Decimal to 64 Bit IEEE 754 Binary: Convert Number 100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116(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;
  • 100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116 ÷ 2 = 50 055 050 505 050 500 000 055 555 500 505 055 055 505 005 555 555 055 505 505 058 + 0;
  • 50 055 050 505 050 500 000 055 555 500 505 055 055 505 005 555 555 055 505 505 058 ÷ 2 = 25 027 525 252 525 250 000 027 777 750 252 527 527 752 502 777 777 527 752 752 529 + 0;
  • 25 027 525 252 525 250 000 027 777 750 252 527 527 752 502 777 777 527 752 752 529 ÷ 2 = 12 513 762 626 262 625 000 013 888 875 126 263 763 876 251 388 888 763 876 376 264 + 1;
  • 12 513 762 626 262 625 000 013 888 875 126 263 763 876 251 388 888 763 876 376 264 ÷ 2 = 6 256 881 313 131 312 500 006 944 437 563 131 881 938 125 694 444 381 938 188 132 + 0;
  • 6 256 881 313 131 312 500 006 944 437 563 131 881 938 125 694 444 381 938 188 132 ÷ 2 = 3 128 440 656 565 656 250 003 472 218 781 565 940 969 062 847 222 190 969 094 066 + 0;
  • 3 128 440 656 565 656 250 003 472 218 781 565 940 969 062 847 222 190 969 094 066 ÷ 2 = 1 564 220 328 282 828 125 001 736 109 390 782 970 484 531 423 611 095 484 547 033 + 0;
  • 1 564 220 328 282 828 125 001 736 109 390 782 970 484 531 423 611 095 484 547 033 ÷ 2 = 782 110 164 141 414 062 500 868 054 695 391 485 242 265 711 805 547 742 273 516 + 1;
  • 782 110 164 141 414 062 500 868 054 695 391 485 242 265 711 805 547 742 273 516 ÷ 2 = 391 055 082 070 707 031 250 434 027 347 695 742 621 132 855 902 773 871 136 758 + 0;
  • 391 055 082 070 707 031 250 434 027 347 695 742 621 132 855 902 773 871 136 758 ÷ 2 = 195 527 541 035 353 515 625 217 013 673 847 871 310 566 427 951 386 935 568 379 + 0;
  • 195 527 541 035 353 515 625 217 013 673 847 871 310 566 427 951 386 935 568 379 ÷ 2 = 97 763 770 517 676 757 812 608 506 836 923 935 655 283 213 975 693 467 784 189 + 1;
  • 97 763 770 517 676 757 812 608 506 836 923 935 655 283 213 975 693 467 784 189 ÷ 2 = 48 881 885 258 838 378 906 304 253 418 461 967 827 641 606 987 846 733 892 094 + 1;
  • 48 881 885 258 838 378 906 304 253 418 461 967 827 641 606 987 846 733 892 094 ÷ 2 = 24 440 942 629 419 189 453 152 126 709 230 983 913 820 803 493 923 366 946 047 + 0;
  • 24 440 942 629 419 189 453 152 126 709 230 983 913 820 803 493 923 366 946 047 ÷ 2 = 12 220 471 314 709 594 726 576 063 354 615 491 956 910 401 746 961 683 473 023 + 1;
  • 12 220 471 314 709 594 726 576 063 354 615 491 956 910 401 746 961 683 473 023 ÷ 2 = 6 110 235 657 354 797 363 288 031 677 307 745 978 455 200 873 480 841 736 511 + 1;
  • 6 110 235 657 354 797 363 288 031 677 307 745 978 455 200 873 480 841 736 511 ÷ 2 = 3 055 117 828 677 398 681 644 015 838 653 872 989 227 600 436 740 420 868 255 + 1;
  • 3 055 117 828 677 398 681 644 015 838 653 872 989 227 600 436 740 420 868 255 ÷ 2 = 1 527 558 914 338 699 340 822 007 919 326 936 494 613 800 218 370 210 434 127 + 1;
