Decimal to 64 Bit IEEE 754 Binary: Convert Number 1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057(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;
  • 1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057 ÷ 2 = 550 000 000 005 505 050 505 000 000 000 000 000 000 000 000 000 000 000 000 000 028 + 1;
  • 550 000 000 005 505 050 505 000 000 000 000 000 000 000 000 000 000 000 000 000 028 ÷ 2 = 275 000 000 002 752 525 252 500 000 000 000 000 000 000 000 000 000 000 000 000 014 + 0;
  • 275 000 000 002 752 525 252 500 000 000 000 000 000 000 000 000 000 000 000 000 014 ÷ 2 = 137 500 000 001 376 262 626 250 000 000 000 000 000 000 000 000 000 000 000 000 007 + 0;
  • 137 500 000 001 376 262 626 250 000 000 000 000 000 000 000 000 000 000 000 000 007 ÷ 2 = 68 750 000 000 688 131 313 125 000 000 000 000 000 000 000 000 000 000 000 000 003 + 1;
  • 68 750 000 000 688 131 313 125 000 000 000 000 000 000 000 000 000 000 000 000 003 ÷ 2 = 34 375 000 000 344 065 656 562 500 000 000 000 000 000 000 000 000 000 000 000 001 + 1;
  • 34 375 000 000 344 065 656 562 500 000 000 000 000 000 000 000 000 000 000 000 001 ÷ 2 = 17 187 500 000 172 032 828 281 250 000 000 000 000 000 000 000 000 000 000 000 000 + 1;
  • 17 187 500 000 172 032 828 281 250 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 8 593 750 000 086 016 414 140 625 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 8 593 750 000 086 016 414 140 625 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 4 296 875 000 043 008 207 070 312 500 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 4 296 875 000 043 008 207 070 312 500 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 2 148 437 500 021 504 103 535 156 250 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 2 148 437 500 021 504 103 535 156 250 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 1 074 218 750 010 752 051 767 578 125 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 1 074 218 750 010 752 051 767 578 125 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 537 109 375 005 376 025 883 789 062 500 000 000 000 000 000 000 000 000 000 000 + 0;
  • 537 109 375 005 376 025 883 789 062 500 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 268 554 687 502 688 012 941 894 531 250 000 000 000 000 000 000 000 000 000 000 + 0;
  • 268 554 687 502 688 012 941 894 531 250 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 134 277 343 751 344 006 470 947 265 625 000 000 000 000 000 000 000 000 000 000 + 0;
  • 134 277 343 751 344 006 470 947 265 625 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 67 138 671 875 672 003 235 473 632 812 500 000 000 000 000 000 000 000 000 000 + 0;
  • 67 138 671 875 672 003 235 473 632 812 500 000 000 000 000 000 000 000 000 000 ÷ 2 = 33 569 335 937 836 001 617 736 816 406 250 000 000 000 000 000 000 000 000 000 + 0;
  • 33 569 335 937 836 001 617 736 816 406 250 000 000 000 000 000 000 000 000 000 ÷ 2 = 16 784 667 968 918 000 808 868 408 203 125 000 000 000 000 000 000 000 000 000 + 0;
  • 16 784 667 968 918 000 808 868 408 203 125 000 000 000 000 000 000 000 000 000 ÷ 2 = 8 392 333 984 459 000 404 434 204 101 562 500 000 000 000 000 000 000 000 000 + 0;
