0.000 020 830 729 321 671 205 134 999 154 509 660 616 677 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 020 830 729 321 671 205 134 999 154 509 660 616 677(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
0.000 020 830 729 321 671 205 134 999 154 509 660 616 677(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: 0.000 020 830 729 321 671 205 134 999 154 509 660 616 677.

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) 0.000 020 830 729 321 671 205 134 999 154 509 660 616 677 × 2 = 0 + 0.000 041 661 458 643 342 410 269 998 309 019 321 233 354;
  • 2) 0.000 041 661 458 643 342 410 269 998 309 019 321 233 354 × 2 = 0 + 0.000 083 322 917 286 684 820 539 996 618 038 642 466 708;
  • 3) 0.000 083 322 917 286 684 820 539 996 618 038 642 466 708 × 2 = 0 + 0.000 166 645 834 573 369 641 079 993 236 077 284 933 416;
  • 4) 0.000 166 645 834 573 369 641 079 993 236 077 284 933 416 × 2 = 0 + 0.000 333 291 669 146 739 282 159 986 472 154 569 866 832;
  • 5) 0.000 333 291 669 146 739 282 159 986 472 154 569 866 832 × 2 = 0 + 0.000 666 583 338 293 478 564 319 972 944 309 139 733 664;
  • 6) 0.000 666 583 338 293 478 564 319 972 944 309 139 733 664 × 2 = 0 + 0.001 333 166 676 586 957 128 639 945 888 618 279 467 328;
  • 7) 0.001 333 166 676 586 957 128 639 945 888 618 279 467 328 × 2 = 0 + 0.002 666 333 353 173 914 257 279 891 777 236 558 934 656;
  • 8) 0.002 666 333 353 173 914 257 279 891 777 236 558 934 656 × 2 = 0 + 0.005 332 666 706 347 828 514 559 783 554 473 117 869 312;
  • 9) 0.005 332 666 706 347 828 514 559 783 554 473 117 869 312 × 2 = 0 + 0.010 665 333 412 695 657 029 119 567 108 946 235 738 624;
  • 10) 0.010 665 333 412 695 657 029 119 567 108 946 235 738 624 × 2 = 0 + 0.021 330 666 825 391 314 058 239 134 217 892 471 477 248;
  • 11) 0.021 330 666 825 391 314 058 239 134 217 892 471 477 248 × 2 = 0 + 0.042 661 333 650 782 628 116 478 268 435 784 942 954 496;
  • 12) 0.042 661 333 650 782 628 116 478 268 435 784 942 954 496 × 2 = 0 + 0.085 322 667 301 565 256 232 956 536 871 569 885 908 992;
  • 13) 0.085 322 667 301 565 256 232 956 536 871 569 885 908 992 × 2 = 0 + 0.170 645 334 603 130 512 465 913 073 743 139 771 817 984;
  • 14) 0.170 645 334 603 130 512 465 913 073 743 139 771 817 984 × 2 = 0 + 0.341 290 669 206 261 024 931 826 147 486 279 543 635 968;
  • 15) 0.341 290 669 206 261 024 931 826 147 486 279 543 635 968 × 2 = 0 + 0.682 581 338 412 522 049 863 652 294 972 559 087 271 936;
  • 16) 0.682 581 338 412 522 049 863 652 294 972 559 087 271 936 × 2 = 1 + 0.365 162 676 825 044 099 727 304 589 945 118 174 543 872;
  • 17) 0.365 162 676 825 044 099 727 304 589 945 118 174 543 872 × 2 = 0 + 0.730 325 353 650 088 199 454 609 179 890 236 349 087 744;
  • 18) 0.730 325 353 650 088 199 454 609 179 890 236 349 087 744 × 2 = 1 + 0.460 650 707 300 176 398 909 218 359 780 472 698 175 488;
  • 19) 0.460 650 707 300 176 398 909 218 359 780 472 698 175 488 × 2 = 0 + 0.921 301 414 600 352 797 818 436 719 560 945 396 350 976;
  • 20) 0.921 301 414 600 352 797 818 436 719 560 945 396 350 976 × 2 = 1 + 0.842 602 829 200 705 595 636 873 439 121 890 792 701 952;
  • 21) 0.842 602 829 200 705 595 636 873 439 121 890 792 701 952 × 2 = 1 + 0.685 205 658 401 411 191 273 746 878 243 781 585 403 904;
  • 22) 0.685 205 658 401 411 191 273 746 878 243 781 585 403 904 × 2 = 1 + 0.370 411 316 802 822 382 547 493 756 487 563 170 807 808;