  • 1 527 558 914 338 699 340 822 007 919 326 936 494 613 800 218 370 210 434 127 ÷ 2 = 763 779 457 169 349 670 411 003 959 663 468 247 306 900 109 185 105 217 063 + 1;
  • 763 779 457 169 349 670 411 003 959 663 468 247 306 900 109 185 105 217 063 ÷ 2 = 381 889 728 584 674 835 205 501 979 831 734 123 653 450 054 592 552 608 531 + 1;
  • 381 889 728 584 674 835 205 501 979 831 734 123 653 450 054 592 552 608 531 ÷ 2 = 190 944 864 292 337 417 602 750 989 915 867 061 826 725 027 296 276 304 265 + 1;
  • 190 944 864 292 337 417 602 750 989 915 867 061 826 725 027 296 276 304 265 ÷ 2 = 95 472 432 146 168 708 801 375 494 957 933 530 913 362 513 648 138 152 132 + 1;
  • 95 472 432 146 168 708 801 375 494 957 933 530 913 362 513 648 138 152 132 ÷ 2 = 47 736 216 073 084 354 400 687 747 478 966 765 456 681 256 824 069 076 066 + 0;
  • 47 736 216 073 084 354 400 687 747 478 966 765 456 681 256 824 069 076 066 ÷ 2 = 23 868 108 036 542 177 200 343 873 739 483 382 728 340 628 412 034 538 033 + 0;
  • 23 868 108 036 542 177 200 343 873 739 483 382 728 340 628 412 034 538 033 ÷ 2 = 11 934 054 018 271 088 600 171 936 869 741 691 364 170 314 206 017 269 016 + 1;
  • 11 934 054 018 271 088 600 171 936 869 741 691 364 170 314 206 017 269 016 ÷ 2 = 5 967 027 009 135 544 300 085 968 434 870 845 682 085 157 103 008 634 508 + 0;
  • 5 967 027 009 135 544 300 085 968 434 870 845 682 085 157 103 008 634 508 ÷ 2 = 2 983 513 504 567 772 150 042 984 217 435 422 841 042 578 551 504 317 254 + 0;
  • 2 983 513 504 567 772 150 042 984 217 435 422 841 042 578 551 504 317 254 ÷ 2 = 1 491 756 752 283 886 075 021 492 108 717 711 420 521 289 275 752 158 627 + 0;
  • 1 491 756 752 283 886 075 021 492 108 717 711 420 521 289 275 752 158 627 ÷ 2 = 745 878 376 141 943 037 510 746 054 358 855 710 260 644 637 876 079 313 + 1;
  • 745 878 376 141 943 037 510 746 054 358 855 710 260 644 637 876 079 313 ÷ 2 = 372 939 188 070 971 518 755 373 027 179 427 855 130 322 318 938 039 656 + 1;
  • 372 939 188 070 971 518 755 373 027 179 427 855 130 322 318 938 039 656 ÷ 2 = 186 469 594 035 485 759 377 686 513 589 713 927 565 161 159 469 019 828 + 0;
  • 186 469 594 035 485 759 377 686 513 589 713 927 565 161 159 469 019 828 ÷ 2 = 93 234 797 017 742 879 688 843 256 794 856 963 782 580 579 734 509 914 + 0;
  • 93 234 797 017 742 879 688 843 256 794 856 963 782 580 579 734 509 914 ÷ 2 = 46 617 398 508 871 439 844 421 628 397 428 481 891 290 289 867 254 957 + 0;
  • 46 617 398 508 871 439 844 421 628 397 428 481 891 290 289 867 254 957 ÷ 2 = 23 308 699 254 435 719 922 210 814 198 714 240 945 645 144 933 627 478 + 1;
  • 23 308 699 254 435 719 922 210 814 198 714 240 945 645 144 933 627 478 ÷ 2 = 11 654 349 627 217 859 961 105 407 099 357 120 472 822 572 466 813 739 + 0;
  • 11 654 349 627 217 859 961 105 407 099 357 120 472 822 572 466 813 739 ÷ 2 = 5 827 174 813 608 929 980 552 703 549 678 560 236 411 286 233 406 869 + 1;
