64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (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;
  • 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 ÷ 2 = 550 000 000 005 049 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 0;
  • 550 000 000 005 049 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 275 000 000 002 524 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 275 000 000 002 524 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 137 500 000 001 262 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 137 500 000 001 262 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 68 750 000 000 631 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 68 750 000 000 631 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 34 375 000 000 315 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 34 375 000 000 315 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 17 187 500 000 157 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 17 187 500 000 157 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 8 593 750 000 078 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 8 593 750 000 078 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 4 296 875 000 039 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 4 296 875 000 039 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 2 148 437 500 019 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 2 148 437 500 019 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 1 074 218 750 009 863 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 1 074 218 750 009 863 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 537 109 375 004 931 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 537 109 375 004 931 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 268 554 687 502 465 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 268 554 687 502 465 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 134 277 343 751 232 910 156 249 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 134 277 343 751 232 910 156 249 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 67 138 671 875 616 455 078 124 999 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 67 138 671 875 616 455 078 124 999 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 33 569 335 937 808 227 539 062 499 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 33 569 335 937 808 227 539 062 499 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 16 784 667 968 904 113 769 531 249 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 16 784 667 968 904 113 769 531 249 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 8 392 333 984 452 056 884 765 624 999 999 999 999 999 999 999 999 999 999 999 + 1;
  • 8 392 333 984 452 056 884 765 624 999 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 4 196 166 992 226 028 442 382 812 499 999 999 999 999 999 999 999 999 999 999 + 1;