  • 23) 0.370 411 316 802 822 382 547 493 756 487 563 170 807 808 × 2 = 0 + 0.740 822 633 605 644 765 094 987 512 975 126 341 615 616;
  • 24) 0.740 822 633 605 644 765 094 987 512 975 126 341 615 616 × 2 = 1 + 0.481 645 267 211 289 530 189 975 025 950 252 683 231 232;
  • 25) 0.481 645 267 211 289 530 189 975 025 950 252 683 231 232 × 2 = 0 + 0.963 290 534 422 579 060 379 950 051 900 505 366 462 464;
  • 26) 0.963 290 534 422 579 060 379 950 051 900 505 366 462 464 × 2 = 1 + 0.926 581 068 845 158 120 759 900 103 801 010 732 924 928;
  • 27) 0.926 581 068 845 158 120 759 900 103 801 010 732 924 928 × 2 = 1 + 0.853 162 137 690 316 241 519 800 207 602 021 465 849 856;
  • 28) 0.853 162 137 690 316 241 519 800 207 602 021 465 849 856 × 2 = 1 + 0.706 324 275 380 632 483 039 600 415 204 042 931 699 712;
  • 29) 0.706 324 275 380 632 483 039 600 415 204 042 931 699 712 × 2 = 1 + 0.412 648 550 761 264 966 079 200 830 408 085 863 399 424;
  • 30) 0.412 648 550 761 264 966 079 200 830 408 085 863 399 424 × 2 = 0 + 0.825 297 101 522 529 932 158 401 660 816 171 726 798 848;
  • 31) 0.825 297 101 522 529 932 158 401 660 816 171 726 798 848 × 2 = 1 + 0.650 594 203 045 059 864 316 803 321 632 343 453 597 696;
  • 32) 0.650 594 203 045 059 864 316 803 321 632 343 453 597 696 × 2 = 1 + 0.301 188 406 090 119 728 633 606 643 264 686 907 195 392;
  • 33) 0.301 188 406 090 119 728 633 606 643 264 686 907 195 392 × 2 = 0 + 0.602 376 812 180 239 457 267 213 286 529 373 814 390 784;
  • 34) 0.602 376 812 180 239 457 267 213 286 529 373 814 390 784 × 2 = 1 + 0.204 753 624 360 478 914 534 426 573 058 747 628 781 568;
  • 35) 0.204 753 624 360 478 914 534 426 573 058 747 628 781 568 × 2 = 0 + 0.409 507 248 720 957 829 068 853 146 117 495 257 563 136;
  • 36) 0.409 507 248 720 957 829 068 853 146 117 495 257 563 136 × 2 = 0 + 0.819 014 497 441 915 658 137 706 292 234 990 515 126 272;
  • 37) 0.819 014 497 441 915 658 137 706 292 234 990 515 126 272 × 2 = 1 + 0.638 028 994 883 831 316 275 412 584 469 981 030 252 544;
  • 38) 0.638 028 994 883 831 316 275 412 584 469 981 030 252 544 × 2 = 1 + 0.276 057 989 767 662 632 550 825 168 939 962 060 505 088;
  • 39) 0.276 057 989 767 662 632 550 825 168 939 962 060 505 088 × 2 = 0 + 0.552 115 979 535 325 265 101 650 337 879 924 121 010 176;
  • 40) 0.552 115 979 535 325 265 101 650 337 879 924 121 010 176 × 2 = 1 + 0.104 231 959 070 650 530 203 300 675 759 848 242 020 352;
  • 41) 0.104 231 959 070 650 530 203 300 675 759 848 242 020 352 × 2 = 0 + 0.208 463 918 141 301 060 406 601 351 519 696 484 040 704;
  • 42) 0.208 463 918 141 301 060 406 601 351 519 696 484 040 704 × 2 = 0 + 0.416 927 836 282 602 120 813 202 703 039 392 968 081 408;
  • 43) 0.416 927 836 282 602 120 813 202 703 039 392 968 081 408 × 2 = 0 + 0.833 855 672 565 204 241 626 405 406 078 785 936 162 816;
  • 44) 0.833 855 672 565 204 241 626 405 406 078 785 936 162 816 × 2 = 1 + 0.667 711 345 130 408 483 252 810 812 157 571 872 325 632;
  • 45) 0.667 711 345 130 408 483 252 810 812 157 571 872 325 632 × 2 = 1 + 0.335 422 690 260 816 966 505 621 624 315 143 744 651 264;
  • 46) 0.335 422 690 260 816 966 505 621 624 315 143 744 651 264 × 2 = 0 + 0.670 845 380 521 633 933 011 243 248 630 287 489 302 528;
  • 47) 0.670 845 380 521 633 933 011 243 248 630 287 489 302 528 × 2 = 1 + 0.341 690 761 043 267 866 022 486 497 260 574 978 605 056;
