0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 436;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 436 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 872;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 872 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 744;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 744 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 488;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 488 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 976;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 976 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 453 952;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 453 952 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 907 904;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 907 904 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 815 808;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 815 808 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 631 616;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 631 616 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 263 232;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 263 232 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 526 464;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 526 464 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 052 928;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 052 928 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 922 105 856;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 922 105 856 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 844 211 712;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 844 211 712 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 688 423 424;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 688 423 424 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 376 846 848;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 376 846 848 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 753 693 696;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 753 693 696 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 507 387 392;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 507 387 392 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 014 774 784;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 014 774 784 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 029 549 568;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 029 549 568 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 059 099 136;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 059 099 136 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 118 198 272;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 118 198 272 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 236 396 544;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 236 396 544 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 472 793 088;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 472 793 088 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 945 586 176;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 945 586 176 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 505 891 172 352;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 505 891 172 352 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 011 782 344 704;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 011 782 344 704 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 023 564 689 408;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 023 564 689 408 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 047 129 378 816;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 047 129 378 816 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 094 258 757 632;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 094 258 757 632 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 188 517 515 264;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 188 517 515 264 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 377 035 030 528;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 377 035 030 528 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 754 070 061 056;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 754 070 061 056 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 353 508 140 122 112;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 353 508 140 122 112 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 707 016 280 244 224;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 707 016 280 244 224 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 414 032 560 488 448;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 414 032 560 488 448 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 828 065 120 976 896;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 828 065 120 976 896 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 656 130 241 953 792;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 656 130 241 953 792 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 312 260 483 907 584;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 312 260 483 907 584 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 624 520 967 815 168;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 624 520 967 815 168 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 249 041 935 630 336;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 249 041 935 630 336 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 498 083 871 260 672;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 498 083 871 260 672 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 996 167 742 521 344;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 996 167 742 521 344 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 992 335 485 042 688;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 992 335 485 042 688 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 691 984 670 970 085 376;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 691 984 670 970 085 376 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 383 969 341 940 170 752;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 383 969 341 940 170 752 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 767 938 683 880 341 504;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 767 938 683 880 341 504 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 535 877 367 760 683 008;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 535 877 367 760 683 008 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 071 754 735 521 366 016;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 071 754 735 521 366 016 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 143 509 471 042 732 032;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 143 509 471 042 732 032 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 287 018 942 085 464 064;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 