  • 4 196 166 992 226 028 442 382 812 499 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 2 098 083 496 113 014 221 191 406 249 999 999 999 999 999 999 999 999 999 999 + 1;
  • 2 098 083 496 113 014 221 191 406 249 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 1 049 041 748 056 507 110 595 703 124 999 999 999 999 999 999 999 999 999 999 + 1;
  • 1 049 041 748 056 507 110 595 703 124 999 999 999 999 999 999 999 999 999 999 ÷ 2 = 524 520 874 028 253 555 297 851 562 499 999 999 999 999 999 999 999 999 999 + 1;
  • 524 520 874 028 253 555 297 851 562 499 999 999 999 999 999 999 999 999 999 ÷ 2 = 262 260 437 014 126 777 648 925 781 249 999 999 999 999 999 999 999 999 999 + 1;
  • 262 260 437 014 126 777 648 925 781 249 999 999 999 999 999 999 999 999 999 ÷ 2 = 131 130 218 507 063 388 824 462 890 624 999 999 999 999 999 999 999 999 999 + 1;
  • 131 130 218 507 063 388 824 462 890 624 999 999 999 999 999 999 999 999 999 ÷ 2 = 65 565 109 253 531 694 412 231 445 312 499 999 999 999 999 999 999 999 999 + 1;
  • 65 565 109 253 531 694 412 231 445 312 499 999 999 999 999 999 999 999 999 ÷ 2 = 32 782 554 626 765 847 206 115 722 656 249 999 999 999 999 999 999 999 999 + 1;
  • 32 782 554 626 765 847 206 115 722 656 249 999 999 999 999 999 999 999 999 ÷ 2 = 16 391 277 313 382 923 603 057 861 328 124 999 999 999 999 999 999 999 999 + 1;
  • 16 391 277 313 382 923 603 057 861 328 124 999 999 999 999 999 999 999 999 ÷ 2 = 8 195 638 656 691 461 801 528 930 664 062 499 999 999 999 999 999 999 999 + 1;
  • 8 195 638 656 691 461 801 528 930 664 062 499 999 999 999 999 999 999 999 ÷ 2 = 4 097 819 328 345 730 900 764 465 332 031 249 999 999 999 999 999 999 999 + 1;
  • 4 097 819 328 345 730 900 764 465 332 031 249 999 999 999 999 999 999 999 ÷ 2 = 2 048 909 664 172 865 450 382 232 666 015 624 999 999 999 999 999 999 999 + 1;
  • 2 048 909 664 172 865 450 382 232 666 015 624 999 999 999 999 999 999 999 ÷ 2 = 1 024 454 832 086 432 725 191 116 333 007 812 499 999 999 999 999 999 999 + 1;
  • 1 024 454 832 086 432 725 191 116 333 007 812 499 999 999 999 999 999 999 ÷ 2 = 512 227 416 043 216 362 595 558 166 503 906 249 999 999 999 999 999 999 + 1;
  • 512 227 416 043 216 362 595 558 166 503 906 249 999 999 999 999 999 999 ÷ 2 = 256 113 708 021 608 181 297 779 083 251 953 124 999 999 999 999 999 999 + 1;
  • 256 113 708 021 608 181 297 779 083 251 953 124 999 999 999 999 999 999 ÷ 2 = 128 056 854 010 804 090 648 889 541 625 976 562 499 999 999 999 999 999 + 1;
  • 128 056 854 010 804 090 648 889 541 625 976 562 499 999 999 999 999 999 ÷ 2 = 64 028 427 005 402 045 324 444 770 812 988 281 249 999 999 999 999 999 + 1;
  • 64 028 427 005 402 045 324 444 770 812 988 281 249 999 999 999 999 999 ÷ 2 = 32 014 213 502 701 022 662 222 385 406 494 140 624 999 999 999 999 999 + 1;
  • 32 014 213 502 701 022 662 222 385 406 494 140 624 999 999 999 999 999 ÷ 2 = 16 007 106 751 350 511 331 111 192 703 247 070 312 499 999 999 999 999 + 1;
  • 16 007 106 751 350 511 331 111 192 703 247 070 312 499 999 999 999 999 ÷ 2 = 8 003 553 375 675 255 665 555 596 351 623 535 156 249 999 999 999 999 + 1;
