0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 794;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 794 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 588;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 588 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 176;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 176 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 352;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 352 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 716 704;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 716 704 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 433 408;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 433 408 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 866 816;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 866 816 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 733 632;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 733 632 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 467 264;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 467 264 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 934 528;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 934 528 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 869 056;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 869 056 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 738 112;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 738 112 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 476 224;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 476 224 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 838 952 448;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 838 952 448 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 677 904 896;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 677 904 896 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 355 809 792;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 355 809 792 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 711 619 584;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 711 619 584 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 423 239 168;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 423 239 168 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 846 478 336;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 846 478 336 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 692 956 672;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 692 956 672 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 385 913 344;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 385 913 344 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 771 826 688;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 771 826 688 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 543 653 376;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 543 653 376 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 087 306 752;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 087 306 752 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 174 613 504;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 174 613 504 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 484 349 227 008;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 484 349 227 008 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 968 698 454 016;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 968 698 454 016 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 937 396 908 032;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 937 396 908 032 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 874 793 816 064;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 874 793 816 064 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 749 587 632 128;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 749 587 632 128 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 499 175 264 256;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 499 175 264 256 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 086 998 350 528 512;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 086 998 350 528 512 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 173 996 701 057 024;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 173 996 701 057 024 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 347 993 402 114 048;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 347 993 402 114 048 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 695 986 804 228 096;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 695 986 804 228 096 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 391 973 608 456 192;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 391 973 608 456 192 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 783 947 216 912 384;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 783 947 216 912 384 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 567 894 433 824 768;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 567 894 433 824 768 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 135 788 867 649 536;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 135 788 867 649 536 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 271 577 735 299 072;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 271 577 735 299 072 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 543 155 470 598 144;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 543 155 470 598 144 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 086 310 941 196 288;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 086 310 941 196 288 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 172 621 882 392 576;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 172 621 882 392 576 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 340 345 243 764 785 152;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 340 345 243 764 785 152 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 680 690 487 529 570 304;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 680 690 487 529 570 304 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 361 380 975 059 140 608;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 361 380 975 059 140 608 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 722 761 950 118 281 216;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 722 761 950 118 281 216 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 445 523 900 236 562 432;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 445 523 900 236 562 432 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 891 047 800 473 124 864;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 891 047 800 473 124 864 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 