  • 5 827 174 813 608 929 980 552 703 549 678 560 236 411 286 233 406 869 ÷ 2 = 2 913 587 406 804 464 990 276 351 774 839 280 118 205 643 116 703 434 + 1;
  • 2 913 587 406 804 464 990 276 351 774 839 280 118 205 643 116 703 434 ÷ 2 = 1 456 793 703 402 232 495 138 175 887 419 640 059 102 821 558 351 717 + 0;
  • 1 456 793 703 402 232 495 138 175 887 419 640 059 102 821 558 351 717 ÷ 2 = 728 396 851 701 116 247 569 087 943 709 820 029 551 410 779 175 858 + 1;
  • 728 396 851 701 116 247 569 087 943 709 820 029 551 410 779 175 858 ÷ 2 = 364 198 425 850 558 123 784 543 971 854 910 014 775 705 389 587 929 + 0;
  • 364 198 425 850 558 123 784 543 971 854 910 014 775 705 389 587 929 ÷ 2 = 182 099 212 925 279 061 892 271 985 927 455 007 387 852 694 793 964 + 1;
  • 182 099 212 925 279 061 892 271 985 927 455 007 387 852 694 793 964 ÷ 2 = 91 049 606 462 639 530 946 135 992 963 727 503 693 926 347 396 982 + 0;
  • 91 049 606 462 639 530 946 135 992 963 727 503 693 926 347 396 982 ÷ 2 = 45 524 803 231 319 765 473 067 996 481 863 751 846 963 173 698 491 + 0;
  • 45 524 803 231 319 765 473 067 996 481 863 751 846 963 173 698 491 ÷ 2 = 22 762 401 615 659 882 736 533 998 240 931 875 923 481 586 849 245 + 1;
  • 22 762 401 615 659 882 736 533 998 240 931 875 923 481 586 849 245 ÷ 2 = 11 381 200 807 829 941 368 266 999 120 465 937 961 740 793 424 622 + 1;
  • 11 381 200 807 829 941 368 266 999 120 465 937 961 740 793 424 622 ÷ 2 = 5 690 600 403 914 970 684 133 499 560 232 968 980 870 396 712 311 + 0;
  • 5 690 600 403 914 970 684 133 499 560 232 968 980 870 396 712 311 ÷ 2 = 2 845 300 201 957 485 342 066 749 780 116 484 490 435 198 356 155 + 1;
  • 2 845 300 201 957 485 342 066 749 780 116 484 490 435 198 356 155 ÷ 2 = 1 422 650 100 978 742 671 033 374 890 058 242 245 217 599 178 077 + 1;
  • 1 422 650 100 978 742 671 033 374 890 058 242 245 217 599 178 077 ÷ 2 = 711 325 050 489 371 335 516 687 445 029 121 122 608 799 589 038 + 1;
  • 711 325 050 489 371 335 516 687 445 029 121 122 608 799 589 038 ÷ 2 = 355 662 525 244 685 667 758 343 722 514 560 561 304 399 794 519 + 0;
  • 355 662 525 244 685 667 758 343 722 514 560 561 304 399 794 519 ÷ 2 = 177 831 262 622 342 833 879 171 861 257 280 280 652 199 897 259 + 1;
  • 177 831 262 622 342 833 879 171 861 257 280 280 652 199 897 259 ÷ 2 = 88 915 631 311 171 416 939 585 930 628 640 140 326 099 948 629 + 1;
  • 88 915 631 311 171 416 939 585 930 628 640 140 326 099 948 629 ÷ 2 = 44 457 815 655 585 708 469 792 965 314 320 070 163 049 974 314 + 1;
  • 44 457 815 655 585 708 469 792 965 314 320 070 163 049 974 314 ÷ 2 = 22 228 907 827 792 854 234 896 482 657 160 035 081 524 987 157 + 0;
  • 22 228 907 827 792 854 234 896 482 657 160 035 081 524 987 157 ÷ 2 = 11 114 453 913 896 427 117 448 241 328 580 017 540 762 493 578 + 1;
  • 11 114 453 913 896 427 117 448 241 328 580 017 540 762 493 578 ÷ 2 = 5 557 226 956 948 213 558 724 120 664 290 008 770 381 246 789 + 0;
