Decimal to 64 Bit IEEE 754 Binary: Convert Number 10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. 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;
  • 10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014 ÷ 2 = 5 000 000 050 050 055 000 000 000 000 000 000 000 000 000 000 000 000 000 000 007 + 0;
  • 5 000 000 050 050 055 000 000 000 000 000 000 000 000 000 000 000 000 000 000 007 ÷ 2 = 2 500 000 025 025 027 500 000 000 000 000 000 000 000 000 000 000 000 000 000 003 + 1;
  • 2 500 000 025 025 027 500 000 000 000 000 000 000 000 000 000 000 000 000 000 003 ÷ 2 = 1 250 000 012 512 513 750 000 000 000 000 000 000 000 000 000 000 000 000 000 001 + 1;
  • 1 250 000 012 512 513 750 000 000 000 000 000 000 000 000 000 000 000 000 000 001 ÷ 2 = 625 000 006 256 256 875 000 000 000 000 000 000 000 000 000 000 000 000 000 000 + 1;
  • 625 000 006 256 256 875 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 312 500 003 128 128 437 500 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 312 500 003 128 128 437 500 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 156 250 001 564 064 218 750 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 156 250 001 564 064 218 750 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 78 125 000 782 032 109 375 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 78 125 000 782 032 109 375 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 39 062 500 391 016 054 687 500 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 39 062 500 391 016 054 687 500 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 19 531 250 195 508 027 343 750 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 19 531 250 195 508 027 343 750 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 9 765 625 097 754 013 671 875 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 9 765 625 097 754 013 671 875 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 4 882 812 548 877 006 835 937 500 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 4 882 812 548 877 006 835 937 500 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 2 441 406 274 438 503 417 968 750 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 2 441 406 274 438 503 417 968 750 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 1 220 703 137 219 251 708 984 375 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 1 220 703 137 219 251 708 984 375 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 610 351 568 609 625 854 492 187 500 000 000 000 000 000 000 000 000 000 000 + 0;
  • 610 351 568 609 625 854 492 187 500 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 305 175 784 304 812 927 246 093 750 000 000 000 000 000 000 000 000 000 000 + 0;
  • 305 175 784 304 812 927 246 093 750 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 152 587 892 152 406 463 623 046 875 000 000 000 000 000 000 000 000 000 000 + 0;
