0.000 020 830 729 321 671 205 134 999 154 519 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 519 2(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
0.000 020 830 729 321 671 205 134 999 154 519 2(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: 0.000 020 830 729 321 671 205 134 999 154 519 2.

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) 0.000 020 830 729 321 671 205 134 999 154 519 2 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 038 4;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 038 4 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 076 8;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 076 8 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 153 6;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 153 6 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 307 2;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 307 2 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 614 4;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 614 4 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 889 228 8;
  • 7) 0.001 333 166 676 586 957 128 639 945 889 228 8 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 778 457 6;
  • 8) 0.002 666 333 353 173 914 257 279 891 778 457 6 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 556 915 2;
  • 9) 0.005 332 666 706 347 828 514 559 783 556 915 2 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 113 830 4;
  • 10) 0.010 665 333 412 695 657 029 119 567 113 830 4 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 227 660 8;
  • 11) 0.021 330 666 825 391 314 058 239 134 227 660 8 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 455 321 6;
  • 12) 0.042 661 333 650 782 628 116 478 268 455 321 6 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 910 643 2;
  • 13) 0.085 322 667 301 565 256 232 956 536 910 643 2 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 821 286 4;
  • 14) 0.170 645 334 603 130 512 465 913 073 821 286 4 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 642 572 8;
  • 15) 0.341 290 669 206 261 024 931 826 147 642 572 8 × 2 = 0 + 0.682 581 338 412 522 049 863 652 295 285 145 6;
  • 16) 0.682 581 338 412 522 049 863 652 295 285 145 6 × 2 = 1 + 0.365 162 676 825 044 099 727 304 590 570 291 2;
  • 17) 0.365 162 676 825 044 099 727 304 590 570 291 2 × 2 = 0 + 0.730 325 353 650 088 199 454 609 181 140 582 4;
  • 18) 0.730 325 353 650 088 199 454 609 181 140 582 4 × 2 = 1 + 0.460 650 707 300 176 398 909 218 362 281 164 8;
  • 19) 0.460 650 707 300 176 398 909 218 362 281 164 8 × 2 = 0 + 0.921 301 414 600 352 797 818 436 724 562 329 6;
  • 20) 0.921 301 414 600 352 797 818 436 724 562 329 6 × 2 = 1 + 0.842 602 829 200 705 595 636 873 449 124 659 2;
  • 21) 0.842 602 829 200 705 595 636 873 449 124 659 2 × 2 = 1 + 0.685 205 658 401 411 191 273 746 898 249 318 4;
  • 22) 0.685 205 658 401 411 191 273 746 898 249 318 4 × 2 = 1 + 0.370 411 316 802 822 382 547 493 796 498 636 8;
  • 23) 0.370 411 316 802 822 382 547 493 796 498 636 8 × 2 = 0 + 0.740 822 633 605 644 765 094 987 592 997 273 6;
  • 24) 0.740 822 633 605 644 765 094 987 592 997 273 6 × 2 = 1 + 0.481 645 267 211 289 530 189 975 185 994 547 2;
  • 25) 0.481 645 267 211 289 530 189 975 185 994 547 2 × 2 = 0 + 0.963 290 534 422 579 060 379 950 371 989 094 4;
  • 26) 0.963 290 534 422 579 060 379 950 371 989 094 4 × 2 = 1 + 0.926 581 068 845 158 120 759 900 743 978 188 8;
  • 27) 0.926 581 068 845 158 120 759 900 743 978 188 8 × 2 = 1 + 0.853 162 137 690 316 241 519 801 487 956 377 6;
  • 28) 0.853 162 137 690 316 241 519 801 487 956 377 6 × 2 = 1 + 0.706 324 275 380 632 483 039 602 975 912 755 2;
  • 29) 0.706 324 275 380 632 483 039 602 975 912 755 2 × 2 = 1 + 0.412 648 550 761 264 966 079 205 951 825 510 4;
  • 30) 0.412 648 550 761 264 966 079 205 951 825 510 4 × 2 = 0 + 0.825 297 101 522 529 932 158 411 903 651 020 8;
  • 31) 0.825 297 101 522 529 932 158 411 903 651 020 8 × 2 = 1 + 0.650 594 203 045 059 864 316 823 807 302 041 6;
  • 32) 0.650 594 203 045 059 864 316 823 807 302 041 6 × 2 = 1 + 0.301 188 406 090 119 728 633 647 614 604 083 2;
  • 33) 0.301 188 406 090 119 728 633 647 614 604 083 2 × 2 = 0 + 0.602 376 812 180 239 457 267 295 229 208 166 4;
  • 34) 0.602 376 812 180 239 457 267 295 229 208 166 4 × 2 = 1 + 0.204 753 624 360 478 914 534 590 458 416 332 8;