  • 5 557 226 956 948 213 558 724 120 664 290 008 770 381 246 789 ÷ 2 = 2 778 613 478 474 106 779 362 060 332 145 004 385 190 623 394 + 1;
  • 2 778 613 478 474 106 779 362 060 332 145 004 385 190 623 394 ÷ 2 = 1 389 306 739 237 053 389 681 030 166 072 502 192 595 311 697 + 0;
  • 1 389 306 739 237 053 389 681 030 166 072 502 192 595 311 697 ÷ 2 = 694 653 369 618 526 694 840 515 083 036 251 096 297 655 848 + 1;
  • 694 653 369 618 526 694 840 515 083 036 251 096 297 655 848 ÷ 2 = 347 326 684 809 263 347 420 257 541 518 125 548 148 827 924 + 0;
  • 347 326 684 809 263 347 420 257 541 518 125 548 148 827 924 ÷ 2 = 173 663 342 404 631 673 710 128 770 759 062 774 074 413 962 + 0;
  • 173 663 342 404 631 673 710 128 770 759 062 774 074 413 962 ÷ 2 = 86 831 671 202 315 836 855 064 385 379 531 387 037 206 981 + 0;
  • 86 831 671 202 315 836 855 064 385 379 531 387 037 206 981 ÷ 2 = 43 415 835 601 157 918 427 532 192 689 765 693 518 603 490 + 1;
  • 43 415 835 601 157 918 427 532 192 689 765 693 518 603 490 ÷ 2 = 21 707 917 800 578 959 213 766 096 344 882 846 759 301 745 + 0;
  • 21 707 917 800 578 959 213 766 096 344 882 846 759 301 745 ÷ 2 = 10 853 958 900 289 479 606 883 048 172 441 423 379 650 872 + 1;
  • 10 853 958 900 289 479 606 883 048 172 441 423 379 650 872 ÷ 2 = 5 426 979 450 144 739 803 441 524 086 220 711 689 825 436 + 0;
  • 5 426 979 450 144 739 803 441 524 086 220 711 689 825 436 ÷ 2 = 2 713 489 725 072 369 901 720 762 043 110 355 844 912 718 + 0;
  • 2 713 489 725 072 369 901 720 762 043 110 355 844 912 718 ÷ 2 = 1 356 744 862 536 184 950 860 381 021 555 177 922 456 359 + 0;
  • 1 356 744 862 536 184 950 860 381 021 555 177 922 456 359 ÷ 2 = 678 372 431 268 092 475 430 190 510 777 588 961 228 179 + 1;
  • 678 372 431 268 092 475 430 190 510 777 588 961 228 179 ÷ 2 = 339 186 215 634 046 237 715 095 255 388 794 480 614 089 + 1;
  • 339 186 215 634 046 237 715 095 255 388 794 480 614 089 ÷ 2 = 169 593 107 817 023 118 857 547 627 694 397 240 307 044 + 1;
  • 169 593 107 817 023 118 857 547 627 694 397 240 307 044 ÷ 2 = 84 796 553 908 511 559 428 773 813 847 198 620 153 522 + 0;
  • 84 796 553 908 511 559 428 773 813 847 198 620 153 522 ÷ 2 = 42 398 276 954 255 779 714 386 906 923 599 310 076 761 + 0;
  • 42 398 276 954 255 779 714 386 906 923 599 310 076 761 ÷ 2 = 21 199 138 477 127 889 857 193 453 461 799 655 038 380 + 1;
  • 21 199 138 477 127 889 857 193 453 461 799 655 038 380 ÷ 2 = 10 599 569 238 563 944 928 596 726 730 899 827 519 190 + 0;
  • 10 599 569 238 563 944 928 596 726 730 899 827 519 190 ÷ 2 = 5 299 784 619 281 972 464 298 363 365 449 913 759 595 + 0;
  • 5 299 784 619 281 972 464 298 363 365 449 913 759 595 ÷ 2 = 2 649 892 309 640 986 232 149 181 682 724 956 879 797 + 1;
  • 2 649 892 309 640 986 232 149 181 682 724 956 879 797 ÷ 2 = 1 324 946 154 820 493 116 074 590 841 362 478 439 898 + 1;
  • 1 324 946 154 820 493 116 074 590 841 362 478 439 898 ÷ 2 = 662 473 077 410 246 558 037 295 420 681 239 219 949 + 0;
