0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 666 62;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 666 62 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 333 24;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 333 24 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 666 48;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 666 48 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 332 96;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 332 96 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 665 92;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 665 92 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 331 84;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 331 84 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 663 68;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 663 68 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 327 36;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 327 36 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 654 72;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 654 72 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 309 44;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 309 44 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 226 618 88;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 226 618 88 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 453 237 76;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 453 237 76 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 906 475 52;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 906 475 52 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 812 951 04;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 812 951 04 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 625 902 08;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 625 902 08 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 251 804 16;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 251 804 16 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 503 608 32;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 503 608 32 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 007 216 64;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 007 216 64 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 014 433 28;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 014 433 28 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 084 028 866 56;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 084 028 866 56 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 168 057 733 12;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 168 057 733 12 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 336 115 466 24;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 336 115 466 24 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 672 230 932 48;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 672 230 932 48 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 344 461 864 96;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 344 461 864 96 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 688 923 729 92;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 688 923 729 92 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 377 847 459 84;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 377 847 459 84 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 755 694 919 68;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 755 694 919 68 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 511 389 839 36;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 511 389 839 36 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 022 779 678 72;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 022 779 678 72 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 270 045 559 357 44;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 270 045 559 357 44 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 540 091 118 714 88;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 540 091 118 714 88 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 080 182 237 429 76;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 080 182 237 429 76 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 160 364 474 859 52;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 160 364 474 859 52 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 320 728 949 719 04;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 320 728 949 719 04 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 641 457 899 438 08;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 641 457 899 438 08 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 282 915 798 876 16;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 282 915 798 876 16 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 565 831 597 752 32;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 565 831 597 752 32 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 131 663 195 504 64;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 131 663 195 504 64 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 322 263 326 391 009 28;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 322 263 326 391 009 28 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 644 526 652 782 018 56;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 644 526 652 782 018 56 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 289 053 305 564 037 12;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 289 053 305 564 037 12 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 578 106 611 128 074 24;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 578 106 611 128 074 24 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 156 213 222 256 148 48;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 156 213 222 256 148 48 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 312 426 444 512 296 96;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 312 426 444 512 296 96 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 624 852 889 024 593 92;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 624 852 889 024 593 92 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 249 705 778 049 187 84;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 249 705 778 049 187 84 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 499 411 556 098 375 68;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 499 411 556 098 375 68 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 404 998 823 112 196 751 36;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 404 998 823 112 196 751 36 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 809 997 646 224 393 502 72;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 809 997 646 224 393 502 72 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 619 995 292 448 787 005 44;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 619 995 292 448 787 005 44 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 239 990 584 897 574 010 88;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 