64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022(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;
  • 100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022 ÷ 2 = 50 000 500 055 500 555 555 550 500 050 555 000 000 000 000 000 000 000 000 000 011 + 0;
  • 50 000 500 055 500 555 555 550 500 050 555 000 000 000 000 000 000 000 000 000 011 ÷ 2 = 25 000 250 027 750 277 777 775 250 025 277 500 000 000 000 000 000 000 000 000 005 + 1;
  • 25 000 250 027 750 277 777 775 250 025 277 500 000 000 000 000 000 000 000 000 005 ÷ 2 = 12 500 125 013 875 138 888 887 625 012 638 750 000 000 000 000 000 000 000 000 002 + 1;
  • 12 500 125 013 875 138 888 887 625 012 638 750 000 000 000 000 000 000 000 000 002 ÷ 2 = 6 250 062 506 937 569 444 443 812 506 319 375 000 000 000 000 000 000 000 000 001 + 0;
  • 6 250 062 506 937 569 444 443 812 506 319 375 000 000 000 000 000 000 000 000 001 ÷ 2 = 3 125 031 253 468 784 722 221 906 253 159 687 500 000 000 000 000 000 000 000 000 + 1;
  • 3 125 031 253 468 784 722 221 906 253 159 687 500 000 000 000 000 000 000 000 000 ÷ 2 = 1 562 515 626 734 392 361 110 953 126 579 843 750 000 000 000 000 000 000 000 000 + 0;
  • 1 562 515 626 734 392 361 110 953 126 579 843 750 000 000 000 000 000 000 000 000 ÷ 2 = 781 257 813 367 196 180 555 476 563 289 921 875 000 000 000 000 000 000 000 000 + 0;
  • 781 257 813 367 196 180 555 476 563 289 921 875 000 000 000 000 000 000 000 000 ÷ 2 = 390 628 906 683 598 090 277 738 281 644 960 937 500 000 000 000 000 000 000 000 + 0;
  • 390 628 906 683 598 090 277 738 281 644 960 937 500 000 000 000 000 000 000 000 ÷ 2 = 195 314 453 341 799 045 138 869 140 822 480 468 750 000 000 000 000 000 000 000 + 0;
  • 195 314 453 341 799 045 138 869 140 822 480 468 750 000 000 000 000 000 000 000 ÷ 2 = 97 657 226 670 899 522 569 434 570 411 240 234 375 000 000 000 000 000 000 000 + 0;
  • 97 657 226 670 899 522 569 434 570 411 240 234 375 000 000 000 000 000 000 000 ÷ 2 = 48 828 613 335 449 761 284 717 285 205 620 117 187 500 000 000 000 000 000 000 + 0;
  • 48 828 613 335 449 761 284 717 285 205 620 117 187 500 000 000 000 000 000 000 ÷ 2 = 24 414 306 667 724 880 642 358 642 602 810 058 593 750 000 000 000 000 000 000 + 0;
  • 24 414 306 667 724 880 642 358 642 602 810 058 593 750 000 000 000 000 000 000 ÷ 2 = 12 207 153 333 862 440 321 179 321 301 405 029 296 875 000 000 000 000 000 000 + 0;
  • 12 207 153 333 862 440 321 179 321 301 405 029 296 875 000 000 000 000 000 000 ÷ 2 = 6 103 576 666 931 220 160 589 660 650 702 514 648 437 500 000 000 000 000 000 + 0;
  • 6 103 576 666 931 220 160 589 660 650 702 514 648 437 500 000 000 000 000 000 ÷ 2 = 3 051 788 333 465 610 080 294 830 325 351 257 324 218 750 000 000 000 000 000 + 0;
  • 3 051 788 333 465 610 080 294 830 325 351 257 324 218 750 000 000 000 000 000 ÷ 2 = 1 525 894 166 732 805 040 147 415 162 675 628 662 109 375 000 000 000 000 000 + 0;