  • 35) 0.204 753 624 360 478 914 534 590 458 416 332 8 × 2 = 0 + 0.409 507 248 720 957 829 069 180 916 832 665 6;
  • 36) 0.409 507 248 720 957 829 069 180 916 832 665 6 × 2 = 0 + 0.819 014 497 441 915 658 138 361 833 665 331 2;
  • 37) 0.819 014 497 441 915 658 138 361 833 665 331 2 × 2 = 1 + 0.638 028 994 883 831 316 276 723 667 330 662 4;
  • 38) 0.638 028 994 883 831 316 276 723 667 330 662 4 × 2 = 1 + 0.276 057 989 767 662 632 553 447 334 661 324 8;
  • 39) 0.276 057 989 767 662 632 553 447 334 661 324 8 × 2 = 0 + 0.552 115 979 535 325 265 106 894 669 322 649 6;
  • 40) 0.552 115 979 535 325 265 106 894 669 322 649 6 × 2 = 1 + 0.104 231 959 070 650 530 213 789 338 645 299 2;
  • 41) 0.104 231 959 070 650 530 213 789 338 645 299 2 × 2 = 0 + 0.208 463 918 141 301 060 427 578 677 290 598 4;
  • 42) 0.208 463 918 141 301 060 427 578 677 290 598 4 × 2 = 0 + 0.416 927 836 282 602 120 855 157 354 581 196 8;
  • 43) 0.416 927 836 282 602 120 855 157 354 581 196 8 × 2 = 0 + 0.833 855 672 565 204 241 710 314 709 162 393 6;
  • 44) 0.833 855 672 565 204 241 710 314 709 162 393 6 × 2 = 1 + 0.667 711 345 130 408 483 420 629 418 324 787 2;
  • 45) 0.667 711 345 130 408 483 420 629 418 324 787 2 × 2 = 1 + 0.335 422 690 260 816 966 841 258 836 649 574 4;
  • 46) 0.335 422 690 260 816 966 841 258 836 649 574 4 × 2 = 0 + 0.670 845 380 521 633 933 682 517 673 299 148 8;
  • 47) 0.670 845 380 521 633 933 682 517 673 299 148 8 × 2 = 1 + 0.341 690 761 043 267 867 365 035 346 598 297 6;
  • 48) 0.341 690 761 043 267 867 365 035 346 598 297 6 × 2 = 0 + 0.683 381 522 086 535 734 730 070 693 196 595 2;
  • 49) 0.683 381 522 086 535 734 730 070 693 196 595 2 × 2 = 1 + 0.366 763 044 173 071 469 460 141 386 393 190 4;
  • 50) 0.366 763 044 173 071 469 460 141 386 393 190 4 × 2 = 0 + 0.733 526 088 346 142 938 920 282 772 786 380 8;
  • 51) 0.733 526 088 346 142 938 920 282 772 786 380 8 × 2 = 1 + 0.467 052 176 692 285 877 840 565 545 572 761 6;
  • 52) 0.467 052 176 692 285 877 840 565 545 572 761 6 × 2 = 0 + 0.934 104 353 384 571 755 681 131 091 145 523 2;
  • 53) 0.934 104 353 384 571 755 681 131 091 145 523 2 × 2 = 1 + 0.868 208 706 769 143 511 362 262 182 291 046 4;
  • 54) 0.868 208 706 769 143 511 362 262 182 291 046 4 × 2 = 1 + 0.736 417 413 538 287 022 724 524 364 582 092 8;
  • 55) 0.736 417 413 538 287 022 724 524 364 582 092 8 × 2 = 1 + 0.472 834 827 076 574 045 449 048 729 164 185 6;
  • 56) 0.472 834 827 076 574 045 449 048 729 164 185 6 × 2 = 0 + 0.945 669 654 153 148 090 898 097 458 328 371 2;
  • 57) 0.945 669 654 153 148 090 898 097 458 328 371 2 × 2 = 1 + 0.891 339 308 306 296 181 796 194 916 656 742 4;
  • 58) 0.891 339 308 306 296 181 796 194 916 656 742 4 × 2 = 1 + 0.782 678 616 612 592 363 592 389 833 313 484 8;
  • 59) 0.782 678 616 612 592 363 592 389 833 313 484 8 × 2 = 1 + 0.565 357 233 225 184 727 184 779 666 626 969 6;
  • 60) 0.565 357 233 225 184 727 184 779 666 626 969 6 × 2 = 1 + 0.130 714 466 450 369 454 369 559 333 253 939 2;
  • 61) 0.130 714 466 450 369 454 369 559 333 253 939 2 × 2 = 0 + 0.261 428 932 900 738 908 739 118 666 507 878 4;
  • 62) 0.261 428 932 900 738 908 739 118 666 507 878 4 × 2 = 0 + 0.522 857 865 801 477 817 478 237 333 015 756 8;
  • 63) 0.522 857 865 801 477 817 478 237 333 015 756 8 × 2 = 1 + 0.045 715 731 602 955 634 956 474 666 031 513 6;
  • 64) 0.045 715 731 602 955 634 956 474 666 031 513 6 × 2 = 0 + 0.091 431 463 205 911 269 912 949 332 063 027 2;
  • 65) 0.091 431 463 205 911 269 912 949 332 063 027 2 × 2 = 0 + 0.182 862 926 411 822 539 825 898 664 126 054 4;
  • 66) 0.182 862 926 411 822 539 825 898 664 126 054 4 × 2 = 0 + 0.365 725 852 823 645 079 651 797 328 252 108 8;
  • 67) 0.365 725 852 823 645 079 651 797 328 252 108 8 × 2 = 0 + 0.731 451 705 647 290 159 303 594 656 504 217 6;
  • 68) 0.731 451 705 647 290 159 303 594 656 504 217 6 × 2 = 1 + 0.462 903 411 294 580 318 607 189 313 008 435 2;

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:


0.000 020 830 729 321 671 205 134 999 154 519 2(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:

0.000 020 830 729 321 671 205 134 999 154 519 2(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:


0.000 020 830 729 321 671 205 134 999 154 519 2(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 0.000 020 830 729 321 671 205 134 999 154 519 2 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