  • 152 587 892 152 406 463 623 046 875 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 76 293 946 076 203 231 811 523 437 500 000 000 000 000 000 000 000 000 000 + 0;
  • 76 293 946 076 203 231 811 523 437 500 000 000 000 000 000 000 000 000 000 ÷ 2 = 38 146 973 038 101 615 905 761 718 750 000 000 000 000 000 000 000 000 000 + 0;
  • 38 146 973 038 101 615 905 761 718 750 000 000 000 000 000 000 000 000 000 ÷ 2 = 19 073 486 519 050 807 952 880 859 375 000 000 000 000 000 000 000 000 000 + 0;
  • 19 073 486 519 050 807 952 880 859 375 000 000 000 000 000 000 000 000 000 ÷ 2 = 9 536 743 259 525 403 976 440 429 687 500 000 000 000 000 000 000 000 000 + 0;
  • 9 536 743 259 525 403 976 440 429 687 500 000 000 000 000 000 000 000 000 ÷ 2 = 4 768 371 629 762 701 988 220 214 843 750 000 000 000 000 000 000 000 000 + 0;
  • 4 768 371 629 762 701 988 220 214 843 750 000 000 000 000 000 000 000 000 ÷ 2 = 2 384 185 814 881 350 994 110 107 421 875 000 000 000 000 000 000 000 000 + 0;
  • 2 384 185 814 881 350 994 110 107 421 875 000 000 000 000 000 000 000 000 ÷ 2 = 1 192 092 907 440 675 497 055 053 710 937 500 000 000 000 000 000 000 000 + 0;
  • 1 192 092 907 440 675 497 055 053 710 937 500 000 000 000 000 000 000 000 ÷ 2 = 596 046 453 720 337 748 527 526 855 468 750 000 000 000 000 000 000 000 + 0;
  • 596 046 453 720 337 748 527 526 855 468 750 000 000 000 000 000 000 000 ÷ 2 = 298 023 226 860 168 874 263 763 427 734 375 000 000 000 000 000 000 000 + 0;
  • 298 023 226 860 168 874 263 763 427 734 375 000 000 000 000 000 000 000 ÷ 2 = 149 011 613 430 084 437 131 881 713 867 187 500 000 000 000 000 000 000 + 0;
  • 149 011 613 430 084 437 131 881 713 867 187 500 000 000 000 000 000 000 ÷ 2 = 74 505 806 715 042 218 565 940 856 933 593 750 000 000 000 000 000 000 + 0;
  • 74 505 806 715 042 218 565 940 856 933 593 750 000 000 000 000 000 000 ÷ 2 = 37 252 903 357 521 109 282 970 428 466 796 875 000 000 000 000 000 000 + 0;
  • 37 252 903 357 521 109 282 970 428 466 796 875 000 000 000 000 000 000 ÷ 2 = 18 626 451 678 760 554 641 485 214 233 398 437 500 000 000 000 000 000 + 0;
  • 18 626 451 678 760 554 641 485 214 233 398 437 500 000 000 000 000 000 ÷ 2 = 9 313 225 839 380 277 320 742 607 116 699 218 750 000 000 000 000 000 + 0;
  • 9 313 225 839 380 277 320 742 607 116 699 218 750 000 000 000 000 000 ÷ 2 = 4 656 612 919 690 138 660 371 303 558 349 609 375 000 000 000 000 000 + 0;
  • 4 656 612 919 690 138 660 371 303 558 349 609 375 000 000 000 000 000 ÷ 2 = 2 328 306 459 845 069 330 185 651 779 174 804 687 500 000 000 000 000 + 0;
  • 2 328 306 459 845 069 330 185 651 779 174 804 687 500 000 000 000 000 ÷ 2 = 1 164 153 229 922 534 665 092 825 889 587 402 343 750 000 000 000 000 + 0;
  • 1 164 153 229 922 534 665 092 825 889 587 402 343 750 000 000 000 000 ÷ 2 = 582 076 614 961 267 332 546 412 944 793 701 171 875 000 000 000 000 + 0;