239 990 584 897 574 010 88 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 479 981 169 795 148 021 76;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 479 981 169 795 148 021 76 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 959 962 339 590 296 043 52;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 959 962 339 590 296 043 52 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 919 924 679 180 592 087 04;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 919 924 679 180 592 087 04 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 839 849 358 361 184 174 08;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 839 849 358 361 184 174 08 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 679 698 716 722 368 348 16;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 679 698 716 722 368 348 16 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 359 397 433 444 736 696 32;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 359 397 433 444 736 696 32 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 718 794 866 889 473 392 64;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 718 794 866 889 473 392 64 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 061 437 589 733 778 946 785 28;
  • 60) 0.834 622 944 239 526 987 075 805 664 061 437 589 733 778 946 785 28 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 122 875 179 467 557 893 570 56;
  • 61) 0.669 245 888 479 053 974 151 611 328 122 875 179 467 557 893 570 56 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 245 750 358 935 115 787 141 12;
  • 62) 0.338 491 776 958 107 948 303 222 656 245 750 358 935 115 787 141 12 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 491 500 717 870 231 574 282 24;
  • 63) 0.676 983 553 916 215 896 606 445 312 491 500 717 870 231 574 282 24 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 983 001 435 740 463 148 564 48;
  • 64) 0.353 967 107 832 431 793 212 890 624 983 001 435 740 463 148 564 48 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 966 002 871 480 926 297 128 96;
  • 65) 0.707 934 215 664 863 586 425 781 249 966 002 871 480 926 297 128 96 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 932 005 742 961 852 594 257 92;
  • 66) 0.415 868 431 329 727 172 851 562 499 932 005 742 961 852 594 257 92 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 864 011 485 923 705 188 515 84;
  • 67) 0.831 736 862 659 454 345 703 124 999 864 011 485 923 705 188 515 84 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 728 022 971 847 410 377 031 68;
  • 68) 0.663 473 725 318 908 691 406 249 999 728 022 971 847 410 377 031 68 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 456 045 943 694 820 754 063 36;
  • 69) 0.326 947 450 637 817 382 812 499 999 456 045 943 694 820 754 063 36 × 2 = 0 + 0.653 894 901 275 634 765 624 999 998 912 091 887 389 641 508 126 72;
  • 70) 0.653 894 901 275 634 765 624 999 998 912 091 887 389 641 508 126 72 × 2 = 1 + 0.307 789 802 551 269 531 249 999 997 824 183 774 779 283 016 253 44;
  • 71) 0.307 789 802 551 269 531 249 999 997 824 183 774 779 283 016 253 44 × 2 = 0 + 0.615 579 605 102 539 062 499 999 995 648 367 549 558 566 032 506 88;
  • 72) 0.615 579 605 102 539 062 499 999 995 648 367 549 558 566 032 506 88 × 2 = 1 + 0.231 159 210 205 078 124 999 999 991 296 735 099 117 132 065 013 76;
  • 73) 0.231 159 210 205 078 124 999 999 991 296 735 099 117 132 065 013 76 × 2 = 0 + 0.462 318 420 410 156 249 999 999 982 593 470 198 234 264 130 027 52;
  • 74) 0.462 318 420 410 156 249 999 999 982 593 470 198 234 264 130 027 52 × 2 = 0 + 0.924 636 840 820 312 499 999 999 965 186 940 396 468 528 260 055 04;
  • 75) 0.924 636 840 820 312 499 999 999 965 186 940 396 468 528 260 055 04 × 2 = 1 + 0.849 273 681 640 624 999 999 999 930 373 880 792 937 056 520 110 08;
  • 76) 0.849 273 681 640 624 999 999 999 930 373 880 792 937 056 520 110 08 × 2 = 1 + 0.698 547 363 281 249 999 999 999 860 747 761 585 874 113 040 220 16;
  • 77) 0.698 547 363 281 249 999 999 999 860 747 761 585 874 113 040 220 16 × 2 = 1 + 0.397 094 726 562 499 999 999 999 721 495 523 171 748 226 080 440 32;
  • 78) 0.397 094 726 562 499 999 999 999 721 495 523 171 748 226 080 440 32 × 2 = 0 + 0.794 189 453 124 999 999 999 999 442 991 046 343 496 452 160 880 64;
  • 79) 0.794 189 453 124 999 999 999 999 442 991 046 343 496 452 160 880 64 × 2 = 1 + 0.588 378 906 249 999 999 999 998 885 982 092 686 992 904 321 761 28;
  • 80) 0.588 378 906 249 999 999 999 998 885 982 092 686 992 904 321 761 28 × 2 = 1 + 0.176 757 812 499 999 999 999 997 771 964 185 373 985 808 643 522 56;
  • 81) 0.176 757 812 499 999 999 999 997 771 964 185 373 985 808 643 522 56 × 2 = 0 + 0.353 515 624 999 999 999 999 995 543 928 370 747 971 617 287 045 12;
  • 82) 0.353 515 624 999 999 999 999 995 543 928 370 747 971 617 287 045 12 × 2 = 0 + 0.707 031 249 999 999 999 999 991 087 856 741 495 943 234 574 090 24;
  • 83) 0.707 031 249 999 999 999 999 991 087 856 741 495 943 234 574 090 24 × 2 = 1 + 0.414 062 499 999 999 999 999 982 175 713 482 991 886 469 148 180 48;
  • 84) 0.414 062 499 999 999 999 999 982 175 713 482 991 886 469 148 180 48 × 2 = 0 + 0.828 124 999 999 999 999 999 964 351 426 965 983 772 938 296 360 96;
  • 85) 0.828 124 999 999 999 999 999 964 351 426 965 983 772 938 296 360 96 × 2 = 1 + 0.656 249 999 999 999 999 999 928 702 853 931 967 545 876 592 721 92;
  • 86) 0.656 249 999 999 999 999 999 928 702 853 931 967 545 876 592 721 92 × 2 = 1 + 0.312 499 999 999 999 999 999 857 405 707 863 935 091 753 185 443 84;
  • 87) 0.312 499 999 999 999 999 999 857 405 707 863 935 091 753 185 443 84 × 2 = 0 + 0.624 999 999 999 999 999 999 714 811 415 727 870 183 506 370 887 68;
  • 88) 0.624 999 999 999 999 999 999 714 811 415 727 870 183 506 370 887 68 × 2 = 1 + 0.249 999 999 999 999 999 999 429 622 831 455 740 367 012 741 775 36;
  • 89) 0.249 999 999 999 999 999 999 429 622 831 455 740 367 012 741 775 36 × 2 = 0 + 0.499 999 999 999 999 999 998 859 245 662 911 480 734 025 483 550 72;
  • 90) 0.499 999 999 999 999 999 998 859 245 662 911 480 734 025 483 550 72 × 2 = 0 + 0.999 999 999 999 999 999 997 718 491 325 822 961 468 050 967 101 44;
  • 91) 0.999 999 999 999 999 999 997 718 491 325 822 961 468 050 967 101 44 × 2 = 1 + 0.999 999 999 999 999 999 995 436 982 651 645 922 936 101 934 202 88;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


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): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(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;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 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) =

984(10) =

011 1101 1000(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. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


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 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 31 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

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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