  • 8 392 333 984 459 000 404 434 204 101 562 500 000 000 000 000 000 000 000 000 ÷ 2 = 4 196 166 992 229 500 202 217 102 050 781 250 000 000 000 000 000 000 000 000 + 0;
  • 4 196 166 992 229 500 202 217 102 050 781 250 000 000 000 000 000 000 000 000 ÷ 2 = 2 098 083 496 114 750 101 108 551 025 390 625 000 000 000 000 000 000 000 000 + 0;
  • 2 098 083 496 114 750 101 108 551 025 390 625 000 000 000 000 000 000 000 000 ÷ 2 = 1 049 041 748 057 375 050 554 275 512 695 312 500 000 000 000 000 000 000 000 + 0;
  • 1 049 041 748 057 375 050 554 275 512 695 312 500 000 000 000 000 000 000 000 ÷ 2 = 524 520 874 028 687 525 277 137 756 347 656 250 000 000 000 000 000 000 000 + 0;
  • 524 520 874 028 687 525 277 137 756 347 656 250 000 000 000 000 000 000 000 ÷ 2 = 262 260 437 014 343 762 638 568 878 173 828 125 000 000 000 000 000 000 000 + 0;
  • 262 260 437 014 343 762 638 568 878 173 828 125 000 000 000 000 000 000 000 ÷ 2 = 131 130 218 507 171 881 319 284 439 086 914 062 500 000 000 000 000 000 000 + 0;
  • 131 130 218 507 171 881 319 284 439 086 914 062 500 000 000 000 000 000 000 ÷ 2 = 65 565 109 253 585 940 659 642 219 543 457 031 250 000 000 000 000 000 000 + 0;
  • 65 565 109 253 585 940 659 642 219 543 457 031 250 000 000 000 000 000 000 ÷ 2 = 32 782 554 626 792 970 329 821 109 771 728 515 625 000 000 000 000 000 000 + 0;
  • 32 782 554 626 792 970 329 821 109 771 728 515 625 000 000 000 000 000 000 ÷ 2 = 16 391 277 313 396 485 164 910 554 885 864 257 812 500 000 000 000 000 000 + 0;
  • 16 391 277 313 396 485 164 910 554 885 864 257 812 500 000 000 000 000 000 ÷ 2 = 8 195 638 656 698 242 582 455 277 442 932 128 906 250 000 000 000 000 000 + 0;
  • 8 195 638 656 698 242 582 455 277 442 932 128 906 250 000 000 000 000 000 ÷ 2 = 4 097 819 328 349 121 291 227 638 721 466 064 453 125 000 000 000 000 000 + 0;
  • 4 097 819 328 349 121 291 227 638 721 466 064 453 125 000 000 000 000 000 ÷ 2 = 2 048 909 664 174 560 645 613 819 360 733 032 226 562 500 000 000 000 000 + 0;
  • 2 048 909 664 174 560 645 613 819 360 733 032 226 562 500 000 000 000 000 ÷ 2 = 1 024 454 832 087 280 322 806 909 680 366 516 113 281 250 000 000 000 000 + 0;
  • 1 024 454 832 087 280 322 806 909 680 366 516 113 281 250 000 000 000 000 ÷ 2 = 512 227 416 043 640 161 403 454 840 183 258 056 640 625 000 000 000 000 + 0;
  • 512 227 416 043 640 161 403 454 840 183 258 056 640 625 000 000 000 000 ÷ 2 = 256 113 708 021 820 080 701 727 420 091 629 028 320 312 500 000 000 000 + 0;
  • 256 113 708 021 820 080 701 727 420 091 629 028 320 312 500 000 000 000 ÷ 2 = 128 056 854 010 910 040 350 863 710 045 814 514 160 156 250 000 000 000 + 0;
  • 128 056 854 010 910 040 350 863 710 045 814 514 160 156 250 000 000 000 ÷ 2 = 64 028 427 005 455 020 175 431 855 022 907 257 080 078 125 000 000 000 + 0;
  • 64 028 427 005 455 020 175 431 855 022 907 257 080 078 125 000 000 000 ÷ 2 = 32 014 213 502 727 510 087 715 927 511 453 628 540 039 062 500 000 000 + 0;
  • 32 014 213 502 727 510 087 715 927 511 453 628 540 039 062 500 000 000 ÷ 2 = 16 007 106 751 363 755 043 857 963 755 726 814 270 019 531 250 000 000 + 0;
  • 16 007 106 751 363 755 043 857 963 755 726 814 270 019 531 250 000 000 ÷ 2 = 8 003 553 375 681 877 521 928 981 877 863 407 135 009 765 625 000 000 + 0;
