64bit IEEE 754: Decimal -> Double Precision Floating Point Binary: 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123(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;
  • 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123 ÷ 2 = 55 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 561 + 1;
  • 55 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 561 ÷ 2 = 27 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 780 + 1;
  • 27 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 780 ÷ 2 = 13 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 890 + 0;
  • 13 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 890 ÷ 2 = 6 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 445 + 0;
  • 6 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 445 ÷ 2 = 3 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 + 1;
  • 3 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 1 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 1 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 868 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 868 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 434 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 434 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 217 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 217 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 108 506 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 108 506 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 54 253 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 54 253 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 27 126 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 27 126 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 13 563 368 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 13 563 368 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 6 781 684 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 6 781 684 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 3 390 842 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 3 390 842 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 1 695 421 006 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 1 695 421 006 944 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 847 710 503 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 847 710 503 472 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 423 855 251 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 423 855 251 736 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 211 927 625 868 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 211 927 625 868 055 555 555 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 105 963 812 934 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 105 963 812 934 027 777 777 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 52 981 906 467 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 52 981 906 467 013 888 888 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 26 490 953 233 506 944 444 444 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 26 490 953 233 506 944 444 444 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 13 245 476 616 753 472 222 222 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 13 245 476 616 753 472 222 222 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 6 622 738 308 376 736 111 111 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 6 622 738 308 376 736 111 111 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 3 311 369 154 188 368 055 555 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 3 311 369 154 188 368 055 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 1 655 684 577 094 184 027 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 1 655 684 577 094 184 027 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 827 842 288 547 092 013 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 827 842 288 547 092 013 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 413 921 144 273 546 006 944 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 413 921 144 273 546 006 944 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 206 960 572 136 773 003 472 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 206 960 572 136 773 003 472 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 103 480 286 068 386 501 736 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 103 480 286 068 386 501 736 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 51 740 143 034 193 250 868 055 555 555 555 555 555 555 555 555 555 555 + 1;
  • 51 740 143 034 193 250 868 055 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 25 870 071 517 096 625 434 027 777 777 777 777 777 777 777 777 777 777 + 1;
  • 25 870 071 517 096 625 434 027 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 12 935 035 758 548 312 717 013 888 888 888 888 888 888 888 888 888 888 + 1;
  • 12 935 035 758 548 312 717 013 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 6 467 517 879 274 156 358 506 944 444 444 444 444 444 444 444 444 444 + 0;
  • 6 467 517 879 274 156 358 506 944 444 444 444 444 444 444 444 444 444 ÷ 2 = 3 233 758 939 637 078 179 253 472 222 222 222 222 222 222 222 222 222 + 0;
  • 3 233 758 939 637 078 179 253 472 222 222 222 222 222 222 222 222 222 ÷ 2 = 1 616 879 469 818 539 089 626 736 111 111 111 111 111 111 111 111 111 + 0;
  • 1 616 879 469 818 539 089 626 736 111 111 111 111 111 111 111 111 111 ÷ 2 = 808 439 734 909 269 544 813 368 055 555 555 555 555 555 555 555 555 + 1;
  • 808 439 734 909 269 544 813 368 055 555 555 555 555 555 555 555 555 ÷ 2 = 404 219 867 454 634 772 406 684 027 777 777 777 777 777 777 777 777 + 1;
  • 404 219 867 454 634 772 406 684 027 777 777 777 777 777 777 777 777 ÷ 2 = 202 109 933 727 317 386 203 342 013 888 888 888 888 888 888 888 888 + 1;
  • 202 109 933 727 317 386 203 342 013 888 888 888 888 888 888 888 888 ÷ 2 = 101 054 966 863 658 693 101 671 006 944 444 444 444 444 444 444 444 + 0;
  • 101 054 966 863 658 693 101 671 006 944 444 444 444 444 444 444 444 ÷ 2 = 50 527 483 431 829 346 550 835 503 472 222 222 222 222 222 222 222 + 0;
  • 50 527 483 431 829 346 550 835 503 472 222 222 222 222 222 222 222 ÷ 2 = 25 263 741 715 914 673 275 417 751 736 111 111 111 111 111 111 111 + 0;
  • 25 263 741 715 914 673 275 417 751 736 111 111 111 111 111 111 111 ÷ 2 = 12 631 870 857 957 336 637 708 875 868 055 555 555 555 555 555 555 + 1;
  • 12 631 870 857 957 336 637 708 875 868 055 555 555 555 555 555 555 ÷ 2 = 6 315 935 428 978 668 318 854 437 934 027 777 777 777 777 777 777 + 1;
  • 6 315 935 428 978 668 318 854 437 934 027 777 777 777 777 777 777 ÷ 2 = 3 157 967 714 489 334 159 427 218 967 013 888 888 888 888 888 888 + 1;
