0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 852;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 852 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 704;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 704 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 408;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 408 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 816;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 816 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 717 632;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 717 632 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 435 264;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 435 264 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 870 528;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 870 528 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 741 056;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 741 056 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 482 112;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 482 112 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 964 224;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 964 224 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 928 448;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 928 448 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 856 896;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 856 896 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 713 792;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 713 792 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 427 584;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 427 584 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 678 855 168;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 678 855 168 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 357 710 336;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 357 710 336 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 715 420 672;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 715 420 672 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 430 841 344;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 430 841 344 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 861 682 688;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 861 682 688 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 723 365 376;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 723 365 376 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 446 730 752;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 446 730 752 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 893 461 504;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 893 461 504 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 786 923 008;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 786 923 008 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 573 846 016;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 573 846 016 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 743 147 692 032;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 743 147 692 032 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 486 295 384 064;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 486 295 384 064 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 972 590 768 128;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 972 590 768 128 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 945 181 536 256;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 945 181 536 256 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 890 363 072 512;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 890 363 072 512 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 780 726 145 024;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 780 726 145 024 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 561 452 290 048;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 561 452 290 048 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 122 904 580 096;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 122 904 580 096 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 245 809 160 192;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 245 809 160 192 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 348 491 618 320 384;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 348 491 618 320 384 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 696 983 236 640 768;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 696 983 236 640 768 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 393 966 473 281 536;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 393 966 473 281 536 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 787 932 946 563 072;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 787 932 946 563 072 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 575 865 893 126 144;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 575 865 893 126 144 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 151 731 786 252 288;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 151 731 786 252 288 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 303 463 572 504 576;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 303 463 572 504 576 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 606 927 145 009 152;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 606 927 145 009 152 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 213 854 290 018 304;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 213 854 290 018 304 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 427 708 580 036 608;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 427 708 580 036 608 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 340 855 417 160 073 216;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 340 855 417 160 073 216 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 681 710 834 320 146 432;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 681 710 834 320 146 432 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 363 421 668 640 292 864;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 363 421 668 640 292 864 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 726 843 337 280 585 728;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 726 843 337 280 585 728 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 453 686 674 561 171 456;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 453 686 674 561 171 456 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 907 373 349 122 342 912;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 907 373 349 122 342 912 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 