  • 582 076 614 961 267 332 546 412 944 793 701 171 875 000 000 000 000 ÷ 2 = 291 038 307 480 633 666 273 206 472 396 850 585 937 500 000 000 000 + 0;
  • 291 038 307 480 633 666 273 206 472 396 850 585 937 500 000 000 000 ÷ 2 = 145 519 153 740 316 833 136 603 236 198 425 292 968 750 000 000 000 + 0;
  • 145 519 153 740 316 833 136 603 236 198 425 292 968 750 000 000 000 ÷ 2 = 72 759 576 870 158 416 568 301 618 099 212 646 484 375 000 000 000 + 0;
  • 72 759 576 870 158 416 568 301 618 099 212 646 484 375 000 000 000 ÷ 2 = 36 379 788 435 079 208 284 150 809 049 606 323 242 187 500 000 000 + 0;
  • 36 379 788 435 079 208 284 150 809 049 606 323 242 187 500 000 000 ÷ 2 = 18 189 894 217 539 604 142 075 404 524 803 161 621 093 750 000 000 + 0;
  • 18 189 894 217 539 604 142 075 404 524 803 161 621 093 750 000 000 ÷ 2 = 9 094 947 108 769 802 071 037 702 262 401 580 810 546 875 000 000 + 0;
  • 9 094 947 108 769 802 071 037 702 262 401 580 810 546 875 000 000 ÷ 2 = 4 547 473 554 384 901 035 518 851 131 200 790 405 273 437 500 000 + 0;
  • 4 547 473 554 384 901 035 518 851 131 200 790 405 273 437 500 000 ÷ 2 = 2 273 736 777 192 450 517 759 425 565 600 395 202 636 718 750 000 + 0;
  • 2 273 736 777 192 450 517 759 425 565 600 395 202 636 718 750 000 ÷ 2 = 1 136 868 388 596 225 258 879 712 782 800 197 601 318 359 375 000 + 0;
  • 1 136 868 388 596 225 258 879 712 782 800 197 601 318 359 375 000 ÷ 2 = 568 434 194 298 112 629 439 856 391 400 098 800 659 179 687 500 + 0;
  • 568 434 194 298 112 629 439 856 391 400 098 800 659 179 687 500 ÷ 2 = 284 217 097 149 056 314 719 928 195 700 049 400 329 589 843 750 + 0;
  • 284 217 097 149 056 314 719 928 195 700 049 400 329 589 843 750 ÷ 2 = 142 108 548 574 528 157 359 964 097 850 024 700 164 794 921 875 + 0;
  • 142 108 548 574 528 157 359 964 097 850 024 700 164 794 921 875 ÷ 2 = 71 054 274 287 264 078 679 982 048 925 012 350 082 397 460 937 + 1;
  • 71 054 274 287 264 078 679 982 048 925 012 350 082 397 460 937 ÷ 2 = 35 527 137 143 632 039 339 991 024 462 506 175 041 198 730 468 + 1;
  • 35 527 137 143 632 039 339 991 024 462 506 175 041 198 730 468 ÷ 2 = 17 763 568 571 816 019 669 995 512 231 253 087 520 599 365 234 + 0;
  • 17 763 568 571 816 019 669 995 512 231 253 087 520 599 365 234 ÷ 2 = 8 881 784 285 908 009 834 997 756 115 626 543 760 299 682 617 + 0;
  • 8 881 784 285 908 009 834 997 756 115 626 543 760 299 682 617 ÷ 2 = 4 440 892 142 954 004 917 498 878 057 813 271 880 149 841 308 + 1;
  • 4 440 892 142 954 004 917 498 878 057 813 271 880 149 841 308 ÷ 2 = 2 220 446 071 477 002 458 749 439 028 906 635 940 074 920 654 + 0;
  • 2 220 446 071 477 002 458 749 439 028 906 635 940 074 920 654 ÷ 2 = 1 110 223 035 738 501 229 374 719 514 453 317 970 037 460 327 + 0;
  • 1 110 223 035 738 501 229 374 719 514 453 317 970 037 460 327 ÷ 2 = 555 111 517 869 250 614 687 359 757 226 658 985 018 730 163 + 1;