  • 8 003 553 375 675 255 665 555 596 351 623 535 156 249 999 999 999 999 ÷ 2 = 4 001 776 687 837 627 832 777 798 175 811 767 578 124 999 999 999 999 + 1;
  • 4 001 776 687 837 627 832 777 798 175 811 767 578 124 999 999 999 999 ÷ 2 = 2 000 888 343 918 813 916 388 899 087 905 883 789 062 499 999 999 999 + 1;
  • 2 000 888 343 918 813 916 388 899 087 905 883 789 062 499 999 999 999 ÷ 2 = 1 000 444 171 959 406 958 194 449 543 952 941 894 531 249 999 999 999 + 1;
  • 1 000 444 171 959 406 958 194 449 543 952 941 894 531 249 999 999 999 ÷ 2 = 500 222 085 979 703 479 097 224 771 976 470 947 265 624 999 999 999 + 1;
  • 500 222 085 979 703 479 097 224 771 976 470 947 265 624 999 999 999 ÷ 2 = 250 111 042 989 851 739 548 612 385 988 235 473 632 812 499 999 999 + 1;
  • 250 111 042 989 851 739 548 612 385 988 235 473 632 812 499 999 999 ÷ 2 = 125 055 521 494 925 869 774 306 192 994 117 736 816 406 249 999 999 + 1;
  • 125 055 521 494 925 869 774 306 192 994 117 736 816 406 249 999 999 ÷ 2 = 62 527 760 747 462 934 887 153 096 497 058 868 408 203 124 999 999 + 1;
  • 62 527 760 747 462 934 887 153 096 497 058 868 408 203 124 999 999 ÷ 2 = 31 263 880 373 731 467 443 576 548 248 529 434 204 101 562 499 999 + 1;
  • 31 263 880 373 731 467 443 576 548 248 529 434 204 101 562 499 999 ÷ 2 = 15 631 940 186 865 733 721 788 274 124 264 717 102 050 781 249 999 + 1;
  • 15 631 940 186 865 733 721 788 274 124 264 717 102 050 781 249 999 ÷ 2 = 7 815 970 093 432 866 860 894 137 062 132 358 551 025 390 624 999 + 1;
  • 7 815 970 093 432 866 860 894 137 062 132 358 551 025 390 624 999 ÷ 2 = 3 907 985 046 716 433 430 447 068 531 066 179 275 512 695 312 499 + 1;
  • 3 907 985 046 716 433 430 447 068 531 066 179 275 512 695 312 499 ÷ 2 = 1 953 992 523 358 216 715 223 534 265 533 089 637 756 347 656 249 + 1;
  • 1 953 992 523 358 216 715 223 534 265 533 089 637 756 347 656 249 ÷ 2 = 976 996 261 679 108 357 611 767 132 766 544 818 878 173 828 124 + 1;
  • 976 996 261 679 108 357 611 767 132 766 544 818 878 173 828 124 ÷ 2 = 488 498 130 839 554 178 805 883 566 383 272 409 439 086 914 062 + 0;
  • 488 498 130 839 554 178 805 883 566 383 272 409 439 086 914 062 ÷ 2 = 244 249 065 419 777 089 402 941 783 191 636 204 719 543 457 031 + 0;
  • 244 249 065 419 777 089 402 941 783 191 636 204 719 543 457 031 ÷ 2 = 122 124 532 709 888 544 701 470 891 595 818 102 359 771 728 515 + 1;
  • 122 124 532 709 888 544 701 470 891 595 818 102 359 771 728 515 ÷ 2 = 61 062 266 354 944 272 350 735 445 797 909 051 179 885 864 257 + 1;
  • 61 062 266 354 944 272 350 735 445 797 909 051 179 885 864 257 ÷ 2 = 30 531 133 177 472 136 175 367 722 898 954 525 589 942 932 128 + 1;
  • 30 531 133 177 472 136 175 367 722 898 954 525 589 942 932 128 ÷ 2 = 15 265 566 588 736 068 087 683 861 449 477 262 794 971 466 064 + 0;
  • 15 265 566 588 736 068 087 683 861 449 477 262 794 971 466 064 ÷ 2 = 7 632 783 294 368 034 043 841 930 724 738 631 397 485 733 032 + 0;