  • 8 003 553 375 681 877 521 928 981 877 863 407 135 009 765 625 000 000 ÷ 2 = 4 001 776 687 840 938 760 964 490 938 931 703 567 504 882 812 500 000 + 0;
  • 4 001 776 687 840 938 760 964 490 938 931 703 567 504 882 812 500 000 ÷ 2 = 2 000 888 343 920 469 380 482 245 469 465 851 783 752 441 406 250 000 + 0;
  • 2 000 888 343 920 469 380 482 245 469 465 851 783 752 441 406 250 000 ÷ 2 = 1 000 444 171 960 234 690 241 122 734 732 925 891 876 220 703 125 000 + 0;
  • 1 000 444 171 960 234 690 241 122 734 732 925 891 876 220 703 125 000 ÷ 2 = 500 222 085 980 117 345 120 561 367 366 462 945 938 110 351 562 500 + 0;
  • 500 222 085 980 117 345 120 561 367 366 462 945 938 110 351 562 500 ÷ 2 = 250 111 042 990 058 672 560 280 683 683 231 472 969 055 175 781 250 + 0;
  • 250 111 042 990 058 672 560 280 683 683 231 472 969 055 175 781 250 ÷ 2 = 125 055 521 495 029 336 280 140 341 841 615 736 484 527 587 890 625 + 0;
  • 125 055 521 495 029 336 280 140 341 841 615 736 484 527 587 890 625 ÷ 2 = 62 527 760 747 514 668 140 070 170 920 807 868 242 263 793 945 312 + 1;
  • 62 527 760 747 514 668 140 070 170 920 807 868 242 263 793 945 312 ÷ 2 = 31 263 880 373 757 334 070 035 085 460 403 934 121 131 896 972 656 + 0;
  • 31 263 880 373 757 334 070 035 085 460 403 934 121 131 896 972 656 ÷ 2 = 15 631 940 186 878 667 035 017 542 730 201 967 060 565 948 486 328 + 0;
  • 15 631 940 186 878 667 035 017 542 730 201 967 060 565 948 486 328 ÷ 2 = 7 815 970 093 439 333 517 508 771 365 100 983 530 282 974 243 164 + 0;
  • 7 815 970 093 439 333 517 508 771 365 100 983 530 282 974 243 164 ÷ 2 = 3 907 985 046 719 666 758 754 385 682 550 491 765 141 487 121 582 + 0;
  • 3 907 985 046 719 666 758 754 385 682 550 491 765 141 487 121 582 ÷ 2 = 1 953 992 523 359 833 379 377 192 841 275 245 882 570 743 560 791 + 0;
  • 1 953 992 523 359 833 379 377 192 841 275 245 882 570 743 560 791 ÷ 2 = 976 996 261 679 916 689 688 596 420 637 622 941 285 371 780 395 + 1;
  • 976 996 261 679 916 689 688 596 420 637 622 941 285 371 780 395 ÷ 2 = 488 498 130 839 958 344 844 298 210 318 811 470 642 685 890 197 + 1;
  • 488 498 130 839 958 344 844 298 210 318 811 470 642 685 890 197 ÷ 2 = 244 249 065 419 979 172 422 149 105 159 405 735 321 342 945 098 + 1;
  • 244 249 065 419 979 172 422 149 105 159 405 735 321 342 945 098 ÷ 2 = 122 124 532 709 989 586 211 074 552 579 702 867 660 671 472 549 + 0;
  • 122 124 532 709 989 586 211 074 552 579 702 867 660 671 472 549 ÷ 2 = 61 062 266 354 994 793 105 537 276 289 851 433 830 335 736 274 + 1;
  • 61 062 266 354 994 793 105 537 276 289 851 433 830 335 736 274 ÷ 2 = 30 531 133 177 497 396 552 768 638 144 925 716 915 167 868 137 + 0;
  • 30 531 133 177 497 396 552 768 638 144 925 716 915 167 868 137 ÷ 2 = 15 265 566 588 748 698 276 384 319 072 462 858 457 583 934 068 + 1;
  • 15 265 566 588 748 698 276 384 319 072 462 858 457 583 934 068 ÷ 2 = 7 632 783 294 374 349 138 192 159 536 231 429 228 791 967 034 + 0;
  • 7 632 783 294 374 349 138 192 159 536 231 429 228 791 967 034 ÷ 2 = 3 816 391 647 187 174 569 096 079 768 115 714 614 395 983 517 + 0;
  • 3 816 391 647 187 174 569 096 079 768 115 714 614 395 983 517 ÷ 2 = 1 908 195 823 593 587 284 548 039 884 057 857 307 197 991 758 + 1;