  • 555 111 517 869 250 614 687 359 757 226 658 985 018 730 163 ÷ 2 = 277 555 758 934 625 307 343 679 878 613 329 492 509 365 081 + 1;
  • 277 555 758 934 625 307 343 679 878 613 329 492 509 365 081 ÷ 2 = 138 777 879 467 312 653 671 839 939 306 664 746 254 682 540 + 1;
  • 138 777 879 467 312 653 671 839 939 306 664 746 254 682 540 ÷ 2 = 69 388 939 733 656 326 835 919 969 653 332 373 127 341 270 + 0;
  • 69 388 939 733 656 326 835 919 969 653 332 373 127 341 270 ÷ 2 = 34 694 469 866 828 163 417 959 984 826 666 186 563 670 635 + 0;
  • 34 694 469 866 828 163 417 959 984 826 666 186 563 670 635 ÷ 2 = 17 347 234 933 414 081 708 979 992 413 333 093 281 835 317 + 1;
  • 17 347 234 933 414 081 708 979 992 413 333 093 281 835 317 ÷ 2 = 8 673 617 466 707 040 854 489 996 206 666 546 640 917 658 + 1;
  • 8 673 617 466 707 040 854 489 996 206 666 546 640 917 658 ÷ 2 = 4 336 808 733 353 520 427 244 998 103 333 273 320 458 829 + 0;
  • 4 336 808 733 353 520 427 244 998 103 333 273 320 458 829 ÷ 2 = 2 168 404 366 676 760 213 622 499 051 666 636 660 229 414 + 1;
  • 2 168 404 366 676 760 213 622 499 051 666 636 660 229 414 ÷ 2 = 1 084 202 183 338 380 106 811 249 525 833 318 330 114 707 + 0;
  • 1 084 202 183 338 380 106 811 249 525 833 318 330 114 707 ÷ 2 = 542 101 091 669 190 053 405 624 762 916 659 165 057 353 + 1;
  • 542 101 091 669 190 053 405 624 762 916 659 165 057 353 ÷ 2 = 271 050 545 834 595 026 702 812 381 458 329 582 528 676 + 1;
  • 271 050 545 834 595 026 702 812 381 458 329 582 528 676 ÷ 2 = 135 525 272 917 297 513 351 406 190 729 164 791 264 338 + 0;
  • 135 525 272 917 297 513 351 406 190 729 164 791 264 338 ÷ 2 = 67 762 636 458 648 756 675 703 095 364 582 395 632 169 + 0;
  • 67 762 636 458 648 756 675 703 095 364 582 395 632 169 ÷ 2 = 33 881 318 229 324 378 337 851 547 682 291 197 816 084 + 1;
  • 33 881 318 229 324 378 337 851 547 682 291 197 816 084 ÷ 2 = 16 940 659 114 662 189 168 925 773 841 145 598 908 042 + 0;
  • 16 940 659 114 662 189 168 925 773 841 145 598 908 042 ÷ 2 = 8 470 329 557 331 094 584 462 886 920 572 799 454 021 + 0;
  • 8 470 329 557 331 094 584 462 886 920 572 799 454 021 ÷ 2 = 4 235 164 778 665 547 292 231 443 460 286 399 727 010 + 1;
  • 4 235 164 778 665 547 292 231 443 460 286 399 727 010 ÷ 2 = 2 117 582 389 332 773 646 115 721 730 143 199 863 505 + 0;
  • 2 117 582 389 332 773 646 115 721 730 143 199 863 505 ÷ 2 = 1 058 791 194 666 386 823 057 860 865 071 599 931 752 + 1;
  • 1 058 791 194 666 386 823 057 860 865 071 599 931 752 ÷ 2 = 529 395 597 333 193 411 528 930 432 535 799 965 876 + 0;
  • 529 395 597 333 193 411 528 930 432 535 799 965 876 ÷ 2 = 264 697 798 666 596 705 764 465 216 267 899 982 938 + 0;
  • 264 697 798 666 596 705 764 465 216 267 899 982 938 ÷ 2 = 132 348 899 333 298 352 882 232 608 133 949 991 469 + 0;