814 746 698 244 685 824;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 814 746 698 244 685 824 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 629 493 396 489 371 648;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 629 493 396 489 371 648 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 258 986 792 978 743 296;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 258 986 792 978 743 296 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 517 973 585 957 486 592;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 517 973 585 957 486 592 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 949 035 947 171 914 973 184;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 949 035 947 171 914 973 184 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 898 071 894 343 829 946 368;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 898 071 894 343 829 946 368 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 796 143 788 687 659 892 736;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 796 143 788 687 659 892 736 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 592 287 577 375 319 785 472;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 592 287 577 375 319 785 472 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 184 575 154 750 639 570 944;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 184 575 154 750 639 570 944 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 369 150 309 501 279 141 888;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 369 150 309 501 279 141 888 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 738 300 619 002 558 283 776;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 738 300 619 002 558 283 776 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 476 601 238 005 116 567 552;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 476 601 238 005 116 567 552 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 953 202 476 010 233 135 104;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 953 202 476 010 233 135 104 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 997 906 404 952 020 466 270 208;
  • 64) 0.353 967 107 832 431 793 212 890 624 997 906 404 952 020 466 270 208 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 995 812 809 904 040 932 540 416;
  • 65) 0.707 934 215 664 863 586 425 781 249 995 812 809 904 040 932 540 416 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 991 625 619 808 081 865 080 832;
  • 66) 0.415 868 431 329 727 172 851 562 499 991 625 619 808 081 865 080 832 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 983 251 239 616 163 730 161 664;
  • 67) 0.831 736 862 659 454 345 703 124 999 983 251 239 616 163 730 161 664 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 966 502 479 232 327 460 323 328;
  • 68) 0.663 473 725 318 908 691 406 249 999 966 502 479 232 327 460 323 328 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 933 004 958 464 654 920 646 656;
  • 69) 0.326 947 450 637 817 382 812 499 999 933 004 958 464 654 920 646 656 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 866 009 916 929 309 841 293 312;
  • 70) 0.653 894 901 275 634 765 624 999 999 866 009 916 929 309 841 293 312 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 732 019 833 858 619 682 586 624;
  • 71) 0.307 789 802 551 269 531 249 999 999 732 019 833 858 619 682 586 624 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 464 039 667 717 239 365 173 248;
  • 72) 0.615 579 605 102 539 062 499 999 999 464 039 667 717 239 365 173 248 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 928 079 335 434 478 730 346 496;
  • 73) 0.231 159 210 205 078 124 999 999 998 928 079 335 434 478 730 346 496 × 2 = 0 + 0.462 318 420 410 156 249 999 999 997 856 158 670 868 957 460 692 992;
  • 74) 0.462 318 420 410 156 249 999 999 997 856 158 670 868 957 460 692 992 × 2 = 0 + 0.924 636 840 820 312 499 999 999 995 712 317 341 737 914 921 385 984;
  • 75) 0.924 636 840 820 312 499 999 999 995 712 317 341 737 914 921 385 984 × 2 = 1 + 0.849 273 681 640 624 999 999 999 991 424 634 683 475 829 842 771 968;
  • 76) 0.849 273 681 640 624 999 999 999 991 424 634 683 475 829 842 771 968 × 2 = 1 + 0.698 547 363 281 249 999 999 999 982 849 269 366 951 659 685 543 936;
  • 77) 0.698 547 363 281 249 999 999 999 982 849 269 366 951 659 685 543 936 × 2 = 1 + 0.397 094 726 562 499 999 999 999 965 698 538 733 903 319 371 087 872;
  • 78) 0.397 094 726 562 499 999 999 999 965 698 538 733 903 319 371 087 872 × 2 = 0 + 0.794 189 453 124 999 999 999 999 931 397 077 467 806 638 742 175 744;
  • 79) 0.794 189 453 124 999 999 999 999 931 397 077 467 806 638 742 175 744 × 2 = 1 + 0.588 378 906 249 999 999 999 999 862 794 154 935 613 277 484 351 488;
  • 80) 0.588 378 906 249 999 999 999 999 862 794 154 935 613 277 484 351 488 × 2 = 1 + 0.176 757 812 499 999 999 999 999 725 588 309 871 226 554 968 702 976;
  • 81) 0.176 757 812 499 999 999 999 999 725 588 309 871 226 554 968 702 976 × 2 = 0 + 0.353 515 624 999 999 999 999 999 451 176 619 742 453 109 937 405 952;
  • 82) 0.353 515 624 999 999 999 999 999 451 176 619 742 453 109 937 405 952 × 2 = 0 + 0.707 031 249 999 999 999 999 998 902 353 239 484 906 219 874 811 904;
  • 83) 0.707 031 249 999 999 999 999 998 902 353 239 484 906 219 874 811 904 × 2 = 1 + 0.414 062 499 999 999 999 999 997 804 706 478 969 812 439 749 623 808;
  • 84) 0.414 062 499 999 999 999 999 997 804 706 478 969 812 439 749 623 808 × 2 = 0 + 0.828 124 999 999 999 999 999 995 609 412 957 939 624 879 499 247 616;
  • 85) 0.828 124 999 999 999 999 999 995 609 412 957 939 624 879 499 247 616 × 2 = 1 + 0.656 249 999 999 999 999 999 991 218 825 915 879 249 758 998 495 232;
  • 86) 0.656 249 999 999 999 999 999 991 218 825 915 879 249 758 998 495 232 × 2 = 1 + 0.312 499 999 999 999 999 999 982 437 651 831 758 499 517 996 990 464;
  • 87) 0.312 499 999 999 999 999 999 982 437 651 831 758 499 517 996 990 464 × 2 = 0 + 0.624 999 999 999 999 999 999 964 875 303 663 516 999 035 993 980 928;
  • 88) 0.624 999 999 999 999 999 999 964 875 303 663 516 999 035 993 980 928 × 2 = 1 + 0.249 999 999 999 999 999 999 929 750 607 327 033 998 071 987 961 856;
  • 89) 0.249 999 999 999 999 999 999 929 750 607 327 033 998 071 987 961 856 × 2 = 0 + 0.499 999 999 999 999 999 999 859 501 214 654 067 996 143 975 923 712;
  • 90) 0.499 999 999 999 999 999 999 859 501 214 654 067 996 143 975 923 712 × 2 = 0 + 0.999 999 999 999 999 999 999 719 002 429 308 135 992 287 951 847 424;
  • 91) 0.999 999 999 999 999 999 999 719 002 429 308 135 992 287 951 847 424 × 2 = 1 + 0.999 999 999 999 999 999 999 438 004 858 616 271 984 575 903 694 848;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 926 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