  • 662 473 077 410 246 558 037 295 420 681 239 219 949 ÷ 2 = 331 236 538 705 123 279 018 647 710 340 619 609 974 + 1;
  • 331 236 538 705 123 279 018 647 710 340 619 609 974 ÷ 2 = 165 618 269 352 561 639 509 323 855 170 309 804 987 + 0;
  • 165 618 269 352 561 639 509 323 855 170 309 804 987 ÷ 2 = 82 809 134 676 280 819 754 661 927 585 154 902 493 + 1;
  • 82 809 134 676 280 819 754 661 927 585 154 902 493 ÷ 2 = 41 404 567 338 140 409 877 330 963 792 577 451 246 + 1;
  • 41 404 567 338 140 409 877 330 963 792 577 451 246 ÷ 2 = 20 702 283 669 070 204 938 665 481 896 288 725 623 + 0;
  • 20 702 283 669 070 204 938 665 481 896 288 725 623 ÷ 2 = 10 351 141 834 535 102 469 332 740 948 144 362 811 + 1;
  • 10 351 141 834 535 102 469 332 740 948 144 362 811 ÷ 2 = 5 175 570 917 267 551 234 666 370 474 072 181 405 + 1;
  • 5 175 570 917 267 551 234 666 370 474 072 181 405 ÷ 2 = 2 587 785 458 633 775 617 333 185 237 036 090 702 + 1;
  • 2 587 785 458 633 775 617 333 185 237 036 090 702 ÷ 2 = 1 293 892 729 316 887 808 666 592 618 518 045 351 + 0;
  • 1 293 892 729 316 887 808 666 592 618 518 045 351 ÷ 2 = 646 946 364 658 443 904 333 296 309 259 022 675 + 1;
  • 646 946 364 658 443 904 333 296 309 259 022 675 ÷ 2 = 323 473 182 329 221 952 166 648 154 629 511 337 + 1;
  • 323 473 182 329 221 952 166 648 154 629 511 337 ÷ 2 = 161 736 591 164 610 976 083 324 077 314 755 668 + 1;
  • 161 736 591 164 610 976 083 324 077 314 755 668 ÷ 2 = 80 868 295 582 305 488 041 662 038 657 377 834 + 0;
  • 80 868 295 582 305 488 041 662 038 657 377 834 ÷ 2 = 40 434 147 791 152 744 020 831 019 328 688 917 + 0;
  • 40 434 147 791 152 744 020 831 019 328 688 917 ÷ 2 = 20 217 073 895 576 372 010 415 509 664 344 458 + 1;
  • 20 217 073 895 576 372 010 415 509 664 344 458 ÷ 2 = 10 108 536 947 788 186 005 207 754 832 172 229 + 0;
  • 10 108 536 947 788 186 005 207 754 832 172 229 ÷ 2 = 5 054 268 473 894 093 002 603 877 416 086 114 + 1;
  • 5 054 268 473 894 093 002 603 877 416 086 114 ÷ 2 = 2 527 134 236 947 046 501 301 938 708 043 057 + 0;
  • 2 527 134 236 947 046 501 301 938 708 043 057 ÷ 2 = 1 263 567 118 473 523 250 650 969 354 021 528 + 1;
  • 1 263 567 118 473 523 250 650 969 354 021 528 ÷ 2 = 631 783 559 236 761 625 325 484 677 010 764 + 0;
  • 631 783 559 236 761 625 325 484 677 010 764 ÷ 2 = 315 891 779 618 380 812 662 742 338 505 382 + 0;
  • 315 891 779 618 380 812 662 742 338 505 382 ÷ 2 = 157 945 889 809 190 406 331 371 169 252 691 + 0;
  • 157 945 889 809 190 406 331 371 169 252 691 ÷ 2 = 78 972 944 904 595 203 165 685 584 626 345 + 1;
  • 78 972 944 904 595 203 165 685 584 626 345 ÷ 2 = 39 486 472 452 297 601 582 842 792 313 172 + 1;
  • 39 486 472 452 297 601 582 842 792 313 172 ÷ 2 = 19 743 236 226 148 800 791 421 396 156 586 + 0;
  • 19 743 236 226 148 800 791 421 396 156 586 ÷ 2 = 9 871 618 113 074 400 395 710 698 078 293 + 0;
  • 9 871 618 113 074 400 395 710 698 078 293 ÷ 2 = 4 935 809 056 537 200 197 855 349 039 146 + 1;