  • 132 348 899 333 298 352 882 232 608 133 949 991 469 ÷ 2 = 66 174 449 666 649 176 441 116 304 066 974 995 734 + 1;
  • 66 174 449 666 649 176 441 116 304 066 974 995 734 ÷ 2 = 33 087 224 833 324 588 220 558 152 033 487 497 867 + 0;
  • 33 087 224 833 324 588 220 558 152 033 487 497 867 ÷ 2 = 16 543 612 416 662 294 110 279 076 016 743 748 933 + 1;
  • 16 543 612 416 662 294 110 279 076 016 743 748 933 ÷ 2 = 8 271 806 208 331 147 055 139 538 008 371 874 466 + 1;
  • 8 271 806 208 331 147 055 139 538 008 371 874 466 ÷ 2 = 4 135 903 104 165 573 527 569 769 004 185 937 233 + 0;
  • 4 135 903 104 165 573 527 569 769 004 185 937 233 ÷ 2 = 2 067 951 552 082 786 763 784 884 502 092 968 616 + 1;
  • 2 067 951 552 082 786 763 784 884 502 092 968 616 ÷ 2 = 1 033 975 776 041 393 381 892 442 251 046 484 308 + 0;
  • 1 033 975 776 041 393 381 892 442 251 046 484 308 ÷ 2 = 516 987 888 020 696 690 946 221 125 523 242 154 + 0;
  • 516 987 888 020 696 690 946 221 125 523 242 154 ÷ 2 = 258 493 944 010 348 345 473 110 562 761 621 077 + 0;
  • 258 493 944 010 348 345 473 110 562 761 621 077 ÷ 2 = 129 246 972 005 174 172 736 555 281 380 810 538 + 1;
  • 129 246 972 005 174 172 736 555 281 380 810 538 ÷ 2 = 64 623 486 002 587 086 368 277 640 690 405 269 + 0;
  • 64 623 486 002 587 086 368 277 640 690 405 269 ÷ 2 = 32 311 743 001 293 543 184 138 820 345 202 634 + 1;
  • 32 311 743 001 293 543 184 138 820 345 202 634 ÷ 2 = 16 155 871 500 646 771 592 069 410 172 601 317 + 0;
  • 16 155 871 500 646 771 592 069 410 172 601 317 ÷ 2 = 8 077 935 750 323 385 796 034 705 086 300 658 + 1;
  • 8 077 935 750 323 385 796 034 705 086 300 658 ÷ 2 = 4 038 967 875 161 692 898 017 352 543 150 329 + 0;
  • 4 038 967 875 161 692 898 017 352 543 150 329 ÷ 2 = 2 019 483 937 580 846 449 008 676 271 575 164 + 1;
  • 2 019 483 937 580 846 449 008 676 271 575 164 ÷ 2 = 1 009 741 968 790 423 224 504 338 135 787 582 + 0;
  • 1 009 741 968 790 423 224 504 338 135 787 582 ÷ 2 = 504 870 984 395 211 612 252 169 067 893 791 + 0;
  • 504 870 984 395 211 612 252 169 067 893 791 ÷ 2 = 252 435 492 197 605 806 126 084 533 946 895 + 1;
  • 252 435 492 197 605 806 126 084 533 946 895 ÷ 2 = 126 217 746 098 802 903 063 042 266 973 447 + 1;
  • 126 217 746 098 802 903 063 042 266 973 447 ÷ 2 = 63 108 873 049 401 451 531 521 133 486 723 + 1;
  • 63 108 873 049 401 451 531 521 133 486 723 ÷ 2 = 31 554 436 524 700 725 765 760 566 743 361 + 1;
  • 31 554 436 524 700 725 765 760 566 743 361 ÷ 2 = 15 777 218 262 350 362 882 880 283 371 680 + 1;
  • 15 777 218 262 350 362 882 880 283 371 680 ÷ 2 = 7 888 609 131 175 181 441 440 141 685 840 + 0;
  • 7 888 609 131 175 181 441 440 141 685 840 ÷ 2 = 3 944 304 565 587 590 720 720 070 842 920 + 0;
  • 3 944 304 565 587 590 720 720 070 842 920 ÷ 2 = 1 972 152 282 793 795 360 360 035 421 460 + 0;
  • 1 972 152 282 793 795 360 360 035 421 460 ÷ 2 = 986 076 141 396 897 680 180 017 710 730 + 0;