  • 7 632 783 294 368 034 043 841 930 724 738 631 397 485 733 032 ÷ 2 = 3 816 391 647 184 017 021 920 965 362 369 315 698 742 866 516 + 0;
  • 3 816 391 647 184 017 021 920 965 362 369 315 698 742 866 516 ÷ 2 = 1 908 195 823 592 008 510 960 482 681 184 657 849 371 433 258 + 0;
  • 1 908 195 823 592 008 510 960 482 681 184 657 849 371 433 258 ÷ 2 = 954 097 911 796 004 255 480 241 340 592 328 924 685 716 629 + 0;
  • 954 097 911 796 004 255 480 241 340 592 328 924 685 716 629 ÷ 2 = 477 048 955 898 002 127 740 120 670 296 164 462 342 858 314 + 1;
  • 477 048 955 898 002 127 740 120 670 296 164 462 342 858 314 ÷ 2 = 238 524 477 949 001 063 870 060 335 148 082 231 171 429 157 + 0;
  • 238 524 477 949 001 063 870 060 335 148 082 231 171 429 157 ÷ 2 = 119 262 238 974 500 531 935 030 167 574 041 115 585 714 578 + 1;
  • 119 262 238 974 500 531 935 030 167 574 041 115 585 714 578 ÷ 2 = 59 631 119 487 250 265 967 515 083 787 020 557 792 857 289 + 0;
  • 59 631 119 487 250 265 967 515 083 787 020 557 792 857 289 ÷ 2 = 29 815 559 743 625 132 983 757 541 893 510 278 896 428 644 + 1;
  • 29 815 559 743 625 132 983 757 541 893 510 278 896 428 644 ÷ 2 = 14 907 779 871 812 566 491 878 770 946 755 139 448 214 322 + 0;
  • 14 907 779 871 812 566 491 878 770 946 755 139 448 214 322 ÷ 2 = 7 453 889 935 906 283 245 939 385 473 377 569 724 107 161 + 0;
  • 7 453 889 935 906 283 245 939 385 473 377 569 724 107 161 ÷ 2 = 3 726 944 967 953 141 622 969 692 736 688 784 862 053 580 + 1;
  • 3 726 944 967 953 141 622 969 692 736 688 784 862 053 580 ÷ 2 = 1 863 472 483 976 570 811 484 846 368 344 392 431 026 790 + 0;
  • 1 863 472 483 976 570 811 484 846 368 344 392 431 026 790 ÷ 2 = 931 736 241 988 285 405 742 423 184 172 196 215 513 395 + 0;
  • 931 736 241 988 285 405 742 423 184 172 196 215 513 395 ÷ 2 = 465 868 120 994 142 702 871 211 592 086 098 107 756 697 + 1;
  • 465 868 120 994 142 702 871 211 592 086 098 107 756 697 ÷ 2 = 232 934 060 497 071 351 435 605 796 043 049 053 878 348 + 1;
  • 232 934 060 497 071 351 435 605 796 043 049 053 878 348 ÷ 2 = 116 467 030 248 535 675 717 802 898 021 524 526 939 174 + 0;
  • 116 467 030 248 535 675 717 802 898 021 524 526 939 174 ÷ 2 = 58 233 515 124 267 837 858 901 449 010 762 263 469 587 + 0;
  • 58 233 515 124 267 837 858 901 449 010 762 263 469 587 ÷ 2 = 29 116 757 562 133 918 929 450 724 505 381 131 734 793 + 1;
  • 29 116 757 562 133 918 929 450 724 505 381 131 734 793 ÷ 2 = 14 558 378 781 066 959 464 725 362 252 690 565 867 396 + 1;
  • 14 558 378 781 066 959 464 725 362 252 690 565 867 396 ÷ 2 = 7 279 189 390 533 479 732 362 681 126 345 282 933 698 + 0;
  • 7 279 189 390 533 479 732 362 681 126 345 282 933 698 ÷ 2 = 3 639 594 695 266 739 866 181 340 563 172 641 466 849 + 0;
  • 3 639 594 695 266 739 866 181 340 563 172 641 466 849 ÷ 2 = 1 819 797 347 633 369 933 090 670 281 586 320 733 424 + 1;
  • 1 819 797 347 633 369 933 090 670 281 586 320 733 424 ÷ 2 = 909 898 673 816 684 966 545 335 140 793 160 366 712 + 0;