  • 1 525 894 166 732 805 040 147 415 162 675 628 662 109 375 000 000 000 000 000 ÷ 2 = 762 947 083 366 402 520 073 707 581 337 814 331 054 687 500 000 000 000 000 + 0;
  • 762 947 083 366 402 520 073 707 581 337 814 331 054 687 500 000 000 000 000 ÷ 2 = 381 473 541 683 201 260 036 853 790 668 907 165 527 343 750 000 000 000 000 + 0;
  • 381 473 541 683 201 260 036 853 790 668 907 165 527 343 750 000 000 000 000 ÷ 2 = 190 736 770 841 600 630 018 426 895 334 453 582 763 671 875 000 000 000 000 + 0;
  • 190 736 770 841 600 630 018 426 895 334 453 582 763 671 875 000 000 000 000 ÷ 2 = 95 368 385 420 800 315 009 213 447 667 226 791 381 835 937 500 000 000 000 + 0;
  • 95 368 385 420 800 315 009 213 447 667 226 791 381 835 937 500 000 000 000 ÷ 2 = 47 684 192 710 400 157 504 606 723 833 613 395 690 917 968 750 000 000 000 + 0;
  • 47 684 192 710 400 157 504 606 723 833 613 395 690 917 968 750 000 000 000 ÷ 2 = 23 842 096 355 200 078 752 303 361 916 806 697 845 458 984 375 000 000 000 + 0;
  • 23 842 096 355 200 078 752 303 361 916 806 697 845 458 984 375 000 000 000 ÷ 2 = 11 921 048 177 600 039 376 151 680 958 403 348 922 729 492 187 500 000 000 + 0;
  • 11 921 048 177 600 039 376 151 680 958 403 348 922 729 492 187 500 000 000 ÷ 2 = 5 960 524 088 800 019 688 075 840 479 201 674 461 364 746 093 750 000 000 + 0;
  • 5 960 524 088 800 019 688 075 840 479 201 674 461 364 746 093 750 000 000 ÷ 2 = 2 980 262 044 400 009 844 037 920 239 600 837 230 682 373 046 875 000 000 + 0;
  • 2 980 262 044 400 009 844 037 920 239 600 837 230 682 373 046 875 000 000 ÷ 2 = 1 490 131 022 200 004 922 018 960 119 800 418 615 341 186 523 437 500 000 + 0;
  • 1 490 131 022 200 004 922 018 960 119 800 418 615 341 186 523 437 500 000 ÷ 2 = 745 065 511 100 002 461 009 480 059 900 209 307 670 593 261 718 750 000 + 0;
  • 745 065 511 100 002 461 009 480 059 900 209 307 670 593 261 718 750 000 ÷ 2 = 372 532 755 550 001 230 504 740 029 950 104 653 835 296 630 859 375 000 + 0;
  • 372 532 755 550 001 230 504 740 029 950 104 653 835 296 630 859 375 000 ÷ 2 = 186 266 377 775 000 615 252 370 014 975 052 326 917 648 315 429 687 500 + 0;
  • 186 266 377 775 000 615 252 370 014 975 052 326 917 648 315 429 687 500 ÷ 2 = 93 133 188 887 500 307 626 185 007 487 526 163 458 824 157 714 843 750 + 0;
  • 93 133 188 887 500 307 626 185 007 487 526 163 458 824 157 714 843 750 ÷ 2 = 46 566 594 443 750 153 813 092 503 743 763 081 729 412 078 857 421 875 + 0;
  • 46 566 594 443 750 153 813 092 503 743 763 081 729 412 078 857 421 875 ÷ 2 = 23 283 297 221 875 076 906 546 251 871 881 540 864 706 039 428 710 937 + 1;
  • 23 283 297 221 875 076 906 546 251 871 881 540 864 706 039 428 710 937 ÷ 2 = 11 641 648 610 937 538 453 273 125 935 940 770 432 353 019 714 355 468 + 1;