  • 3 157 967 714 489 334 159 427 218 967 013 888 888 888 888 888 888 ÷ 2 = 1 578 983 857 244 667 079 713 609 483 506 944 444 444 444 444 444 + 0;
  • 1 578 983 857 244 667 079 713 609 483 506 944 444 444 444 444 444 ÷ 2 = 789 491 928 622 333 539 856 804 741 753 472 222 222 222 222 222 + 0;
  • 789 491 928 622 333 539 856 804 741 753 472 222 222 222 222 222 ÷ 2 = 394 745 964 311 166 769 928 402 370 876 736 111 111 111 111 111 + 0;
  • 394 745 964 311 166 769 928 402 370 876 736 111 111 111 111 111 ÷ 2 = 197 372 982 155 583 384 964 201 185 438 368 055 555 555 555 555 + 1;
  • 197 372 982 155 583 384 964 201 185 438 368 055 555 555 555 555 ÷ 2 = 98 686 491 077 791 692 482 100 592 719 184 027 777 777 777 777 + 1;
  • 98 686 491 077 791 692 482 100 592 719 184 027 777 777 777 777 ÷ 2 = 49 343 245 538 895 846 241 050 296 359 592 013 888 888 888 888 + 1;
  • 49 343 245 538 895 846 241 050 296 359 592 013 888 888 888 888 ÷ 2 = 24 671 622 769 447 923 120 525 148 179 796 006 944 444 444 444 + 0;
  • 24 671 622 769 447 923 120 525 148 179 796 006 944 444 444 444 ÷ 2 = 12 335 811 384 723 961 560 262 574 089 898 003 472 222 222 222 + 0;
  • 12 335 811 384 723 961 560 262 574 089 898 003 472 222 222 222 ÷ 2 = 6 167 905 692 361 980 780 131 287 044 949 001 736 111 111 111 + 0;
  • 6 167 905 692 361 980 780 131 287 044 949 001 736 111 111 111 ÷ 2 = 3 083 952 846 180 990 390 065 643 522 474 500 868 055 555 555 + 1;
  • 3 083 952 846 180 990 390 065 643 522 474 500 868 055 555 555 ÷ 2 = 1 541 976 423 090 495 195 032 821 761 237 250 434 027 777 777 + 1;
  • 1 541 976 423 090 495 195 032 821 761 237 250 434 027 777 777 ÷ 2 = 770 988 211 545 247 597 516 410 880 618 625 217 013 888 888 + 1;
  • 770 988 211 545 247 597 516 410 880 618 625 217 013 888 888 ÷ 2 = 385 494 105 772 623 798 758 205 440 309 312 608 506 944 444 + 0;
  • 385 494 105 772 623 798 758 205 440 309 312 608 506 944 444 ÷ 2 = 192 747 052 886 311 899 379 102 720 154 656 304 253 472 222 + 0;
  • 192 747 052 886 311 899 379 102 720 154 656 304 253 472 222 ÷ 2 = 96 373 526 443 155 949 689 551 360 077 328 152 126 736 111 + 0;
  • 96 373 526 443 155 949 689 551 360 077 328 152 126 736 111 ÷ 2 = 48 186 763 221 577 974 844 775 680 038 664 076 063 368 055 + 1;
  • 48 186 763 221 577 974 844 775 680 038 664 076 063 368 055 ÷ 2 = 24 093 381 610 788 987 422 387 840 019 332 038 031 684 027 + 1;
  • 24 093 381 610 788 987 422 387 840 019 332 038 031 684 027 ÷ 2 = 12 046 690 805 394 493 711 193 920 009 666 019 015 842 013 + 1;
  • 12 046 690 805 394 493 711 193 920 009 666 019 015 842 013 ÷ 2 = 6 023 345 402 697 246 855 596 960 004 833 009 507 921 006 + 1;
  • 6 023 345 402 697 246 855 596 960 004 833 009 507 921 006 ÷ 2 = 3 011 672 701 348 623 427 798 480 002 416 504 753 960 503 + 0;
  • 3 011 672 701 348 623 427 798 480 002 416 504 753 960 503 ÷ 2 = 1 505 836 350 674 311 713 899 240 001 208 252 376 980 251 + 1;
  • 1 505 836 350 674 311 713 899 240 001 208 252 376 980 251 ÷ 2 = 752 918 175 337 155 856 949 620 000 604 126 188 490 125 + 1;
  • 752 918 175 337 155 856 949 620 000 604 126 188 490 125 ÷ 2 = 376 459 087 668 577 928 474 810 000 302 063 094 245 062 + 1;
  • 376 459 087 668 577 928 474 810 000 302 063 094 245 062 ÷ 2 = 188 229 543 834 288 964 237 405 000 151 031 547 122 531 + 0;