  • 986 076 141 396 897 680 180 017 710 730 ÷ 2 = 493 038 070 698 448 840 090 008 855 365 + 0;
  • 493 038 070 698 448 840 090 008 855 365 ÷ 2 = 246 519 035 349 224 420 045 004 427 682 + 1;
  • 246 519 035 349 224 420 045 004 427 682 ÷ 2 = 123 259 517 674 612 210 022 502 213 841 + 0;
  • 123 259 517 674 612 210 022 502 213 841 ÷ 2 = 61 629 758 837 306 105 011 251 106 920 + 1;
  • 61 629 758 837 306 105 011 251 106 920 ÷ 2 = 30 814 879 418 653 052 505 625 553 460 + 0;
  • 30 814 879 418 653 052 505 625 553 460 ÷ 2 = 15 407 439 709 326 526 252 812 776 730 + 0;
  • 15 407 439 709 326 526 252 812 776 730 ÷ 2 = 7 703 719 854 663 263 126 406 388 365 + 0;
  • 7 703 719 854 663 263 126 406 388 365 ÷ 2 = 3 851 859 927 331 631 563 203 194 182 + 1;
  • 3 851 859 927 331 631 563 203 194 182 ÷ 2 = 1 925 929 963 665 815 781 601 597 091 + 0;
  • 1 925 929 963 665 815 781 601 597 091 ÷ 2 = 962 964 981 832 907 890 800 798 545 + 1;
  • 962 964 981 832 907 890 800 798 545 ÷ 2 = 481 482 490 916 453 945 400 399 272 + 1;
  • 481 482 490 916 453 945 400 399 272 ÷ 2 = 240 741 245 458 226 972 700 199 636 + 0;
  • 240 741 245 458 226 972 700 199 636 ÷ 2 = 120 370 622 729 113 486 350 099 818 + 0;
  • 120 370 622 729 113 486 350 099 818 ÷ 2 = 60 185 311 364 556 743 175 049 909 + 0;
  • 60 185 311 364 556 743 175 049 909 ÷ 2 = 30 092 655 682 278 371 587 524 954 + 1;
  • 30 092 655 682 278 371 587 524 954 ÷ 2 = 15 046 327 841 139 185 793 762 477 + 0;
  • 15 046 327 841 139 185 793 762 477 ÷ 2 = 7 523 163 920 569 592 896 881 238 + 1;
  • 7 523 163 920 569 592 896 881 238 ÷ 2 = 3 761 581 960 284 796 448 440 619 + 0;
  • 3 761 581 960 284 796 448 440 619 ÷ 2 = 1 880 790 980 142 398 224 220 309 + 1;
  • 1 880 790 980 142 398 224 220 309 ÷ 2 = 940 395 490 071 199 112 110 154 + 1;
  • 940 395 490 071 199 112 110 154 ÷ 2 = 470 197 745 035 599 556 055 077 + 0;
  • 470 197 745 035 599 556 055 077 ÷ 2 = 235 098 872 517 799 778 027 538 + 1;
  • 235 098 872 517 799 778 027 538 ÷ 2 = 117 549 436 258 899 889 013 769 + 0;
  • 117 549 436 258 899 889 013 769 ÷ 2 = 58 774 718 129 449 944 506 884 + 1;
  • 58 774 718 129 449 944 506 884 ÷ 2 = 29 387 359 064 724 972 253 442 + 0;
  • 29 387 359 064 724 972 253 442 ÷ 2 = 14 693 679 532 362 486 126 721 + 0;
  • 14 693 679 532 362 486 126 721 ÷ 2 = 7 346 839 766 181 243 063 360 + 1;
  • 7 346 839 766 181 243 063 360 ÷ 2 = 3 673 419 883 090 621 531 680 + 0;
  • 3 673 419 883 090 621 531 680 ÷ 2 = 1 836 709 941 545 310 765 840 + 0;
  • 1 836 709 941 545 310 765 840 ÷ 2 = 918 354 970 772 655 382 920 + 0;
  • 918 354 970 772 655 382 920 ÷ 2 = 459 177 485 386 327 691 460 + 0;
  • 459 177 485 386 327 691 460 ÷ 2 = 229 588 742 693 163 845 730 + 0;
  • 229 588 742 693 163 845 730 ÷ 2 = 114 794 371 346 581 922 865 + 0;
  • 114 794 371 346 581 922 865 ÷ 2 = 57 397 185 673 290 961 432 + 1;