  • 909 898 673 816 684 966 545 335 140 793 160 366 712 ÷ 2 = 454 949 336 908 342 483 272 667 570 396 580 183 356 + 0;
  • 454 949 336 908 342 483 272 667 570 396 580 183 356 ÷ 2 = 227 474 668 454 171 241 636 333 785 198 290 091 678 + 0;
  • 227 474 668 454 171 241 636 333 785 198 290 091 678 ÷ 2 = 113 737 334 227 085 620 818 166 892 599 145 045 839 + 0;
  • 113 737 334 227 085 620 818 166 892 599 145 045 839 ÷ 2 = 56 868 667 113 542 810 409 083 446 299 572 522 919 + 1;
  • 56 868 667 113 542 810 409 083 446 299 572 522 919 ÷ 2 = 28 434 333 556 771 405 204 541 723 149 786 261 459 + 1;
  • 28 434 333 556 771 405 204 541 723 149 786 261 459 ÷ 2 = 14 217 166 778 385 702 602 270 861 574 893 130 729 + 1;
  • 14 217 166 778 385 702 602 270 861 574 893 130 729 ÷ 2 = 7 108 583 389 192 851 301 135 430 787 446 565 364 + 1;
  • 7 108 583 389 192 851 301 135 430 787 446 565 364 ÷ 2 = 3 554 291 694 596 425 650 567 715 393 723 282 682 + 0;
  • 3 554 291 694 596 425 650 567 715 393 723 282 682 ÷ 2 = 1 777 145 847 298 212 825 283 857 696 861 641 341 + 0;
  • 1 777 145 847 298 212 825 283 857 696 861 641 341 ÷ 2 = 888 572 923 649 106 412 641 928 848 430 820 670 + 1;
  • 888 572 923 649 106 412 641 928 848 430 820 670 ÷ 2 = 444 286 461 824 553 206 320 964 424 215 410 335 + 0;
  • 444 286 461 824 553 206 320 964 424 215 410 335 ÷ 2 = 222 143 230 912 276 603 160 482 212 107 705 167 + 1;
  • 222 143 230 912 276 603 160 482 212 107 705 167 ÷ 2 = 111 071 615 456 138 301 580 241 106 053 852 583 + 1;
  • 111 071 615 456 138 301 580 241 106 053 852 583 ÷ 2 = 55 535 807 728 069 150 790 120 553 026 926 291 + 1;
  • 55 535 807 728 069 150 790 120 553 026 926 291 ÷ 2 = 27 767 903 864 034 575 395 060 276 513 463 145 + 1;
  • 27 767 903 864 034 575 395 060 276 513 463 145 ÷ 2 = 13 883 951 932 017 287 697 530 138 256 731 572 + 1;
  • 13 883 951 932 017 287 697 530 138 256 731 572 ÷ 2 = 6 941 975 966 008 643 848 765 069 128 365 786 + 0;
  • 6 941 975 966 008 643 848 765 069 128 365 786 ÷ 2 = 3 470 987 983 004 321 924 382 534 564 182 893 + 0;
  • 3 470 987 983 004 321 924 382 534 564 182 893 ÷ 2 = 1 735 493 991 502 160 962 191 267 282 091 446 + 1;
  • 1 735 493 991 502 160 962 191 267 282 091 446 ÷ 2 = 867 746 995 751 080 481 095 633 641 045 723 + 0;
  • 867 746 995 751 080 481 095 633 641 045 723 ÷ 2 = 433 873 497 875 540 240 547 816 820 522 861 + 1;
  • 433 873 497 875 540 240 547 816 820 522 861 ÷ 2 = 216 936 748 937 770 120 273 908 410 261 430 + 1;
  • 216 936 748 937 770 120 273 908 410 261 430 ÷ 2 = 108 468 374 468 885 060 136 954 205 130 715 + 0;
  • 108 468 374 468 885 060 136 954 205 130 715 ÷ 2 = 54 234 187 234 442 530 068 477 102 565 357 + 1;
  • 54 234 187 234 442 530 068 477 102 565 357 ÷ 2 = 27 117 093 617 221 265 034 238 551 282 678 + 1;
  • 27 117 093 617 221 265 034 238 551 282 678 ÷ 2 = 13 558 546 808 610 632 517 119 275 641 339 + 0;
  • 13 558 546 808 610 632 517 119 275 641 339 ÷ 2 = 6 779 273 404 305 316 258 559 637 820 669 + 1;