  • 188 229 543 834 288 964 237 405 000 151 031 547 122 531 ÷ 2 = 94 114 771 917 144 482 118 702 500 075 515 773 561 265 + 1;
  • 94 114 771 917 144 482 118 702 500 075 515 773 561 265 ÷ 2 = 47 057 385 958 572 241 059 351 250 037 757 886 780 632 + 1;
  • 47 057 385 958 572 241 059 351 250 037 757 886 780 632 ÷ 2 = 23 528 692 979 286 120 529 675 625 018 878 943 390 316 + 0;
  • 23 528 692 979 286 120 529 675 625 018 878 943 390 316 ÷ 2 = 11 764 346 489 643 060 264 837 812 509 439 471 695 158 + 0;
  • 11 764 346 489 643 060 264 837 812 509 439 471 695 158 ÷ 2 = 5 882 173 244 821 530 132 418 906 254 719 735 847 579 + 0;
  • 5 882 173 244 821 530 132 418 906 254 719 735 847 579 ÷ 2 = 2 941 086 622 410 765 066 209 453 127 359 867 923 789 + 1;
  • 2 941 086 622 410 765 066 209 453 127 359 867 923 789 ÷ 2 = 1 470 543 311 205 382 533 104 726 563 679 933 961 894 + 1;
  • 1 470 543 311 205 382 533 104 726 563 679 933 961 894 ÷ 2 = 735 271 655 602 691 266 552 363 281 839 966 980 947 + 0;
  • 735 271 655 602 691 266 552 363 281 839 966 980 947 ÷ 2 = 367 635 827 801 345 633 276 181 640 919 983 490 473 + 1;
  • 367 635 827 801 345 633 276 181 640 919 983 490 473 ÷ 2 = 183 817 913 900 672 816 638 090 820 459 991 745 236 + 1;
  • 183 817 913 900 672 816 638 090 820 459 991 745 236 ÷ 2 = 91 908 956 950 336 408 319 045 410 229 995 872 618 + 0;
  • 91 908 956 950 336 408 319 045 410 229 995 872 618 ÷ 2 = 45 954 478 475 168 204 159 522 705 114 997 936 309 + 0;
  • 45 954 478 475 168 204 159 522 705 114 997 936 309 ÷ 2 = 22 977 239 237 584 102 079 761 352 557 498 968 154 + 1;
  • 22 977 239 237 584 102 079 761 352 557 498 968 154 ÷ 2 = 11 488 619 618 792 051 039 880 676 278 749 484 077 + 0;
  • 11 488 619 618 792 051 039 880 676 278 749 484 077 ÷ 2 = 5 744 309 809 396 025 519 940 338 139 374 742 038 + 1;
  • 5 744 309 809 396 025 519 940 338 139 374 742 038 ÷ 2 = 2 872 154 904 698 012 759 970 169 069 687 371 019 + 0;
  • 2 872 154 904 698 012 759 970 169 069 687 371 019 ÷ 2 = 1 436 077 452 349 006 379 985 084 534 843 685 509 + 1;
  • 1 436 077 452 349 006 379 985 084 534 843 685 509 ÷ 2 = 718 038 726 174 503 189 992 542 267 421 842 754 + 1;
  • 718 038 726 174 503 189 992 542 267 421 842 754 ÷ 2 = 359 019 363 087 251 594 996 271 133 710 921 377 + 0;
  • 359 019 363 087 251 594 996 271 133 710 921 377 ÷ 2 = 179 509 681 543 625 797 498 135 566 855 460 688 + 1;
  • 179 509 681 543 625 797 498 135 566 855 460 688 ÷ 2 = 89 754 840 771 812 898 749 067 783 427 730 344 + 0;
  • 89 754 840 771 812 898 749 067 783 427 730 344 ÷ 2 = 44 877 420 385 906 449 374 533 891 713 865 172 + 0;
  • 44 877 420 385 906 449 374 533 891 713 865 172 ÷ 2 = 22 438 710 192 953 224 687 266 945 856 932 586 + 0;
  • 22 438 710 192 953 224 687 266 945 856 932 586 ÷ 2 = 11 219 355 096 476 612 343 633 472 928 466 293 + 0;
  • 11 219 355 096 476 612 343 633 472 928 466 293 ÷ 2 = 5 609 677 548 238 306 171 816 736 464 233 146 + 1;
  • 5 609 677 548 238 306 171 816 736 464 233 146 ÷ 2 = 2 804 838 774 119 153 085 908 368 232 116 573 + 0;
  • 2 804 838 774 119 153 085 908 368 232 116 573 ÷ 2 = 1 402 419 387 059 576 542 954 184 116 058 286 + 1;
  • 1 402 419 387 059 576 542 954 184 116 058 286 ÷ 2 = 701 209 693 529 788 271 477 092 058 029 143 + 0;