  • 57 397 185 673 290 961 432 ÷ 2 = 28 698 592 836 645 480 716 + 0;
  • 28 698 592 836 645 480 716 ÷ 2 = 14 349 296 418 322 740 358 + 0;
  • 14 349 296 418 322 740 358 ÷ 2 = 7 174 648 209 161 370 179 + 0;
  • 7 174 648 209 161 370 179 ÷ 2 = 3 587 324 104 580 685 089 + 1;
  • 3 587 324 104 580 685 089 ÷ 2 = 1 793 662 052 290 342 544 + 1;
  • 1 793 662 052 290 342 544 ÷ 2 = 896 831 026 145 171 272 + 0;
  • 896 831 026 145 171 272 ÷ 2 = 448 415 513 072 585 636 + 0;
  • 448 415 513 072 585 636 ÷ 2 = 224 207 756 536 292 818 + 0;
  • 224 207 756 536 292 818 ÷ 2 = 112 103 878 268 146 409 + 0;
  • 112 103 878 268 146 409 ÷ 2 = 56 051 939 134 073 204 + 1;
  • 56 051 939 134 073 204 ÷ 2 = 28 025 969 567 036 602 + 0;
  • 28 025 969 567 036 602 ÷ 2 = 14 012 984 783 518 301 + 0;
  • 14 012 984 783 518 301 ÷ 2 = 7 006 492 391 759 150 + 1;
  • 7 006 492 391 759 150 ÷ 2 = 3 503 246 195 879 575 + 0;
  • 3 503 246 195 879 575 ÷ 2 = 1 751 623 097 939 787 + 1;
  • 1 751 623 097 939 787 ÷ 2 = 875 811 548 969 893 + 1;
  • 875 811 548 969 893 ÷ 2 = 437 905 774 484 946 + 1;
  • 437 905 774 484 946 ÷ 2 = 218 952 887 242 473 + 0;
  • 218 952 887 242 473 ÷ 2 = 109 476 443 621 236 + 1;
  • 109 476 443 621 236 ÷ 2 = 54 738 221 810 618 + 0;
  • 54 738 221 810 618 ÷ 2 = 27 369 110 905 309 + 0;
  • 27 369 110 905 309 ÷ 2 = 13 684 555 452 654 + 1;
  • 13 684 555 452 654 ÷ 2 = 6 842 277 726 327 + 0;
  • 6 842 277 726 327 ÷ 2 = 3 421 138 863 163 + 1;
  • 3 421 138 863 163 ÷ 2 = 1 710 569 431 581 + 1;
  • 1 710 569 431 581 ÷ 2 = 855 284 715 790 + 1;
  • 855 284 715 790 ÷ 2 = 427 642 357 895 + 0;
  • 427 642 357 895 ÷ 2 = 213 821 178 947 + 1;
  • 213 821 178 947 ÷ 2 = 106 910 589 473 + 1;
  • 106 910 589 473 ÷ 2 = 53 455 294 736 + 1;
  • 53 455 294 736 ÷ 2 = 26 727 647 368 + 0;
  • 26 727 647 368 ÷ 2 = 13 363 823 684 + 0;
  • 13 363 823 684 ÷ 2 = 6 681 911 842 + 0;
  • 6 681 911 842 ÷ 2 = 3 340 955 921 + 0;
  • 3 340 955 921 ÷ 2 = 1 670 477 960 + 1;
  • 1 670 477 960 ÷ 2 = 835 238 980 + 0;
  • 835 238 980 ÷ 2 = 417 619 490 + 0;
  • 417 619 490 ÷ 2 = 208 809 745 + 0;
  • 208 809 745 ÷ 2 = 104 404 872 + 1;
  • 104 404 872 ÷ 2 = 52 202 436 + 0;
  • 52 202 436 ÷ 2 = 26 101 218 + 0;
  • 26 101 218 ÷ 2 = 13 050 609 + 0;
  • 13 050 609 ÷ 2 = 6 525 304 + 1;
  • 6 525 304 ÷ 2 = 3 262 652 + 0;
  • 3 262 652 ÷ 2 = 1 631 326 + 0;
  • 1 631 326 ÷ 2 = 815 663 + 0;
  • 815 663 ÷ 2 = 407 831 + 1;
  • 407 831 ÷ 2 = 203 915 + 1;
  • 203 915 ÷ 2 = 101 957 + 1;
  • 101 957 ÷ 2 = 50 978 + 1;
  • 50 978 ÷ 2 = 25 489 + 0;
  • 25 489 ÷ 2 = 12 744 + 1;
  • 12 744 ÷ 2 = 6 372 + 0;
  • 6 372 ÷ 2 = 3 186 + 0;
  • 3 186 ÷ 2 = 1 593 + 0;
  • 1 593 ÷ 2 = 796 + 1;
  • 796 ÷ 2 = 398 + 0;
  • 398 ÷ 2 = 199 + 0;
  • 199 ÷ 2 = 99 + 1;
  • 99 ÷ 2 = 49 + 1;
  • 49 ÷ 2 = 24 + 1;
  • 24 ÷ 2 = 12 + 0;
  • 12 ÷ 2 = 6 + 0;
  • 6 ÷ 2 = 3 + 0;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