  • 6 779 273 404 305 316 258 559 637 820 669 ÷ 2 = 3 389 636 702 152 658 129 279 818 910 334 + 1;
  • 3 389 636 702 152 658 129 279 818 910 334 ÷ 2 = 1 694 818 351 076 329 064 639 909 455 167 + 0;
  • 1 694 818 351 076 329 064 639 909 455 167 ÷ 2 = 847 409 175 538 164 532 319 954 727 583 + 1;
  • 847 409 175 538 164 532 319 954 727 583 ÷ 2 = 423 704 587 769 082 266 159 977 363 791 + 1;
  • 423 704 587 769 082 266 159 977 363 791 ÷ 2 = 211 852 293 884 541 133 079 988 681 895 + 1;
  • 211 852 293 884 541 133 079 988 681 895 ÷ 2 = 105 926 146 942 270 566 539 994 340 947 + 1;
  • 105 926 146 942 270 566 539 994 340 947 ÷ 2 = 52 963 073 471 135 283 269 997 170 473 + 1;
  • 52 963 073 471 135 283 269 997 170 473 ÷ 2 = 26 481 536 735 567 641 634 998 585 236 + 1;
  • 26 481 536 735 567 641 634 998 585 236 ÷ 2 = 13 240 768 367 783 820 817 499 292 618 + 0;
  • 13 240 768 367 783 820 817 499 292 618 ÷ 2 = 6 620 384 183 891 910 408 749 646 309 + 0;
  • 6 620 384 183 891 910 408 749 646 309 ÷ 2 = 3 310 192 091 945 955 204 374 823 154 + 1;
  • 3 310 192 091 945 955 204 374 823 154 ÷ 2 = 1 655 096 045 972 977 602 187 411 577 + 0;
  • 1 655 096 045 972 977 602 187 411 577 ÷ 2 = 827 548 022 986 488 801 093 705 788 + 1;
  • 827 548 022 986 488 801 093 705 788 ÷ 2 = 413 774 011 493 244 400 546 852 894 + 0;
  • 413 774 011 493 244 400 546 852 894 ÷ 2 = 206 887 005 746 622 200 273 426 447 + 0;
  • 206 887 005 746 622 200 273 426 447 ÷ 2 = 103 443 502 873 311 100 136 713 223 + 1;
  • 103 443 502 873 311 100 136 713 223 ÷ 2 = 51 721 751 436 655 550 068 356 611 + 1;
  • 51 721 751 436 655 550 068 356 611 ÷ 2 = 25 860 875 718 327 775 034 178 305 + 1;
  • 25 860 875 718 327 775 034 178 305 ÷ 2 = 12 930 437 859 163 887 517 089 152 + 1;
  • 12 930 437 859 163 887 517 089 152 ÷ 2 = 6 465 218 929 581 943 758 544 576 + 0;
  • 6 465 218 929 581 943 758 544 576 ÷ 2 = 3 232 609 464 790 971 879 272 288 + 0;
  • 3 232 609 464 790 971 879 272 288 ÷ 2 = 1 616 304 732 395 485 939 636 144 + 0;
  • 1 616 304 732 395 485 939 636 144 ÷ 2 = 808 152 366 197 742 969 818 072 + 0;
  • 808 152 366 197 742 969 818 072 ÷ 2 = 404 076 183 098 871 484 909 036 + 0;
  • 404 076 183 098 871 484 909 036 ÷ 2 = 202 038 091 549 435 742 454 518 + 0;
  • 202 038 091 549 435 742 454 518 ÷ 2 = 101 019 045 774 717 871 227 259 + 0;
  • 101 019 045 774 717 871 227 259 ÷ 2 = 50 509 522 887 358 935 613 629 + 1;
  • 50 509 522 887 358 935 613 629 ÷ 2 = 25 254 761 443 679 467 806 814 + 1;
  • 25 254 761 443 679 467 806 814 ÷ 2 = 12 627 380 721 839 733 903 407 + 0;
  • 12 627 380 721 839 733 903 407 ÷ 2 = 6 313 690 360 919 866 951 703 + 1;
  • 6 313 690 360 919 866 951 703 ÷ 2 = 3 156 845 180 459 933 475 851 + 1;
  • 3 156 845 180 459 933 475 851 ÷ 2 = 1 578 422 590 229 966 737 925 + 1;
  • 1 578 422 590 229 966 737 925 ÷ 2 = 789 211 295 114 983 368 962 + 1;
  • 789 211 295 114 983 368 962 ÷ 2 = 394 605 647 557 491 684 481 + 0;
  • 394 605 647 557 491 684 481 ÷ 2 = 197 302 823 778 745 842 240 + 1;