  • 48) 0.341 690 761 043 267 866 022 486 497 260 574 978 605 056 × 2 = 0 + 0.683 381 522 086 535 732 044 972 994 521 149 957 210 112;
  • 49) 0.683 381 522 086 535 732 044 972 994 521 149 957 210 112 × 2 = 1 + 0.366 763 044 173 071 464 089 945 989 042 299 914 420 224;
  • 50) 0.366 763 044 173 071 464 089 945 989 042 299 914 420 224 × 2 = 0 + 0.733 526 088 346 142 928 179 891 978 084 599 828 840 448;
  • 51) 0.733 526 088 346 142 928 179 891 978 084 599 828 840 448 × 2 = 1 + 0.467 052 176 692 285 856 359 783 956 169 199 657 680 896;
  • 52) 0.467 052 176 692 285 856 359 783 956 169 199 657 680 896 × 2 = 0 + 0.934 104 353 384 571 712 719 567 912 338 399 315 361 792;
  • 53) 0.934 104 353 384 571 712 719 567 912 338 399 315 361 792 × 2 = 1 + 0.868 208 706 769 143 425 439 135 824 676 798 630 723 584;
  • 54) 0.868 208 706 769 143 425 439 135 824 676 798 630 723 584 × 2 = 1 + 0.736 417 413 538 286 850 878 271 649 353 597 261 447 168;
  • 55) 0.736 417 413 538 286 850 878 271 649 353 597 261 447 168 × 2 = 1 + 0.472 834 827 076 573 701 756 543 298 707 194 522 894 336;
  • 56) 0.472 834 827 076 573 701 756 543 298 707 194 522 894 336 × 2 = 0 + 0.945 669 654 153 147 403 513 086 597 414 389 045 788 672;
  • 57) 0.945 669 654 153 147 403 513 086 597 414 389 045 788 672 × 2 = 1 + 0.891 339 308 306 294 807 026 173 194 828 778 091 577 344;
  • 58) 0.891 339 308 306 294 807 026 173 194 828 778 091 577 344 × 2 = 1 + 0.782 678 616 612 589 614 052 346 389 657 556 183 154 688;
  • 59) 0.782 678 616 612 589 614 052 346 389 657 556 183 154 688 × 2 = 1 + 0.565 357 233 225 179 228 104 692 779 315 112 366 309 376;
  • 60) 0.565 357 233 225 179 228 104 692 779 315 112 366 309 376 × 2 = 1 + 0.130 714 466 450 358 456 209 385 558 630 224 732 618 752;
  • 61) 0.130 714 466 450 358 456 209 385 558 630 224 732 618 752 × 2 = 0 + 0.261 428 932 900 716 912 418 771 117 260 449 465 237 504;
  • 62) 0.261 428 932 900 716 912 418 771 117 260 449 465 237 504 × 2 = 0 + 0.522 857 865 801 433 824 837 542 234 520 898 930 475 008;
  • 63) 0.522 857 865 801 433 824 837 542 234 520 898 930 475 008 × 2 = 1 + 0.045 715 731 602 867 649 675 084 469 041 797 860 950 016;
  • 64) 0.045 715 731 602 867 649 675 084 469 041 797 860 950 016 × 2 = 0 + 0.091 431 463 205 735 299 350 168 938 083 595 721 900 032;
  • 65) 0.091 431 463 205 735 299 350 168 938 083 595 721 900 032 × 2 = 0 + 0.182 862 926 411 470 598 700 337 876 167 191 443 800 064;
  • 66) 0.182 862 926 411 470 598 700 337 876 167 191 443 800 064 × 2 = 0 + 0.365 725 852 822 941 197 400 675 752 334 382 887 600 128;
  • 67) 0.365 725 852 822 941 197 400 675 752 334 382 887 600 128 × 2 = 0 + 0.731 451 705 645 882 394 801 351 504 668 765 775 200 256;
  • 68) 0.731 451 705 645 882 394 801 351 504 668 765 775 200 256 × 2 = 1 + 0.462 903 411 291 764 789 602 703 009 337 531 550 400 512;

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:


0.000 020 830 729 321 671 205 134 999 154 509 660 616 677(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:

0.000 020 830 729 321 671 205 134 999 154 509 660 616 677(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:


0.000 020 830 729 321 671 205 134 999 154 509 660 616 677(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 0.000 020 830 729 321 671 205 134 999 154 509 660 616 677 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