  • 4 935 809 056 537 200 197 855 349 039 146 ÷ 2 = 2 467 904 528 268 600 098 927 674 519 573 + 0;
  • 2 467 904 528 268 600 098 927 674 519 573 ÷ 2 = 1 233 952 264 134 300 049 463 837 259 786 + 1;
  • 1 233 952 264 134 300 049 463 837 259 786 ÷ 2 = 616 976 132 067 150 024 731 918 629 893 + 0;
  • 616 976 132 067 150 024 731 918 629 893 ÷ 2 = 308 488 066 033 575 012 365 959 314 946 + 1;
  • 308 488 066 033 575 012 365 959 314 946 ÷ 2 = 154 244 033 016 787 506 182 979 657 473 + 0;
  • 154 244 033 016 787 506 182 979 657 473 ÷ 2 = 77 122 016 508 393 753 091 489 828 736 + 1;
  • 77 122 016 508 393 753 091 489 828 736 ÷ 2 = 38 561 008 254 196 876 545 744 914 368 + 0;
  • 38 561 008 254 196 876 545 744 914 368 ÷ 2 = 19 280 504 127 098 438 272 872 457 184 + 0;
  • 19 280 504 127 098 438 272 872 457 184 ÷ 2 = 9 640 252 063 549 219 136 436 228 592 + 0;
  • 9 640 252 063 549 219 136 436 228 592 ÷ 2 = 4 820 126 031 774 609 568 218 114 296 + 0;
  • 4 820 126 031 774 609 568 218 114 296 ÷ 2 = 2 410 063 015 887 304 784 109 057 148 + 0;
  • 2 410 063 015 887 304 784 109 057 148 ÷ 2 = 1 205 031 507 943 652 392 054 528 574 + 0;
  • 1 205 031 507 943 652 392 054 528 574 ÷ 2 = 602 515 753 971 826 196 027 264 287 + 0;
  • 602 515 753 971 826 196 027 264 287 ÷ 2 = 301 257 876 985 913 098 013 632 143 + 1;
  • 301 257 876 985 913 098 013 632 143 ÷ 2 = 150 628 938 492 956 549 006 816 071 + 1;
  • 150 628 938 492 956 549 006 816 071 ÷ 2 = 75 314 469 246 478 274 503 408 035 + 1;
  • 75 314 469 246 478 274 503 408 035 ÷ 2 = 37 657 234 623 239 137 251 704 017 + 1;
  • 37 657 234 623 239 137 251 704 017 ÷ 2 = 18 828 617 311 619 568 625 852 008 + 1;
  • 18 828 617 311 619 568 625 852 008 ÷ 2 = 9 414 308 655 809 784 312 926 004 + 0;
  • 9 414 308 655 809 784 312 926 004 ÷ 2 = 4 707 154 327 904 892 156 463 002 + 0;
  • 4 707 154 327 904 892 156 463 002 ÷ 2 = 2 353 577 163 952 446 078 231 501 + 0;
  • 2 353 577 163 952 446 078 231 501 ÷ 2 = 1 176 788 581 976 223 039 115 750 + 1;
  • 1 176 788 581 976 223 039 115 750 ÷ 2 = 588 394 290 988 111 519 557 875 + 0;
  • 588 394 290 988 111 519 557 875 ÷ 2 = 294 197 145 494 055 759 778 937 + 1;
  • 294 197 145 494 055 759 778 937 ÷ 2 = 147 098 572 747 027 879 889 468 + 1;
  • 147 098 572 747 027 879 889 468 ÷ 2 = 73 549 286 373 513 939 944 734 + 0;
  • 73 549 286 373 513 939 944 734 ÷ 2 = 36 774 643 186 756 969 972 367 + 0;
  • 36 774 643 186 756 969 972 367 ÷ 2 = 18 387 321 593 378 484 986 183 + 1;
  • 18 387 321 593 378 484 986 183 ÷ 2 = 9 193 660 796 689 242 493 091 + 1;
  • 9 193 660 796 689 242 493 091 ÷ 2 = 4 596 830 398 344 621 246 545 + 1;
  • 4 596 830 398 344 621 246 545 ÷ 2 = 2 298 415 199 172 310 623 272 + 1;
  • 2 298 415 199 172 310 623 272 ÷ 2 = 1 149 207 599 586 155 311 636 + 0;
  • 1 149 207 599 586 155 311 636 ÷ 2 = 574 603 799 793 077 655 818 + 0;
  • 574 603 799 793 077 655 818 ÷ 2 = 287 301 899 896 538 827 909 + 0;
  • 287 301 899 896 538 827 909 ÷ 2 = 143 650 949 948 269 413 954 + 1;