  • 701 209 693 529 788 271 477 092 058 029 143 ÷ 2 = 350 604 846 764 894 135 738 546 029 014 571 + 1;
  • 350 604 846 764 894 135 738 546 029 014 571 ÷ 2 = 175 302 423 382 447 067 869 273 014 507 285 + 1;
  • 175 302 423 382 447 067 869 273 014 507 285 ÷ 2 = 87 651 211 691 223 533 934 636 507 253 642 + 1;
  • 87 651 211 691 223 533 934 636 507 253 642 ÷ 2 = 43 825 605 845 611 766 967 318 253 626 821 + 0;
  • 43 825 605 845 611 766 967 318 253 626 821 ÷ 2 = 21 912 802 922 805 883 483 659 126 813 410 + 1;
  • 21 912 802 922 805 883 483 659 126 813 410 ÷ 2 = 10 956 401 461 402 941 741 829 563 406 705 + 0;
  • 10 956 401 461 402 941 741 829 563 406 705 ÷ 2 = 5 478 200 730 701 470 870 914 781 703 352 + 1;
  • 5 478 200 730 701 470 870 914 781 703 352 ÷ 2 = 2 739 100 365 350 735 435 457 390 851 676 + 0;
  • 2 739 100 365 350 735 435 457 390 851 676 ÷ 2 = 1 369 550 182 675 367 717 728 695 425 838 + 0;
  • 1 369 550 182 675 367 717 728 695 425 838 ÷ 2 = 684 775 091 337 683 858 864 347 712 919 + 0;
  • 684 775 091 337 683 858 864 347 712 919 ÷ 2 = 342 387 545 668 841 929 432 173 856 459 + 1;
  • 342 387 545 668 841 929 432 173 856 459 ÷ 2 = 171 193 772 834 420 964 716 086 928 229 + 1;
  • 171 193 772 834 420 964 716 086 928 229 ÷ 2 = 85 596 886 417 210 482 358 043 464 114 + 1;
  • 85 596 886 417 210 482 358 043 464 114 ÷ 2 = 42 798 443 208 605 241 179 021 732 057 + 0;
  • 42 798 443 208 605 241 179 021 732 057 ÷ 2 = 21 399 221 604 302 620 589 510 866 028 + 1;
  • 21 399 221 604 302 620 589 510 866 028 ÷ 2 = 10 699 610 802 151 310 294 755 433 014 + 0;
  • 10 699 610 802 151 310 294 755 433 014 ÷ 2 = 5 349 805 401 075 655 147 377 716 507 + 0;
  • 5 349 805 401 075 655 147 377 716 507 ÷ 2 = 2 674 902 700 537 827 573 688 858 253 + 1;
  • 2 674 902 700 537 827 573 688 858 253 ÷ 2 = 1 337 451 350 268 913 786 844 429 126 + 1;
  • 1 337 451 350 268 913 786 844 429 126 ÷ 2 = 668 725 675 134 456 893 422 214 563 + 0;
  • 668 725 675 134 456 893 422 214 563 ÷ 2 = 334 362 837 567 228 446 711 107 281 + 1;
  • 334 362 837 567 228 446 711 107 281 ÷ 2 = 167 181 418 783 614 223 355 553 640 + 1;
  • 167 181 418 783 614 223 355 553 640 ÷ 2 = 83 590 709 391 807 111 677 776 820 + 0;
  • 83 590 709 391 807 111 677 776 820 ÷ 2 = 41 795 354 695 903 555 838 888 410 + 0;
  • 41 795 354 695 903 555 838 888 410 ÷ 2 = 20 897 677 347 951 777 919 444 205 + 0;
  • 20 897 677 347 951 777 919 444 205 ÷ 2 = 10 448 838 673 975 888 959 722 102 + 1;
  • 10 448 838 673 975 888 959 722 102 ÷ 2 = 5 224 419 336 987 944 479 861 051 + 0;
  • 5 224 419 336 987 944 479 861 051 ÷ 2 = 2 612 209 668 493 972 239 930 525 + 1;
  • 2 612 209 668 493 972 239 930 525 ÷ 2 = 1 306 104 834 246 986 119 965 262 + 1;
  • 1 306 104 834 246 986 119 965 262 ÷ 2 = 653 052 417 123 493 059 982 631 + 0;
  • 653 052 417 123 493 059 982 631 ÷ 2 = 326 526 208 561 746 529 991 315 + 1;
  • 326 526 208 561 746 529 991 315 ÷ 2 = 163 263 104 280 873 264 995 657 + 1;
  • 163 263 104 280 873 264 995 657 ÷ 2 = 81 631 552 140 436 632 497 828 + 1;
  • 81 631 552 140 436 632 497 828 ÷ 2 = 40 815 776 070 218 316 248 914 + 0;
  • 40 815 776 070 218 316 248 914 ÷ 2 = 20 407 888 035 109 158 124 457 + 0;
  • 20 407 888 035 109 158 124 457 ÷ 2 = 10 203 944 017 554 579 062 228 + 1;
  • 10 203 944 017 554 579 062 228 ÷ 2 = 5 101 972 008 777 289 531 114 + 0;