  • 11 641 648 610 937 538 453 273 125 935 940 770 432 353 019 714 355 468 ÷ 2 = 5 820 824 305 468 769 226 636 562 967 970 385 216 176 509 857 177 734 + 0;
  • 5 820 824 305 468 769 226 636 562 967 970 385 216 176 509 857 177 734 ÷ 2 = 2 910 412 152 734 384 613 318 281 483 985 192 608 088 254 928 588 867 + 0;
  • 2 910 412 152 734 384 613 318 281 483 985 192 608 088 254 928 588 867 ÷ 2 = 1 455 206 076 367 192 306 659 140 741 992 596 304 044 127 464 294 433 + 1;
  • 1 455 206 076 367 192 306 659 140 741 992 596 304 044 127 464 294 433 ÷ 2 = 727 603 038 183 596 153 329 570 370 996 298 152 022 063 732 147 216 + 1;
  • 727 603 038 183 596 153 329 570 370 996 298 152 022 063 732 147 216 ÷ 2 = 363 801 519 091 798 076 664 785 185 498 149 076 011 031 866 073 608 + 0;
  • 363 801 519 091 798 076 664 785 185 498 149 076 011 031 866 073 608 ÷ 2 = 181 900 759 545 899 038 332 392 592 749 074 538 005 515 933 036 804 + 0;
  • 181 900 759 545 899 038 332 392 592 749 074 538 005 515 933 036 804 ÷ 2 = 90 950 379 772 949 519 166 196 296 374 537 269 002 757 966 518 402 + 0;
  • 90 950 379 772 949 519 166 196 296 374 537 269 002 757 966 518 402 ÷ 2 = 45 475 189 886 474 759 583 098 148 187 268 634 501 378 983 259 201 + 0;
  • 45 475 189 886 474 759 583 098 148 187 268 634 501 378 983 259 201 ÷ 2 = 22 737 594 943 237 379 791 549 074 093 634 317 250 689 491 629 600 + 1;
  • 22 737 594 943 237 379 791 549 074 093 634 317 250 689 491 629 600 ÷ 2 = 11 368 797 471 618 689 895 774 537 046 817 158 625 344 745 814 800 + 0;
  • 11 368 797 471 618 689 895 774 537 046 817 158 625 344 745 814 800 ÷ 2 = 5 684 398 735 809 344 947 887 268 523 408 579 312 672 372 907 400 + 0;
  • 5 684 398 735 809 344 947 887 268 523 408 579 312 672 372 907 400 ÷ 2 = 2 842 199 367 904 672 473 943 634 261 704 289 656 336 186 453 700 + 0;
  • 2 842 199 367 904 672 473 943 634 261 704 289 656 336 186 453 700 ÷ 2 = 1 421 099 683 952 336 236 971 817 130 852 144 828 168 093 226 850 + 0;
  • 1 421 099 683 952 336 236 971 817 130 852 144 828 168 093 226 850 ÷ 2 = 710 549 841 976 168 118 485 908 565 426 072 414 084 046 613 425 + 0;
  • 710 549 841 976 168 118 485 908 565 426 072 414 084 046 613 425 ÷ 2 = 355 274 920 988 084 059 242 954 282 713 036 207 042 023 306 712 + 1;
  • 355 274 920 988 084 059 242 954 282 713 036 207 042 023 306 712 ÷ 2 = 177 637 460 494 042 029 621 477 141 356 518 103 521 011 653 356 + 0;
  • 177 637 460 494 042 029 621 477 141 356 518 103 521 011 653 356 ÷ 2 = 88 818 730 247 021 014 810 738 570 678 259 051 760 505 826 678 + 0;
  • 88 818 730 247 021 014 810 738 570 678 259 051 760 505 826 678 ÷ 2 = 44 409 365 123 510 507 405 369 285 339 129 525 880 252 913 339 + 0;
  • 44 409 365 123 510 507 405 369 285 339 129 525 880 252 913 339 ÷ 2 = 22 204 682 561 755 253 702 684 642 669 564 762 940 126 456 669 + 1;
  • 22 204 682 561 755 253 702 684 642 669 564 762 940 126 456 669 ÷ 2 = 11 102 341 280 877 626 851 342 321 334 782 381 470 063 228 334 + 1;