782 095 600 946 249 728;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 782 095 600 946 249 728 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 564 191 201 892 499 456;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 564 191 201 892 499 456 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 128 382 403 784 998 912;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 128 382 403 784 998 912 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 256 764 807 569 997 824;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 256 764 807 569 997 824 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 948 513 529 615 139 995 648;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 948 513 529 615 139 995 648 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 897 027 059 230 279 991 296;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 897 027 059 230 279 991 296 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 794 054 118 460 559 982 592;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 794 054 118 460 559 982 592 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 588 108 236 921 119 965 184;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 588 108 236 921 119 965 184 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 176 216 473 842 239 930 368;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 176 216 473 842 239 930 368 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 352 432 947 684 479 860 736;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 352 432 947 684 479 860 736 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 704 865 895 368 959 721 472;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 704 865 895 368 959 721 472 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 409 731 790 737 919 442 944;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 409 731 790 737 919 442 944 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 819 463 581 475 838 885 888;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 819 463 581 475 838 885 888 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 997 638 927 162 951 677 771 776;
  • 64) 0.353 967 107 832 431 793 212 890 624 997 638 927 162 951 677 771 776 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 995 277 854 325 903 355 543 552;
  • 65) 0.707 934 215 664 863 586 425 781 249 995 277 854 325 903 355 543 552 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 990 555 708 651 806 711 087 104;
  • 66) 0.415 868 431 329 727 172 851 562 499 990 555 708 651 806 711 087 104 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 981 111 417 303 613 422 174 208;
  • 67) 0.831 736 862 659 454 345 703 124 999 981 111 417 303 613 422 174 208 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 962 222 834 607 226 844 348 416;
  • 68) 0.663 473 725 318 908 691 406 249 999 962 222 834 607 226 844 348 416 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 924 445 669 214 453 688 696 832;
  • 69) 0.326 947 450 637 817 382 812 499 999 924 445 669 214 453 688 696 832 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 848 891 338 428 907 377 393 664;
  • 70) 0.653 894 901 275 634 765 624 999 999 848 891 338 428 907 377 393 664 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 697 782 676 857 814 754 787 328;
  • 71) 0.307 789 802 551 269 531 249 999 999 697 782 676 857 814 754 787 328 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 395 565 353 715 629 509 574 656;
  • 72) 0.615 579 605 102 539 062 499 999 999 395 565 353 715 629 509 574 656 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 791 130 707 431 259 019 149 312;
  • 73) 0.231 159 210 205 078 124 999 999 998 791 130 707 431 259 019 149 312 × 2 = 0 + 0.462 318 420 410 156 249 999 999 997 582 261 414 862 518 038 298 624;
  • 74) 0.462 318 420 410 156 249 999 999 997 582 261 414 862 518 038 298 624 × 2 = 0 + 0.924 636 840 820 312 499 999 999 995 164 522 829 725 036 076 597 248;
  • 75) 0.924 636 840 820 312 499 999 999 995 164 522 829 725 036 076 597 248 × 2 = 1 + 0.849 273 681 640 624 999 999 999 990 329 045 659 450 072 153 194 496;
  • 76) 0.849 273 681 640 624 999 999 999 990 329 045 659 450 072 153 194 496 × 2 = 1 + 0.698 547 363 281 249 999 999 999 980 658 091 318 900 144 306 388 992;
  • 77) 0.698 547 363 281 249 999 999 999 980 658 091 318 900 144 306 388 992 × 2 = 1 + 0.397 094 726 562 499 999 999 999 961 316 182 637 800 288 612 777 984;
  • 78) 0.397 094 726 562 499 999 999 999 961 316 182 637 800 288 612 777 984 × 2 = 0 + 0.794 189 453 124 999 999 999 999 922 632 365 275 600 577 225 555 968;
  • 79) 0.794 189 453 124 999 999 999 999 922 632 365 275 600 577 225 555 968 × 2 = 1 + 0.588 378 906 249 999 999 999 999 845 264 730 551 201 154 451 111 936;
  • 80) 0.588 378 906 249 999 999 999 999 845 264 730 551 201 154 451 111 936 × 2 = 1 + 0.176 757 812 499 999 999 999 999 690 529 461 102 402 308 902 223 872;
  • 81) 0.176 757 812 499 999 999 999 999 690 529 461 102 402 308 902 223 872 × 2 = 0 + 0.353 515 624 999 999 999 999 999 381 058 922 204 804 617 804 447 744;
  • 82) 0.353 515 624 999 999 999 999 999 381 058 922 204 804 617 804 447 744 × 2 = 0 + 0.707 031 249 999 999 999 999 998 762 117 844 409 609 235 608 895 488;
  • 83) 0.707 031 249 999 999 999 999 998 762 117 844 409 609 235 608 895 488 × 2 = 1 + 0.414 062 499 999 999 999 999 997 524 235 688 819 218 471 217 790 976;
  • 84) 0.414 062 499 999 999 999 999 997 524 235 688 819 218 471 217 790 976 × 2 = 0 + 0.828 124 999 999 999 999 999 995 048 471 377 638 436 942 435 581 952;
  • 85) 0.828 124 999 999 999 999 999 995 048 471 377 638 436 942 435 581 952 × 2 = 1 + 0.656 249 999 999 999 999 999 990 096 942 755 276 873 884 871 163 904;
  • 86) 0.656 249 999 999 999 999 999 990 096 942 755 276 873 884 871 163 904 × 2 = 1 + 0.312 499 999 999 999 999 999 980 193 885 510 553 747 769 742 327 808;
  • 87) 0.312 499 999 999 999 999 999 980 193 885 510 553 747 769 742 327 808 × 2 = 0 + 0.624 999 999 999 999 999 999 960 387 771 021 107 495 539 484 655 616;
  • 88) 0.624 999 999 999 999 999 999 960 387 771 021 107 495 539 484 655 616 × 2 = 1 + 0.249 999 999 999 999 999 999 920 775 542 042 214 991 078 969 311 232;
  • 89) 0.249 999 999 999 999 999 999 920 775 542 042 214 991 078 969 311 232 × 2 = 0 + 0.499 999 999 999 999 999 999 841 551 084 084 429 982 157 938 622 464;
  • 90) 0.499 999 999 999 999 999 999 841 551 084 084 429 982 157 938 622 464 × 2 = 0 + 0.999 999 999 999 999 999 999 683 102 168 168 859 964 315 877 244 928;
  • 91) 0.999 999 999 999 999 999 999 683 102 168 168 859 964 315 877 244 928 × 2 = 1 + 0.999 999 999 999 999 999 999 366 204 336 337 719 928 631 754 489 856;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 897 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