  • 5 101 972 008 777 289 531 114 ÷ 2 = 2 550 986 004 388 644 765 557 + 0;
  • 2 550 986 004 388 644 765 557 ÷ 2 = 1 275 493 002 194 322 382 778 + 1;
  • 1 275 493 002 194 322 382 778 ÷ 2 = 637 746 501 097 161 191 389 + 0;
  • 637 746 501 097 161 191 389 ÷ 2 = 318 873 250 548 580 595 694 + 1;
  • 318 873 250 548 580 595 694 ÷ 2 = 159 436 625 274 290 297 847 + 0;
  • 159 436 625 274 290 297 847 ÷ 2 = 79 718 312 637 145 148 923 + 1;
  • 79 718 312 637 145 148 923 ÷ 2 = 39 859 156 318 572 574 461 + 1;
  • 39 859 156 318 572 574 461 ÷ 2 = 19 929 578 159 286 287 230 + 1;
  • 19 929 578 159 286 287 230 ÷ 2 = 9 964 789 079 643 143 615 + 0;
  • 9 964 789 079 643 143 615 ÷ 2 = 4 982 394 539 821 571 807 + 1;
  • 4 982 394 539 821 571 807 ÷ 2 = 2 491 197 269 910 785 903 + 1;
  • 2 491 197 269 910 785 903 ÷ 2 = 1 245 598 634 955 392 951 + 1;
  • 1 245 598 634 955 392 951 ÷ 2 = 622 799 317 477 696 475 + 1;
  • 622 799 317 477 696 475 ÷ 2 = 311 399 658 738 848 237 + 1;
  • 311 399 658 738 848 237 ÷ 2 = 155 699 829 369 424 118 + 1;
  • 155 699 829 369 424 118 ÷ 2 = 77 849 914 684 712 059 + 0;
  • 77 849 914 684 712 059 ÷ 2 = 38 924 957 342 356 029 + 1;
  • 38 924 957 342 356 029 ÷ 2 = 19 462 478 671 178 014 + 1;
  • 19 462 478 671 178 014 ÷ 2 = 9 731 239 335 589 007 + 0;
  • 9 731 239 335 589 007 ÷ 2 = 4 865 619 667 794 503 + 1;
  • 4 865 619 667 794 503 ÷ 2 = 2 432 809 833 897 251 + 1;
  • 2 432 809 833 897 251 ÷ 2 = 1 216 404 916 948 625 + 1;
  • 1 216 404 916 948 625 ÷ 2 = 608 202 458 474 312 + 1;
  • 608 202 458 474 312 ÷ 2 = 304 101 229 237 156 + 0;
  • 304 101 229 237 156 ÷ 2 = 152 050 614 618 578 + 0;
  • 152 050 614 618 578 ÷ 2 = 76 025 307 309 289 + 0;
  • 76 025 307 309 289 ÷ 2 = 38 012 653 654 644 + 1;
  • 38 012 653 654 644 ÷ 2 = 19 006 326 827 322 + 0;
  • 19 006 326 827 322 ÷ 2 = 9 503 163 413 661 + 0;
  • 9 503 163 413 661 ÷ 2 = 4 751 581 706 830 + 1;
  • 4 751 581 706 830 ÷ 2 = 2 375 790 853 415 + 0;
  • 2 375 790 853 415 ÷ 2 = 1 187 895 426 707 + 1;
  • 1 187 895 426 707 ÷ 2 = 593 947 713 353 + 1;
  • 593 947 713 353 ÷ 2 = 296 973 856 676 + 1;
  • 296 973 856 676 ÷ 2 = 148 486 928 338 + 0;
  • 148 486 928 338 ÷ 2 = 74 243 464 169 + 0;
  • 74 243 464 169 ÷ 2 = 37 121 732 084 + 1;
  • 37 121 732 084 ÷ 2 = 18 560 866 042 + 0;
  • 18 560 866 042 ÷ 2 = 9 280 433 021 + 0;
  • 9 280 433 021 ÷ 2 = 4 640 216 510 + 1;
  • 4 640 216 510 ÷ 2 = 2 320 108 255 + 0;
  • 2 320 108 255 ÷ 2 = 1 160 054 127 + 1;
  • 1 160 054 127 ÷ 2 = 580 027 063 + 1;
  • 580 027 063 ÷ 2 = 290 013 531 + 1;
  • 290 013 531 ÷ 2 = 145 006 765 + 1;
  • 145 006 765 ÷ 2 = 72 503 382 + 1;
  • 72 503 382 ÷ 2 = 36 251 691 + 0;
  • 36 251 691 ÷ 2 = 18 125 845 + 1;
  • 18 125 845 ÷ 2 = 9 062 922 + 1;
  • 9 062 922 ÷ 2 = 4 531 461 + 0;
  • 4 531 461 ÷ 2 = 2 265 730 + 1;
  • 2 265 730 ÷ 2 = 1 132 865 + 0;
  • 1 132 865 ÷ 2 = 566 432 + 1;
  • 566 432 ÷ 2 = 283 216 + 0;
  • 283 216 ÷ 2 = 141 608 + 0;
  • 141 608 ÷ 2 = 70 804 + 0;
  • 70 804 ÷ 2 = 35 402 + 0;
  • 35 402 ÷ 2 = 17 701 + 0;
  • 17 701 ÷ 2 = 8 850 + 1;
  • 8 850 ÷ 2 = 4 425 + 0;
  • 4 425 ÷ 2 = 2 212 + 1;
  • 2 212 ÷ 2 = 1 106 + 0;
  • 1 106 ÷ 2 = 553 + 0;
  • 553 ÷ 2 = 276 + 1;
  • 276 ÷ 2 = 138 + 0;
  • 138 ÷ 2 = 69 + 0;
  • 69 ÷ 2 = 34 + 1;
  • 34 ÷ 2 = 17 + 0;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 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.