  • 11 102 341 280 877 626 851 342 321 334 782 381 470 063 228 334 ÷ 2 = 5 551 170 640 438 813 425 671 160 667 391 190 735 031 614 167 + 0;
  • 5 551 170 640 438 813 425 671 160 667 391 190 735 031 614 167 ÷ 2 = 2 775 585 320 219 406 712 835 580 333 695 595 367 515 807 083 + 1;
  • 2 775 585 320 219 406 712 835 580 333 695 595 367 515 807 083 ÷ 2 = 1 387 792 660 109 703 356 417 790 166 847 797 683 757 903 541 + 1;
  • 1 387 792 660 109 703 356 417 790 166 847 797 683 757 903 541 ÷ 2 = 693 896 330 054 851 678 208 895 083 423 898 841 878 951 770 + 1;
  • 693 896 330 054 851 678 208 895 083 423 898 841 878 951 770 ÷ 2 = 346 948 165 027 425 839 104 447 541 711 949 420 939 475 885 + 0;
  • 346 948 165 027 425 839 104 447 541 711 949 420 939 475 885 ÷ 2 = 173 474 082 513 712 919 552 223 770 855 974 710 469 737 942 + 1;
  • 173 474 082 513 712 919 552 223 770 855 974 710 469 737 942 ÷ 2 = 86 737 041 256 856 459 776 111 885 427 987 355 234 868 971 + 0;
  • 86 737 041 256 856 459 776 111 885 427 987 355 234 868 971 ÷ 2 = 43 368 520 628 428 229 888 055 942 713 993 677 617 434 485 + 1;
  • 43 368 520 628 428 229 888 055 942 713 993 677 617 434 485 ÷ 2 = 21 684 260 314 214 114 944 027 971 356 996 838 808 717 242 + 1;
  • 21 684 260 314 214 114 944 027 971 356 996 838 808 717 242 ÷ 2 = 10 842 130 157 107 057 472 013 985 678 498 419 404 358 621 + 0;
  • 10 842 130 157 107 057 472 013 985 678 498 419 404 358 621 ÷ 2 = 5 421 065 078 553 528 736 006 992 839 249 209 702 179 310 + 1;
  • 5 421 065 078 553 528 736 006 992 839 249 209 702 179 310 ÷ 2 = 2 710 532 539 276 764 368 003 496 419 624 604 851 089 655 + 0;
  • 2 710 532 539 276 764 368 003 496 419 624 604 851 089 655 ÷ 2 = 1 355 266 269 638 382 184 001 748 209 812 302 425 544 827 + 1;
  • 1 355 266 269 638 382 184 001 748 209 812 302 425 544 827 ÷ 2 = 677 633 134 819 191 092 000 874 104 906 151 212 772 413 + 1;
  • 677 633 134 819 191 092 000 874 104 906 151 212 772 413 ÷ 2 = 338 816 567 409 595 546 000 437 052 453 075 606 386 206 + 1;
  • 338 816 567 409 595 546 000 437 052 453 075 606 386 206 ÷ 2 = 169 408 283 704 797 773 000 218 526 226 537 803 193 103 + 0;
  • 169 408 283 704 797 773 000 218 526 226 537 803 193 103 ÷ 2 = 84 704 141 852 398 886 500 109 263 113 268 901 596 551 + 1;
  • 84 704 141 852 398 886 500 109 263 113 268 901 596 551 ÷ 2 = 42 352 070 926 199 443 250 054 631 556 634 450 798 275 + 1;
  • 42 352 070 926 199 443 250 054 631 556 634 450 798 275 ÷ 2 = 21 176 035 463 099 721 625 027 315 778 317 225 399 137 + 1;
  • 21 176 035 463 099 721 625 027 315 778 317 225 399 137 ÷ 2 = 10 588 017 731 549 860 812 513 657 889 158 612 699 568 + 1;
  • 10 588 017 731 549 860 812 513 657 889 158 612 699 568 ÷ 2 = 5 294 008 865 774 930 406 256 828 944 579 306 349 784 + 0;
  • 5 294 008 865 774 930 406 256 828 944 579 306 349 784 ÷ 2 = 2 647 004 432 887 465 203 128 414 472 289 653 174 892 + 0;