  • 1 908 195 823 593 587 284 548 039 884 057 857 307 197 991 758 ÷ 2 = 954 097 911 796 793 642 274 019 942 028 928 653 598 995 879 + 0;
  • 954 097 911 796 793 642 274 019 942 028 928 653 598 995 879 ÷ 2 = 477 048 955 898 396 821 137 009 971 014 464 326 799 497 939 + 1;
  • 477 048 955 898 396 821 137 009 971 014 464 326 799 497 939 ÷ 2 = 238 524 477 949 198 410 568 504 985 507 232 163 399 748 969 + 1;
  • 238 524 477 949 198 410 568 504 985 507 232 163 399 748 969 ÷ 2 = 119 262 238 974 599 205 284 252 492 753 616 081 699 874 484 + 1;
  • 119 262 238 974 599 205 284 252 492 753 616 081 699 874 484 ÷ 2 = 59 631 119 487 299 602 642 126 246 376 808 040 849 937 242 + 0;
  • 59 631 119 487 299 602 642 126 246 376 808 040 849 937 242 ÷ 2 = 29 815 559 743 649 801 321 063 123 188 404 020 424 968 621 + 0;
  • 29 815 559 743 649 801 321 063 123 188 404 020 424 968 621 ÷ 2 = 14 907 779 871 824 900 660 531 561 594 202 010 212 484 310 + 1;
  • 14 907 779 871 824 900 660 531 561 594 202 010 212 484 310 ÷ 2 = 7 453 889 935 912 450 330 265 780 797 101 005 106 242 155 + 0;
  • 7 453 889 935 912 450 330 265 780 797 101 005 106 242 155 ÷ 2 = 3 726 944 967 956 225 165 132 890 398 550 502 553 121 077 + 1;
  • 3 726 944 967 956 225 165 132 890 398 550 502 553 121 077 ÷ 2 = 1 863 472 483 978 112 582 566 445 199 275 251 276 560 538 + 1;
  • 1 863 472 483 978 112 582 566 445 199 275 251 276 560 538 ÷ 2 = 931 736 241 989 056 291 283 222 599 637 625 638 280 269 + 0;
  • 931 736 241 989 056 291 283 222 599 637 625 638 280 269 ÷ 2 = 465 868 120 994 528 145 641 611 299 818 812 819 140 134 + 1;
  • 465 868 120 994 528 145 641 611 299 818 812 819 140 134 ÷ 2 = 232 934 060 497 264 072 820 805 649 909 406 409 570 067 + 0;
  • 232 934 060 497 264 072 820 805 649 909 406 409 570 067 ÷ 2 = 116 467 030 248 632 036 410 402 824 954 703 204 785 033 + 1;
  • 116 467 030 248 632 036 410 402 824 954 703 204 785 033 ÷ 2 = 58 233 515 124 316 018 205 201 412 477 351 602 392 516 + 1;
  • 58 233 515 124 316 018 205 201 412 477 351 602 392 516 ÷ 2 = 29 116 757 562 158 009 102 600 706 238 675 801 196 258 + 0;
  • 29 116 757 562 158 009 102 600 706 238 675 801 196 258 ÷ 2 = 14 558 378 781 079 004 551 300 353 119 337 900 598 129 + 0;
  • 14 558 378 781 079 004 551 300 353 119 337 900 598 129 ÷ 2 = 7 279 189 390 539 502 275 650 176 559 668 950 299 064 + 1;
  • 7 279 189 390 539 502 275 650 176 559 668 950 299 064 ÷ 2 = 3 639 594 695 269 751 137 825 088 279 834 475 149 532 + 0;
  • 3 639 594 695 269 751 137 825 088 279 834 475 149 532 ÷ 2 = 1 819 797 347 634 875 568 912 544 139 917 237 574 766 + 0;
  • 1 819 797 347 634 875 568 912 544 139 917 237 574 766 ÷ 2 = 909 898 673 817 437 784 456 272 069 958 618 787 383 + 0;
  • 909 898 673 817 437 784 456 272 069 958 618 787 383 ÷ 2 = 454 949 336 908 718 892 228 136 034 979 309 393 691 + 1;
  • 454 949 336 908 718 892 228 136 034 979 309 393 691 ÷ 2 = 227 474 668 454 359 446 114 068 017 489 654 696 845 + 1;
  • 227 474 668 454 359 446 114 068 017 489 654 696 845 ÷ 2 = 113 737 334 227 179 723 057 034 008 744 827 348 422 + 1;
  • 113 737 334 227 179 723 057 034 008 744 827 348 422 ÷ 2 = 56 868 667 113 589 861 528 517 004 372 413 674 211 + 0;