111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123(10) =


100 0101 0010 0101 0000 0101 0110 1111 1010 0100 1110 1001 0001 1110 1101 1111 1011 1010 1001 0011 1011 0100 0110 1100 1011 1000 1010 1110 1010 0001 0110 1010 0110 1100 0110 1110 1111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1101 0011(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 206 positions to the left, so that only one non zero digit remains to the left of it:


111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123(10) =


100 0101 0010 0101 0000 0101 0110 1111 1010 0100 1110 1001 0001 1110 1101 1111 1011 1010 1001 0011 1011 0100 0110 1100 1011 1000 1010 1110 1010 0001 0110 1010 0110 1100 0110 1110 1111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1101 0011(2) =


100 0101 0010 0101 0000 0101 0110 1111 1010 0100 1110 1001 0001 1110 1101 1111 1011 1010 1001 0011 1011 0100 0110 1100 1011 1000 1010 1110 1010 0001 0110 1010 0110 1100 0110 1110 1111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1101 0011(2) × 20 =


1.0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 0100 0111 1011 0111 1110 1110 1010 0100 1110 1101 0001 1011 0010 1110 0010 1011 1010 1000 0101 1010 1001 1011 0001 1011 1011 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0100 11(2) × 2206


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): 206


Mantissa (not normalized):
1.0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 0100 0111 1011 0111 1110 1110 1010 0100 1110 1101 0001 1011 0010 1110 0010 1011 1010 1000 0101 1010 1001 1011 0001 1011 1011 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0100 11


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


206 + 2(11-1) - 1 =


(206 + 1 023)(10) =


1 229(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;
  • 1 229 ÷ 2 = 614 + 1;
  • 614 ÷ 2 = 307 + 0;
  • 307 ÷ 2 = 153 + 1;
  • 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) =


1229(10) =


100 1100 1101(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. 0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 0100 0111 10 1101 1111 1011 1010 1001 0011 1011 0100 0110 1100 1011 1000 1010 1110 1010 0001 0110 1010 0110 1100 0110 1110 1111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1101 0011 =


0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 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 1100 1101


Mantissa (52 bits) =
0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 0100 0111


The base ten decimal number 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1100 1101 - 0001 0100 1001 0100 0001 0101 1011 1110 1001 0011 1010 0100 0111

(64 bits IEEE 754)
  • Sign (1 bit):

    • 0

      63
  • Exponent (11 bits):

    • 1

      62
    • 0

      61
    • 0

      60
    • 1

      59
    • 1

      58
    • 0

      57
    • 0

      56
    • 1

      55
    • 1

      54
    • 0

      53
    • 1

      52
  • Mantissa (52 bits):

    • 0

      51
    • 0

      50
    • 0

      49
    • 1

      48
    • 0

      47
    • 1

      46
    • 0

      45
    • 0

      44
    • 1

      43
    • 0

      42
    • 0

      41
    • 1

      40
    • 0

      39
    • 1

      38
    • 0

      37
    • 0

      36
    • 0

      35
    • 0

      34
    • 0

      33
    • 1

      32
    • 0

      31
    • 1

      30
    • 0

      29
    • 1

      28
    • 1

      27
    • 0

      26
    • 1

      25
    • 1

      24
    • 1

      23
    • 1

      22
    • 1

      21
    • 0

      20
    • 1

      19
    • 0

      18
    • 0

      17
    • 1

      16
    • 0

      15
    • 0

      14
    • 1

      13
    • 1

      12
    • 1

      11
    • 0

      10
    • 1

      9
    • 0

      8
    • 0

      7
    • 1

      6
    • 0

      5
    • 0

      4
    • 0

      3
    • 1

      2
    • 1

      1
    • 1

      0

The latest decimal numbers converted from base ten to 64 bit double precision IEEE 754 floating point binary standard representation

Number 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 123 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 699 999 999 999 999 999 970 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 135 433 072 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 5 235 080 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 1 011 000 111 111 111 111 111 111 111 111 111 111 111 111 111 048 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 14.34 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 2 629 589 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 12 345 678 994 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:57 UTC (GMT)
Number 76 561 198 335 073 291 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:56 UTC (GMT)
Number 261 274 converted from decimal system (written in base ten) to 64 bit double precision IEEE 754 binary floating point representation standard Feb 27 04:56 UTC (GMT)
All base ten decimal numbers converted to 64 bit double precision IEEE 754 binary floating point

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