535 730 986 453 502 944 870 4;
  • 49) 0.683 381 522 086 535 730 986 453 502 944 870 4 × 2 = 1 + 0.366 763 044 173 071 461 972 907 005 889 740 8;
  • 50) 0.366 763 044 173 071 461 972 907 005 889 740 8 × 2 = 0 + 0.733 526 088 346 142 923 945 814 011 779 481 6;
  • 51) 0.733 526 088 346 142 923 945 814 011 779 481 6 × 2 = 1 + 0.467 052 176 692 285 847 891 628 023 558 963 2;
  • 52) 0.467 052 176 692 285 847 891 628 023 558 963 2 × 2 = 0 + 0.934 104 353 384 571 695 783 256 047 117 926 4;
  • 53) 0.934 104 353 384 571 695 783 256 047 117 926 4 × 2 = 1 + 0.868 208 706 769 143 391 566 512 094 235 852 8;
  • 54) 0.868 208 706 769 143 391 566 512 094 235 852 8 × 2 = 1 + 0.736 417 413 538 286 783 133 024 188 471 705 6;
  • 55) 0.736 417 413 538 286 783 133 024 188 471 705 6 × 2 = 1 + 0.472 834 827 076 573 566 266 048 376 943 411 2;
  • 56) 0.472 834 827 076 573 566 266 048 376 943 411 2 × 2 = 0 + 0.945 669 654 153 147 132 532 096 753 886 822 4;
  • 57) 0.945 669 654 153 147 132 532 096 753 886 822 4 × 2 = 1 + 0.891 339 308 306 294 265 064 193 507 773 644 8;
  • 58) 0.891 339 308 306 294 265 064 193 507 773 644 8 × 2 = 1 + 0.782 678 616 612 588 530 128 387 015 547 289 6;
  • 59) 0.782 678 616 612 588 530 128 387 015 547 289 6 × 2 = 1 + 0.565 357 233 225 177 060 256 774 031 094 579 2;
  • 60) 0.565 357 233 225 177 060 256 774 031 094 579 2 × 2 = 1 + 0.130 714 466 450 354 120 513 548 062 189 158 4;
  • 61) 0.130 714 466 450 354 120 513 548 062 189 158 4 × 2 = 0 + 0.261 428 932 900 708 241 027 096 124 378 316 8;
  • 62) 0.261 428 932 900 708 241 027 096 124 378 316 8 × 2 = 0 + 0.522 857 865 801 416 482 054 192 248 756 633 6;
  • 63) 0.522 857 865 801 416 482 054 192 248 756 633 6 × 2 = 1 + 0.045 715 731 602 832 964 108 384 497 513 267 2;
  • 64) 0.045 715 731 602 832 964 108 384 497 513 267 2 × 2 = 0 + 0.091 431 463 205 665 928 216 768 995 026 534 4;
  • 65) 0.091 431 463 205 665 928 216 768 995 026 534 4 × 2 = 0 + 0.182 862 926 411 331 856 433 537 990 053 068 8;
  • 66) 0.182 862 926 411 331 856 433 537 990 053 068 8 × 2 = 0 + 0.365 725 852 822 663 712 867 075 980 106 137 6;
  • 67) 0.365 725 852 822 663 712 867 075 980 106 137 6 × 2 = 0 + 0.731 451 705 645 327 425 734 151 960 212 275 2;
  • 68) 0.731 451 705 645 327 425 734 151 960 212 275 2 × 2 = 1 + 0.462 903 411 290 654 851 468 303 920 424 550 4;

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:


0.000 020 830 729 321 671 205 134 999 154 505 9(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:

0.000 020 830 729 321 671 205 134 999 154 505 9(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:


0.000 020 830 729 321 671 205 134 999 154 505 9(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 0.000 020 830 729 321 671 205 134 999 154 505 9 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