  • 2 647 004 432 887 465 203 128 414 472 289 653 174 892 ÷ 2 = 1 323 502 216 443 732 601 564 207 236 144 826 587 446 + 0;
  • 1 323 502 216 443 732 601 564 207 236 144 826 587 446 ÷ 2 = 661 751 108 221 866 300 782 103 618 072 413 293 723 + 0;
  • 661 751 108 221 866 300 782 103 618 072 413 293 723 ÷ 2 = 330 875 554 110 933 150 391 051 809 036 206 646 861 + 1;
  • 330 875 554 110 933 150 391 051 809 036 206 646 861 ÷ 2 = 165 437 777 055 466 575 195 525 904 518 103 323 430 + 1;
  • 165 437 777 055 466 575 195 525 904 518 103 323 430 ÷ 2 = 82 718 888 527 733 287 597 762 952 259 051 661 715 + 0;
  • 82 718 888 527 733 287 597 762 952 259 051 661 715 ÷ 2 = 41 359 444 263 866 643 798 881 476 129 525 830 857 + 1;
  • 41 359 444 263 866 643 798 881 476 129 525 830 857 ÷ 2 = 20 679 722 131 933 321 899 440 738 064 762 915 428 + 1;
  • 20 679 722 131 933 321 899 440 738 064 762 915 428 ÷ 2 = 10 339 861 065 966 660 949 720 369 032 381 457 714 + 0;
  • 10 339 861 065 966 660 949 720 369 032 381 457 714 ÷ 2 = 5 169 930 532 983 330 474 860 184 516 190 728 857 + 0;
  • 5 169 930 532 983 330 474 860 184 516 190 728 857 ÷ 2 = 2 584 965 266 491 665 237 430 092 258 095 364 428 + 1;
  • 2 584 965 266 491 665 237 430 092 258 095 364 428 ÷ 2 = 1 292 482 633 245 832 618 715 046 129 047 682 214 + 0;
  • 1 292 482 633 245 832 618 715 046 129 047 682 214 ÷ 2 = 646 241 316 622 916 309 357 523 064 523 841 107 + 0;
  • 646 241 316 622 916 309 357 523 064 523 841 107 ÷ 2 = 323 120 658 311 458 154 678 761 532 261 920 553 + 1;
  • 323 120 658 311 458 154 678 761 532 261 920 553 ÷ 2 = 161 560 329 155 729 077 339 380 766 130 960 276 + 1;
  • 161 560 329 155 729 077 339 380 766 130 960 276 ÷ 2 = 80 780 164 577 864 538 669 690 383 065 480 138 + 0;
  • 80 780 164 577 864 538 669 690 383 065 480 138 ÷ 2 = 40 390 082 288 932 269 334 845 191 532 740 069 + 0;
  • 40 390 082 288 932 269 334 845 191 532 740 069 ÷ 2 = 20 195 041 144 466 134 667 422 595 766 370 034 + 1;
  • 20 195 041 144 466 134 667 422 595 766 370 034 ÷ 2 = 10 097 520 572 233 067 333 711 297 883 185 017 + 0;
  • 10 097 520 572 233 067 333 711 297 883 185 017 ÷ 2 = 5 048 760 286 116 533 666 855 648 941 592 508 + 1;
  • 5 048 760 286 116 533 666 855 648 941 592 508 ÷ 2 = 2 524 380 143 058 266 833 427 824 470 796 254 + 0;
  • 2 524 380 143 058 266 833 427 824 470 796 254 ÷ 2 = 1 262 190 071 529 133 416 713 912 235 398 127 + 0;
  • 1 262 190 071 529 133 416 713 912 235 398 127 ÷ 2 = 631 095 035 764 566 708 356 956 117 699 063 + 1;
  • 631 095 035 764 566 708 356 956 117 699 063 ÷ 2 = 315 547 517 882 283 354 178 478 058 849 531 + 1;
  • 315 547 517 882 283 354 178 478 058 849 531 ÷ 2 = 157 773 758 941 141 677 089 239 029 424 765 + 1;
  • 157 773 758 941 141 677 089 239 029 424 765 ÷ 2 = 78 886 879 470 570 838 544 619 514 712 382 + 1;