  • 197 302 823 778 745 842 240 ÷ 2 = 98 651 411 889 372 921 120 + 0;
  • 98 651 411 889 372 921 120 ÷ 2 = 49 325 705 944 686 460 560 + 0;
  • 49 325 705 944 686 460 560 ÷ 2 = 24 662 852 972 343 230 280 + 0;
  • 24 662 852 972 343 230 280 ÷ 2 = 12 331 426 486 171 615 140 + 0;
  • 12 331 426 486 171 615 140 ÷ 2 = 6 165 713 243 085 807 570 + 0;
  • 6 165 713 243 085 807 570 ÷ 2 = 3 082 856 621 542 903 785 + 0;
  • 3 082 856 621 542 903 785 ÷ 2 = 1 541 428 310 771 451 892 + 1;
  • 1 541 428 310 771 451 892 ÷ 2 = 770 714 155 385 725 946 + 0;
  • 770 714 155 385 725 946 ÷ 2 = 385 357 077 692 862 973 + 0;
  • 385 357 077 692 862 973 ÷ 2 = 192 678 538 846 431 486 + 1;
  • 192 678 538 846 431 486 ÷ 2 = 96 339 269 423 215 743 + 0;
  • 96 339 269 423 215 743 ÷ 2 = 48 169 634 711 607 871 + 1;
  • 48 169 634 711 607 871 ÷ 2 = 24 084 817 355 803 935 + 1;
  • 24 084 817 355 803 935 ÷ 2 = 12 042 408 677 901 967 + 1;
  • 12 042 408 677 901 967 ÷ 2 = 6 021 204 338 950 983 + 1;
  • 6 021 204 338 950 983 ÷ 2 = 3 010 602 169 475 491 + 1;
  • 3 010 602 169 475 491 ÷ 2 = 1 505 301 084 737 745 + 1;
  • 1 505 301 084 737 745 ÷ 2 = 752 650 542 368 872 + 1;
  • 752 650 542 368 872 ÷ 2 = 376 325 271 184 436 + 0;
  • 376 325 271 184 436 ÷ 2 = 188 162 635 592 218 + 0;
  • 188 162 635 592 218 ÷ 2 = 94 081 317 796 109 + 0;
  • 94 081 317 796 109 ÷ 2 = 47 040 658 898 054 + 1;
  • 47 040 658 898 054 ÷ 2 = 23 520 329 449 027 + 0;
  • 23 520 329 449 027 ÷ 2 = 11 760 164 724 513 + 1;
  • 11 760 164 724 513 ÷ 2 = 5 880 082 362 256 + 1;
  • 5 880 082 362 256 ÷ 2 = 2 940 041 181 128 + 0;
  • 2 940 041 181 128 ÷ 2 = 1 470 020 590 564 + 0;
  • 1 470 020 590 564 ÷ 2 = 735 010 295 282 + 0;
  • 735 010 295 282 ÷ 2 = 367 505 147 641 + 0;
  • 367 505 147 641 ÷ 2 = 183 752 573 820 + 1;
  • 183 752 573 820 ÷ 2 = 91 876 286 910 + 0;
  • 91 876 286 910 ÷ 2 = 45 938 143 455 + 0;
  • 45 938 143 455 ÷ 2 = 22 969 071 727 + 1;
  • 22 969 071 727 ÷ 2 = 11 484 535 863 + 1;
  • 11 484 535 863 ÷ 2 = 5 742 267 931 + 1;
  • 5 742 267 931 ÷ 2 = 2 871 133 965 + 1;
  • 2 871 133 965 ÷ 2 = 1 435 566 982 + 1;
  • 1 435 566 982 ÷ 2 = 717 783 491 + 0;
  • 717 783 491 ÷ 2 = 358 891 745 + 1;
  • 358 891 745 ÷ 2 = 179 445 872 + 1;
  • 179 445 872 ÷ 2 = 89 722 936 + 0;
  • 89 722 936 ÷ 2 = 44 861 468 + 0;
  • 44 861 468 ÷ 2 = 22 430 734 + 0;
  • 22 430 734 ÷ 2 = 11 215 367 + 0;
  • 11 215 367 ÷ 2 = 5 607 683 + 1;
  • 5 607 683 ÷ 2 = 2 803 841 + 1;
  • 2 803 841 ÷ 2 = 1 401 920 + 1;
  • 1 401 920 ÷ 2 = 700 960 + 0;
  • 700 960 ÷ 2 = 350 480 + 0;
  • 350 480 ÷ 2 = 175 240 + 0;
  • 175 240 ÷ 2 = 87 620 + 0;
  • 87 620 ÷ 2 = 43 810 + 0;
  • 43 810 ÷ 2 = 21 905 + 0;
  • 21 905 ÷ 2 = 10 952 + 1;
  • 10 952 ÷ 2 = 5 476 + 0;
  • 5 476 ÷ 2 = 2 738 + 0;
  • 2 738 ÷ 2 = 1 369 + 0;
  • 1 369 ÷ 2 = 684 + 1;
  • 684 ÷ 2 = 342 + 0;
  • 342 ÷ 2 = 171 + 0;
  • 171 ÷ 2 = 85 + 1;
  • 85 ÷ 2 = 42 + 1;
  • 42 ÷ 2 = 21 + 0;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 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.