287 018 942 085 464 064 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 574 037 884 170 928 128;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 574 037 884 170 928 128 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 977 148 075 768 341 856 256;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 977 148 075 768 341 856 256 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 954 296 151 536 683 712 512;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 954 296 151 536 683 712 512 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 908 592 303 073 367 425 024;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 908 592 303 073 367 425 024 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 817 184 606 146 734 850 048;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 817 184 606 146 734 850 048 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 634 369 212 293 469 700 096;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 634 369 212 293 469 700 096 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 268 738 424 586 939 400 192;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 268 738 424 586 939 400 192 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 537 476 849 173 878 800 384;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 537 476 849 173 878 800 384 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 074 953 698 347 757 600 768;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 074 953 698 347 757 600 768 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 149 907 396 695 515 201 536;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 149 907 396 695 515 201 536 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 299 814 793 391 030 403 072;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 299 814 793 391 030 403 072 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 599 629 586 782 060 806 144;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 599 629 586 782 060 806 144 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 001 199 259 173 564 121 612 288;
  • 65) 0.707 934 215 664 863 586 425 781 250 001 199 259 173 564 121 612 288 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 002 398 518 347 128 243 224 576;
  • 66) 0.415 868 431 329 727 172 851 562 500 002 398 518 347 128 243 224 576 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 004 797 036 694 256 486 449 152;
  • 67) 0.831 736 862 659 454 345 703 125 000 004 797 036 694 256 486 449 152 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 009 594 073 388 512 972 898 304;
  • 68) 0.663 473 725 318 908 691 406 250 000 009 594 073 388 512 972 898 304 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 019 188 146 777 025 945 796 608;
  • 69) 0.326 947 450 637 817 382 812 500 000 019 188 146 777 025 945 796 608 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 038 376 293 554 051 891 593 216;
  • 70) 0.653 894 901 275 634 765 625 000 000 038 376 293 554 051 891 593 216 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 076 752 587 108 103 783 186 432;
  • 71) 0.307 789 802 551 269 531 250 000 000 076 752 587 108 103 783 186 432 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 153 505 174 216 207 566 372 864;
  • 72) 0.615 579 605 102 539 062 500 000 000 153 505 174 216 207 566 372 864 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 307 010 348 432 415 132 745 728;
  • 73) 0.231 159 210 205 078 125 000 000 000 307 010 348 432 415 132 745 728 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 614 020 696 864 830 265 491 456;
  • 74) 0.462 318 420 410 156 250 000 000 000 614 020 696 864 830 265 491 456 × 2 = 0 + 0.924 636 840 820 312 500 000 000 001 228 041 393 729 660 530 982 912;
  • 75) 0.924 636 840 820 312 500 000 000 001 228 041 393 729 660 530 982 912 × 2 = 1 + 0.849 273 681 640 625 000 000 000 002 456 082 787 459 321 061 965 824;
  • 76) 0.849 273 681 640 625 000 000 000 002 456 082 787 459 321 061 965 824 × 2 = 1 + 0.698 547 363 281 250 000 000 000 004 912 165 574 918 642 123 931 648;
  • 77) 0.698 547 363 281 250 000 000 000 004 912 165 574 918 642 123 931 648 × 2 = 1 + 0.397 094 726 562 500 000 000 000 009 824 331 149 837 284 247 863 296;
  • 78) 0.397 094 726 562 500 000 000 000 009 824 331 149 837 284 247 863 296 × 2 = 0 + 0.794 189 453 125 000 000 000 000 019 648 662 299 674 568 495 726 592;
  • 79) 0.794 189 453 125 000 000 000 000 019 648 662 299 674 568 495 726 592 × 2 = 1 + 0.588 378 906 250 000 000 000 000 039 297 324 599 349 136 991 453 184;
  • 80) 0.588 378 906 250 000 000 000 000 039 297 324 599 349 136 991 453 184 × 2 = 1 + 0.176 757 812 500 000 000 000 000 078 594 649 198 698 273 982 906 368;
  • 81) 0.176 757 812 500 000 000 000 000 078 594 649 198 698 273 982 906 368 × 2 = 0 + 0.353 515 625 000 000 000 000 000 157 189 298 397 396 547 965 812 736;
  • 82) 0.353 515 625 000 000 000 000 000 157 189 298 397 396 547 965 812 736 × 2 = 0 + 0.707 031 250 000 000 000 000 000 314 378 596 794 793 095 931 625 472;
  • 83) 0.707 031 250 000 000 000 000 000 314 378 596 794 793 095 931 625 472 × 2 = 1 + 0.414 062 500 000 000 000 000 000 628 757 193 589 586 191 863 250 944;
  • 84) 0.414 062 500 000 000 000 000 000 628 757 193 589 586 191 863 250 944 × 2 = 0 + 0.828 125 000 000 000 000 000 001 257 514 387 179 172 383 726 501 888;
  • 85) 0.828 125 000 000 000 000 000 001 257 514 387 179 172 383 726 501 888 × 2 = 1 + 0.656 250 000 000 000 000 000 002 515 028 774 358 344 767 453 003 776;
  • 86) 0.656 250 000 000 000 000 000 002 515 028 774 358 344 767 453 003 776 × 2 = 1 + 0.312 500 000 000 000 000 000 005 030 057 548 716 689 534 906 007 552;
  • 87) 0.312 500 000 000 000 000 000 005 030 057 548 716 689 534 906 007 552 × 2 = 0 + 0.625 000 000 000 000 000 000 010 060 115 097 433 379 069 812 015 104;
  • 88) 0.625 000 000 000 000 000 000 010 060 115 097 433 379 069 812 015 104 × 2 = 1 + 0.250 000 000 000 000 000 000 020 120 230 194 866 758 139 624 030 208;
  • 89) 0.250 000 000 000 000 000 000 020 120 230 194 866 758 139 624 030 208 × 2 = 0 + 0.500 000 000 000 000 000 000 040 240 460 389 733 516 279 248 060 416;
  • 90) 0.500 000 000 000 000 000 000 040 240 460 389 733 516 279 248 060 416 × 2 = 1 + 0.000 000 000 000 000 000 000 080 480 920 779 467 032 558 496 120 832;
  • 91) 0.000 000 000 000 000 000 000 080 480 920 779 467 032 558 496 120 832 × 2 = 0 + 0.000 000 000 000 000 000 000 160 961 841 558 934 065 116 992 241 664;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(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 1010


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 1010 =

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


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 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 218 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 1010

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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