  • 78 886 879 470 570 838 544 619 514 712 382 ÷ 2 = 39 443 439 735 285 419 272 309 757 356 191 + 0;
  • 39 443 439 735 285 419 272 309 757 356 191 ÷ 2 = 19 721 719 867 642 709 636 154 878 678 095 + 1;
  • 19 721 719 867 642 709 636 154 878 678 095 ÷ 2 = 9 860 859 933 821 354 818 077 439 339 047 + 1;
  • 9 860 859 933 821 354 818 077 439 339 047 ÷ 2 = 4 930 429 966 910 677 409 038 719 669 523 + 1;
  • 4 930 429 966 910 677 409 038 719 669 523 ÷ 2 = 2 465 214 983 455 338 704 519 359 834 761 + 1;
  • 2 465 214 983 455 338 704 519 359 834 761 ÷ 2 = 1 232 607 491 727 669 352 259 679 917 380 + 1;
  • 1 232 607 491 727 669 352 259 679 917 380 ÷ 2 = 616 303 745 863 834 676 129 839 958 690 + 0;
  • 616 303 745 863 834 676 129 839 958 690 ÷ 2 = 308 151 872 931 917 338 064 919 979 345 + 0;
  • 308 151 872 931 917 338 064 919 979 345 ÷ 2 = 154 075 936 465 958 669 032 459 989 672 + 1;
  • 154 075 936 465 958 669 032 459 989 672 ÷ 2 = 77 037 968 232 979 334 516 229 994 836 + 0;
  • 77 037 968 232 979 334 516 229 994 836 ÷ 2 = 38 518 984 116 489 667 258 114 997 418 + 0;
  • 38 518 984 116 489 667 258 114 997 418 ÷ 2 = 19 259 492 058 244 833 629 057 498 709 + 0;
  • 19 259 492 058 244 833 629 057 498 709 ÷ 2 = 9 629 746 029 122 416 814 528 749 354 + 1;
  • 9 629 746 029 122 416 814 528 749 354 ÷ 2 = 4 814 873 014 561 208 407 264 374 677 + 0;
  • 4 814 873 014 561 208 407 264 374 677 ÷ 2 = 2 407 436 507 280 604 203 632 187 338 + 1;
  • 2 407 436 507 280 604 203 632 187 338 ÷ 2 = 1 203 718 253 640 302 101 816 093 669 + 0;
  • 1 203 718 253 640 302 101 816 093 669 ÷ 2 = 601 859 126 820 151 050 908 046 834 + 1;
  • 601 859 126 820 151 050 908 046 834 ÷ 2 = 300 929 563 410 075 525 454 023 417 + 0;
  • 300 929 563 410 075 525 454 023 417 ÷ 2 = 150 464 781 705 037 762 727 011 708 + 1;
  • 150 464 781 705 037 762 727 011 708 ÷ 2 = 75 232 390 852 518 881 363 505 854 + 0;
  • 75 232 390 852 518 881 363 505 854 ÷ 2 = 37 616 195 426 259 440 681 752 927 + 0;
  • 37 616 195 426 259 440 681 752 927 ÷ 2 = 18 808 097 713 129 720 340 876 463 + 1;
  • 18 808 097 713 129 720 340 876 463 ÷ 2 = 9 404 048 856 564 860 170 438 231 + 1;
  • 9 404 048 856 564 860 170 438 231 ÷ 2 = 4 702 024 428 282 430 085 219 115 + 1;
  • 4 702 024 428 282 430 085 219 115 ÷ 2 = 2 351 012 214 141 215 042 609 557 + 1;
  • 2 351 012 214 141 215 042 609 557 ÷ 2 = 1 175 506 107 070 607 521 304 778 + 1;
  • 1 175 506 107 070 607 521 304 778 ÷ 2 = 587 753 053 535 303 760 652 389 + 0;
  • 587 753 053 535 303 760 652 389 ÷ 2 = 293 876 526 767 651 880 326 194 + 1;
  • 293 876 526 767 651 880 326 194 ÷ 2 = 146 938 263 383 825 940 163 097 + 0;
  • 146 938 263 383 825 940 163 097 ÷ 2 = 73 469 131 691 912 970 081 548 + 1;
  • 73 469 131 691 912 970 081 548 ÷ 2 = 36 734 565 845 956 485 040 774 + 0;
  • 36 734 565 845 956 485 040 774 ÷ 2 = 18 367 282 922 978 242 520 387 + 0;