1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998(10) =

10 1010 1100 1000 1000 0001 1100 0011 0111 1100 1000 0110 1000 1111 1110 1001 0000 0010 1111 0110 0000 0011 1100 1010 0111 1110 1101 1011 0100 1111 1010 0111 1000 0100 1100 1100 1001 0101 0000 0111 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 209 positions to the left, so that only one non zero digit remains to the left of it:


1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998(10) =

10 1010 1100 1000 1000 0001 1100 0011 0111 1100 1000 0110 1000 1111 1110 1001 0000 0010 1111 0110 0000 0011 1100 1010 0111 1110 1101 1011 0100 1111 1010 0111 1000 0100 1100 1100 1001 0101 0000 0111 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(2) =

10 1010 1100 1000 1000 0001 1100 0011 0111 1100 1000 0110 1000 1111 1110 1001 0000 0010 1111 0110 0000 0011 1100 1010 0111 1110 1101 1011 0100 1111 1010 0111 1000 0100 1100 1100 1001 0101 0000 0111 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110(2) × 20 =

1.0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111 1111 0100 1000 0001 0111 1011 0000 0001 1110 0101 0011 1111 0110 1101 1010 0111 1101 0011 1100 0010 0110 0110 0100 1010 1000 0011 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0(2) × 2209


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): 209

Mantissa (not normalized):
1.0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111 1111 0100 1000 0001 0111 1011 0000 0001 1110 0101 0011 1111 0110 1101 1010 0111 1101 0011 1100 0010 0110 0110 0100 1010 1000 0011 1001 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 0


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

209 + 2(11-1) - 1 =

(209 + 1 023)(10) =

1 232(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;
  • 1 232 ÷ 2 = 616 + 0;
  • 616 ÷ 2 = 308 + 0;
  • 308 ÷ 2 = 154 + 0;
  • 154 ÷ 2 = 77 + 0;
  • 77 ÷ 2 = 38 + 1;
  • 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) =

1232(10) =

100 1101 0000(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. 0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111 1 1110 1001 0000 0010 1111 0110 0000 0011 1100 1010 0111 1110 1101 1011 0100 1111 1010 0111 1000 0100 1100 1100 1001 0101 0000 0111 0011 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1111 1110 =

0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111


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 1101 0000

Mantissa (52 bits) =
0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111


The base ten decimal number 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1101 0000 - 0101 0110 0100 0100 0000 1110 0001 1011 1110 0100 0011 0100 0111

More operations with decimal numbers converted to 64 bit double precision IEEE 754 binary floating point representation:

» Number 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

» Number 1 100 000 000 010 099 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

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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