  • 143 650 949 948 269 413 954 ÷ 2 = 71 825 474 974 134 706 977 + 0;
  • 71 825 474 974 134 706 977 ÷ 2 = 35 912 737 487 067 353 488 + 1;
  • 35 912 737 487 067 353 488 ÷ 2 = 17 956 368 743 533 676 744 + 0;
  • 17 956 368 743 533 676 744 ÷ 2 = 8 978 184 371 766 838 372 + 0;
  • 8 978 184 371 766 838 372 ÷ 2 = 4 489 092 185 883 419 186 + 0;
  • 4 489 092 185 883 419 186 ÷ 2 = 2 244 546 092 941 709 593 + 0;
  • 2 244 546 092 941 709 593 ÷ 2 = 1 122 273 046 470 854 796 + 1;
  • 1 122 273 046 470 854 796 ÷ 2 = 561 136 523 235 427 398 + 0;
  • 561 136 523 235 427 398 ÷ 2 = 280 568 261 617 713 699 + 0;
  • 280 568 261 617 713 699 ÷ 2 = 140 284 130 808 856 849 + 1;
  • 140 284 130 808 856 849 ÷ 2 = 70 142 065 404 428 424 + 1;
  • 70 142 065 404 428 424 ÷ 2 = 35 071 032 702 214 212 + 0;
  • 35 071 032 702 214 212 ÷ 2 = 17 535 516 351 107 106 + 0;
  • 17 535 516 351 107 106 ÷ 2 = 8 767 758 175 553 553 + 0;
  • 8 767 758 175 553 553 ÷ 2 = 4 383 879 087 776 776 + 1;
  • 4 383 879 087 776 776 ÷ 2 = 2 191 939 543 888 388 + 0;
  • 2 191 939 543 888 388 ÷ 2 = 1 095 969 771 944 194 + 0;
  • 1 095 969 771 944 194 ÷ 2 = 547 984 885 972 097 + 0;
  • 547 984 885 972 097 ÷ 2 = 273 992 442 986 048 + 1;
  • 273 992 442 986 048 ÷ 2 = 136 996 221 493 024 + 0;
  • 136 996 221 493 024 ÷ 2 = 68 498 110 746 512 + 0;
  • 68 498 110 746 512 ÷ 2 = 34 249 055 373 256 + 0;
  • 34 249 055 373 256 ÷ 2 = 17 124 527 686 628 + 0;
  • 17 124 527 686 628 ÷ 2 = 8 562 263 843 314 + 0;
  • 8 562 263 843 314 ÷ 2 = 4 281 131 921 657 + 0;
  • 4 281 131 921 657 ÷ 2 = 2 140 565 960 828 + 1;
  • 2 140 565 960 828 ÷ 2 = 1 070 282 980 414 + 0;
  • 1 070 282 980 414 ÷ 2 = 535 141 490 207 + 0;
  • 535 141 490 207 ÷ 2 = 267 570 745 103 + 1;
  • 267 570 745 103 ÷ 2 = 133 785 372 551 + 1;
  • 133 785 372 551 ÷ 2 = 66 892 686 275 + 1;
  • 66 892 686 275 ÷ 2 = 33 446 343 137 + 1;
  • 33 446 343 137 ÷ 2 = 16 723 171 568 + 1;
  • 16 723 171 568 ÷ 2 = 8 361 585 784 + 0;
  • 8 361 585 784 ÷ 2 = 4 180 792 892 + 0;
  • 4 180 792 892 ÷ 2 = 2 090 396 446 + 0;
  • 2 090 396 446 ÷ 2 = 1 045 198 223 + 0;
  • 1 045 198 223 ÷ 2 = 522 599 111 + 1;
  • 522 599 111 ÷ 2 = 261 299 555 + 1;
  • 261 299 555 ÷ 2 = 130 649 777 + 1;
  • 130 649 777 ÷ 2 = 65 324 888 + 1;
  • 65 324 888 ÷ 2 = 32 662 444 + 0;
  • 32 662 444 ÷ 2 = 16 331 222 + 0;
  • 16 331 222 ÷ 2 = 8 165 611 + 0;
  • 8 165 611 ÷ 2 = 4 082 805 + 1;
  • 4 082 805 ÷ 2 = 2 041 402 + 1;
  • 2 041 402 ÷ 2 = 1 020 701 + 0;
  • 1 020 701 ÷ 2 = 510 350 + 1;
  • 510 350 ÷ 2 = 255 175 + 0;
  • 255 175 ÷ 2 = 127 587 + 1;
  • 127 587 ÷ 2 = 63 793 + 1;
  • 63 793 ÷ 2 = 31 896 + 1;
  • 31 896 ÷ 2 = 15 948 + 0;
  • 15 948 ÷ 2 = 7 974 + 0;
  • 7 974 ÷ 2 = 3 987 + 0;
  • 3 987 ÷ 2 = 1 993 + 1;
  • 1 993 ÷ 2 = 996 + 1;
  • 996 ÷ 2 = 498 + 0;
  • 498 ÷ 2 = 249 + 0;
  • 249 ÷ 2 = 124 + 1;
  • 124 ÷ 2 = 62 + 0;
  • 62 ÷ 2 = 31 + 0;
  • 31 ÷ 2 = 15 + 1;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 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.