  • 18 367 282 922 978 242 520 387 ÷ 2 = 9 183 641 461 489 121 260 193 + 1;
  • 9 183 641 461 489 121 260 193 ÷ 2 = 4 591 820 730 744 560 630 096 + 1;
  • 4 591 820 730 744 560 630 096 ÷ 2 = 2 295 910 365 372 280 315 048 + 0;
  • 2 295 910 365 372 280 315 048 ÷ 2 = 1 147 955 182 686 140 157 524 + 0;
  • 1 147 955 182 686 140 157 524 ÷ 2 = 573 977 591 343 070 078 762 + 0;
  • 573 977 591 343 070 078 762 ÷ 2 = 286 988 795 671 535 039 381 + 0;
  • 286 988 795 671 535 039 381 ÷ 2 = 143 494 397 835 767 519 690 + 1;
  • 143 494 397 835 767 519 690 ÷ 2 = 71 747 198 917 883 759 845 + 0;
  • 71 747 198 917 883 759 845 ÷ 2 = 35 873 599 458 941 879 922 + 1;
  • 35 873 599 458 941 879 922 ÷ 2 = 17 936 799 729 470 939 961 + 0;
  • 17 936 799 729 470 939 961 ÷ 2 = 8 968 399 864 735 469 980 + 1;
  • 8 968 399 864 735 469 980 ÷ 2 = 4 484 199 932 367 734 990 + 0;
  • 4 484 199 932 367 734 990 ÷ 2 = 2 242 099 966 183 867 495 + 0;
  • 2 242 099 966 183 867 495 ÷ 2 = 1 121 049 983 091 933 747 + 1;
  • 1 121 049 983 091 933 747 ÷ 2 = 560 524 991 545 966 873 + 1;
  • 560 524 991 545 966 873 ÷ 2 = 280 262 495 772 983 436 + 1;
  • 280 262 495 772 983 436 ÷ 2 = 140 131 247 886 491 718 + 0;
  • 140 131 247 886 491 718 ÷ 2 = 70 065 623 943 245 859 + 0;
  • 70 065 623 943 245 859 ÷ 2 = 35 032 811 971 622 929 + 1;
  • 35 032 811 971 622 929 ÷ 2 = 17 516 405 985 811 464 + 1;
  • 17 516 405 985 811 464 ÷ 2 = 8 758 202 992 905 732 + 0;
  • 8 758 202 992 905 732 ÷ 2 = 4 379 101 496 452 866 + 0;
  • 4 379 101 496 452 866 ÷ 2 = 2 189 550 748 226 433 + 0;
  • 2 189 550 748 226 433 ÷ 2 = 1 094 775 374 113 216 + 1;
  • 1 094 775 374 113 216 ÷ 2 = 547 387 687 056 608 + 0;
  • 547 387 687 056 608 ÷ 2 = 273 693 843 528 304 + 0;
  • 273 693 843 528 304 ÷ 2 = 136 846 921 764 152 + 0;
  • 136 846 921 764 152 ÷ 2 = 68 423 460 882 076 + 0;
  • 68 423 460 882 076 ÷ 2 = 34 211 730 441 038 + 0;
  • 34 211 730 441 038 ÷ 2 = 17 105 865 220 519 + 0;
  • 17 105 865 220 519 ÷ 2 = 8 552 932 610 259 + 1;
  • 8 552 932 610 259 ÷ 2 = 4 276 466 305 129 + 1;
  • 4 276 466 305 129 ÷ 2 = 2 138 233 152 564 + 1;
  • 2 138 233 152 564 ÷ 2 = 1 069 116 576 282 + 0;
  • 1 069 116 576 282 ÷ 2 = 534 558 288 141 + 0;
  • 534 558 288 141 ÷ 2 = 267 279 144 070 + 1;
  • 267 279 144 070 ÷ 2 = 133 639 572 035 + 0;
  • 133 639 572 035 ÷ 2 = 66 819 786 017 + 1;
  • 66 819 786 017 ÷ 2 = 33 409 893 008 + 1;
  • 33 409 893 008 ÷ 2 = 16 704 946 504 + 0;
  • 16 704 946 504 ÷ 2 = 8 352 473 252 + 0;
  • 8 352 473 252 ÷ 2 = 4 176 236 626 + 0;
  • 4 176 236 626 ÷ 2 = 2 088 118 313 + 0;
  • 2 088 118 313 ÷ 2 = 1 044 059 156 + 1;
  • 1 044 059 156 ÷ 2 = 522 029 578 + 0;
  • 522 029 578 ÷ 2 = 261 014 789 + 0;
  • 261 014 789 ÷ 2 = 130 507 394 + 1;
  • 130 507 394 ÷ 2 = 65 253 697 + 0;
  • 65 253 697 ÷ 2 = 32 626 848 + 1;
  • 32 626 848 ÷ 2 = 16 313 424 + 0;
  • 16 313 424 ÷ 2 = 8 156 712 + 0;
  • 8 156 712 ÷ 2 = 4 078 356 + 0;