2. Construct the base 2 representation of the positive number.

Take all the remainders starting from the bottom of the list constructed above.


10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014(10) =


110 0011 1001 0001 0111 1000 1000 1000 1000 0111 0111 0100 1011 1010 0100 0011 0001 0000 0010 0101 0110 1010 0011 0100 0101 0000 0111 1100 1010 1010 0010 1101 0001 0100 1001 1010 1100 1110 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 202 positions to the left, so that only one non zero digit remains to the left of it:


10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014(10) =


110 0011 1001 0001 0111 1000 1000 1000 1000 0111 0111 0100 1011 1010 0100 0011 0001 0000 0010 0101 0110 1010 0011 0100 0101 0000 0111 1100 1010 1010 0010 1101 0001 0100 1001 1010 1100 1110 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110(2) =


110 0011 1001 0001 0111 1000 1000 1000 1000 0111 0111 0100 1011 1010 0100 0011 0001 0000 0010 0101 0110 1010 0011 0100 0101 0000 0111 1100 1010 1010 0010 1101 0001 0100 1001 1010 1100 1110 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110(2) × 20 =


1.1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110 1001 0000 1100 0100 0000 1001 0101 1010 1000 1101 0001 0100 0001 1111 0010 1010 1000 1011 0100 0101 0010 0110 1011 0011 1001 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 10(2) × 2202


4. 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): 202


Mantissa (not normalized):
1.1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110 1001 0000 1100 0100 0000 1001 0101 1010 1000 1101 0001 0100 0001 1111 0010 1010 1000 1011 0100 0101 0010 0110 1011 0011 1001 0011 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 10


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


202 + 2(11-1) - 1 =


(202 + 1 023)(10) =


1 225(10)


6. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 225 ÷ 2 = 612 + 1;
  • 612 ÷ 2 = 306 + 0;
  • 306 ÷ 2 = 153 + 0;
  • 153 ÷ 2 = 76 + 1;
  • 76 ÷ 2 = 38 + 0;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

7. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =


1225(10) =


100 1100 1001(2)


8. 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, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


1. 1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110 10 0100 0011 0001 0000 0010 0101 0110 1010 0011 0100 0101 0000 0111 1100 1010 1010 0010 1101 0001 0100 1001 1010 1100 1110 0100 1100 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 =


1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110


9. 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) =
100 1100 1001


Mantissa (52 bits) =
1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110


The base ten decimal number 10 000 000 100 100 110 000 000 000 000 000 000 000 000 000 000 000 000 000 000 014 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1100 1001 - 1000 1110 0100 0101 1110 0010 0010 0010 0001 1101 1101 0010 1110

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