100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116(10) =


11 1110 0100 1100 0111 0101 1000 1111 0000 1111 1001 0000 0010 0010 0011 0010 0001 0100 0111 1001 1010 0011 1110 0000 0010 1010 1001 1000 1010 1001 1101 1101 1010 1100 1001 1100 0101 0001 0101 0111 0111 0110 0101 0110 1000 1100 0100 1111 1111 0110 0100 0100(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 205 positions to the left, so that only one non zero digit remains to the left of it:


100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116(10) =


11 1110 0100 1100 0111 0101 1000 1111 0000 1111 1001 0000 0010 0010 0011 0010 0001 0100 0111 1001 1010 0011 1110 0000 0010 1010 1001 1000 1010 1001 1101 1101 1010 1100 1001 1100 0101 0001 0101 0111 0111 0110 0101 0110 1000 1100 0100 1111 1111 0110 0100 0100(2) =


11 1110 0100 1100 0111 0101 1000 1111 0000 1111 1001 0000 0010 0010 0011 0010 0001 0100 0111 1001 1010 0011 1110 0000 0010 1010 1001 1000 1010 1001 1101 1101 1010 1100 1001 1100 0101 0001 0101 0111 0111 0110 0101 0110 1000 1100 0100 1111 1111 0110 0100 0100(2) × 20 =


1.1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001 0001 1001 0000 1010 0011 1100 1101 0001 1111 0000 0001 0101 0100 1100 0101 0100 1110 1110 1101 0110 0100 1110 0010 1000 1010 1011 1011 1011 0010 1011 0100 0110 0010 0111 1111 1011 0010 0010 0(2) × 2205


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): 205


Mantissa (not normalized):
1.1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001 0001 1001 0000 1010 0011 1100 1101 0001 1111 0000 0001 0101 0100 1100 0101 0100 1110 1110 1101 0110 0100 1110 0010 1000 1010 1011 1011 1011 0010 1011 0100 0110 0010 0111 1111 1011 0010 0010 0


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


205 + 2(11-1) - 1 =


(205 + 1 023)(10) =


1 228(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;
  • 1 228 ÷ 2 = 614 + 0;
  • 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) =


1228(10) =


100 1100 1100(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. 1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001 0 0011 0010 0001 0100 0111 1001 1010 0011 1110 0000 0010 1010 1001 1000 1010 1001 1101 1101 1010 1100 1001 1100 0101 0001 0101 0111 0111 0110 0101 0110 1000 1100 0100 1111 1111 0110 0100 0100 =


1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001


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 1100


Mantissa (52 bits) =
1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001


The base ten decimal number 100 110 101 010 101 000 000 111 111 001 010 110 111 010 011 111 110 111 011 010 116 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 1100 1100 - 1111 0010 0110 0011 1010 1100 0111 1000 0111 1100 1000 0001 0001

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