  • 4 078 356 ÷ 2 = 2 039 178 + 0;
  • 2 039 178 ÷ 2 = 1 019 589 + 0;
  • 1 019 589 ÷ 2 = 509 794 + 1;
  • 509 794 ÷ 2 = 254 897 + 0;
  • 254 897 ÷ 2 = 127 448 + 1;
  • 127 448 ÷ 2 = 63 724 + 0;
  • 63 724 ÷ 2 = 31 862 + 0;
  • 31 862 ÷ 2 = 15 931 + 0;
  • 15 931 ÷ 2 = 7 965 + 1;
  • 7 965 ÷ 2 = 3 982 + 1;
  • 3 982 ÷ 2 = 1 991 + 0;
  • 1 991 ÷ 2 = 995 + 1;
  • 995 ÷ 2 = 497 + 1;
  • 497 ÷ 2 = 248 + 1;
  • 248 ÷ 2 = 124 + 0;
  • 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.


100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022(10) =


11 1110 0011 1011 0001 0100 0001 0100 1000 0110 1001 1100 0000 1000 1100 1110 0101 0100 0011 0010 1011 1110 0101 0101 0001 0011 1110 1111 0010 1001 1001 0011 0110 0001 1110 1110 1011 0101 1101 1000 1000 0010 0001 1001 1000 0000 0000 0000 0000 0000 0001 0110(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:


100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022(10) =


11 1110 0011 1011 0001 0100 0001 0100 1000 0110 1001 1100 0000 1000 1100 1110 0101 0100 0011 0010 1011 1110 0101 0101 0001 0011 1110 1111 0010 1001 1001 0011 0110 0001 1110 1110 1011 0101 1101 1000 1000 0010 0001 1001 1000 0000 0000 0000 0000 0000 0001 0110(2) =


11 1110 0011 1011 0001 0100 0001 0100 1000 0110 1001 1100 0000 1000 1100 1110 0101 0100 0011 0010 1011 1110 0101 0101 0001 0011 1110 1111 0010 1001 1001 0011 0110 0001 1110 1110 1011 0101 1101 1000 1000 0010 0001 1001 1000 0000 0000 0000 0000 0000 0001 0110(2) × 20 =


1.1111 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100 0110 0111 0010 1010 0001 1001 0101 1111 0010 1010 1000 1001 1111 0111 1001 0100 1100 1001 1011 0000 1111 0111 0101 1010 1110 1100 0100 0001 0000 1100 1100 0000 0000 0000 0000 0000 0000 1011 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 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100 0110 0111 0010 1010 0001 1001 0101 1111 0010 1010 1000 1001 1111 0111 1001 0100 1100 1001 1011 0000 1111 0111 0101 1010 1110 1100 0100 0001 0000 1100 1100 0000 0000 0000 0000 0000 0000 1011 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 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100 0 1100 1110 0101 0100 0011 0010 1011 1110 0101 0101 0001 0011 1110 1111 0010 1001 1001 0011 0110 0001 1110 1110 1011 0101 1101 1000 1000 0010 0001 1001 1000 0000 0000 0000 0000 0000 0001 0110 =


1111 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100


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 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100


The base ten decimal number 100 001 000 111 001 111 111 101 000 101 110 000 000 000 000 000 000 000 000 000 022 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1100 1100 - 1111 0001 1101 1000 1010 0000 1010 0100 0011 0100 1110 0000 0100

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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