  • 56 868 667 113 589 861 528 517 004 372 413 674 211 ÷ 2 = 28 434 333 556 794 930 764 258 502 186 206 837 105 + 1;
  • 28 434 333 556 794 930 764 258 502 186 206 837 105 ÷ 2 = 14 217 166 778 397 465 382 129 251 093 103 418 552 + 1;
  • 14 217 166 778 397 465 382 129 251 093 103 418 552 ÷ 2 = 7 108 583 389 198 732 691 064 625 546 551 709 276 + 0;
  • 7 108 583 389 198 732 691 064 625 546 551 709 276 ÷ 2 = 3 554 291 694 599 366 345 532 312 773 275 854 638 + 0;
  • 3 554 291 694 599 366 345 532 312 773 275 854 638 ÷ 2 = 1 777 145 847 299 683 172 766 156 386 637 927 319 + 0;
  • 1 777 145 847 299 683 172 766 156 386 637 927 319 ÷ 2 = 888 572 923 649 841 586 383 078 193 318 963 659 + 1;
  • 888 572 923 649 841 586 383 078 193 318 963 659 ÷ 2 = 444 286 461 824 920 793 191 539 096 659 481 829 + 1;
  • 444 286 461 824 920 793 191 539 096 659 481 829 ÷ 2 = 222 143 230 912 460 396 595 769 548 329 740 914 + 1;
  • 222 143 230 912 460 396 595 769 548 329 740 914 ÷ 2 = 111 071 615 456 230 198 297 884 774 164 870 457 + 0;
  • 111 071 615 456 230 198 297 884 774 164 870 457 ÷ 2 = 55 535 807 728 115 099 148 942 387 082 435 228 + 1;
  • 55 535 807 728 115 099 148 942 387 082 435 228 ÷ 2 = 27 767 903 864 057 549 574 471 193 541 217 614 + 0;
  • 27 767 903 864 057 549 574 471 193 541 217 614 ÷ 2 = 13 883 951 932 028 774 787 235 596 770 608 807 + 0;
  • 13 883 951 932 028 774 787 235 596 770 608 807 ÷ 2 = 6 941 975 966 014 387 393 617 798 385 304 403 + 1;
  • 6 941 975 966 014 387 393 617 798 385 304 403 ÷ 2 = 3 470 987 983 007 193 696 808 899 192 652 201 + 1;
  • 3 470 987 983 007 193 696 808 899 192 652 201 ÷ 2 = 1 735 493 991 503 596 848 404 449 596 326 100 + 1;
  • 1 735 493 991 503 596 848 404 449 596 326 100 ÷ 2 = 867 746 995 751 798 424 202 224 798 163 050 + 0;
  • 867 746 995 751 798 424 202 224 798 163 050 ÷ 2 = 433 873 497 875 899 212 101 112 399 081 525 + 0;
  • 433 873 497 875 899 212 101 112 399 081 525 ÷ 2 = 216 936 748 937 949 606 050 556 199 540 762 + 1;
  • 216 936 748 937 949 606 050 556 199 540 762 ÷ 2 = 108 468 374 468 974 803 025 278 099 770 381 + 0;
  • 108 468 374 468 974 803 025 278 099 770 381 ÷ 2 = 54 234 187 234 487 401 512 639 049 885 190 + 1;
  • 54 234 187 234 487 401 512 639 049 885 190 ÷ 2 = 27 117 093 617 243 700 756 319 524 942 595 + 0;
  • 27 117 093 617 243 700 756 319 524 942 595 ÷ 2 = 13 558 546 808 621 850 378 159 762 471 297 + 1;
  • 13 558 546 808 621 850 378 159 762 471 297 ÷ 2 = 6 779 273 404 310 925 189 079 881 235 648 + 1;
  • 6 779 273 404 310 925 189 079 881 235 648 ÷ 2 = 3 389 636 702 155 462 594 539 940 617 824 + 0;
  • 3 389 636 702 155 462 594 539 940 617 824 ÷ 2 = 1 694 818 351 077 731 297 269 970 308 912 + 0;
  • 1 694 818 351 077 731 297 269 970 308 912 ÷ 2 = 847 409 175 538 865 648 634 985 154 456 + 0;
  • 847 409 175 538 865 648 634 985 154 456 ÷ 2 = 423 704 587 769 432 824 317 492 577 228 + 0;
  • 423 704 587 769 432 824 317 492 577 228 ÷ 2 = 211 852 293 884 716 412 158 746 288 614 + 0;
  • 211 852 293 884 716 412 158 746 288 614 ÷ 2 = 105 926 146 942 358 206 079 373 144 307 + 0;
  • 105 926 146 942 358 206 079 373 144 307 ÷ 2 = 52 963 073 471 179 103 039 686 572 153 + 1;
  • 52 963 073 471 179 103 039 686 572 153 ÷ 2 = 26 481 536 735 589 551 519 843 286 076 + 1;
  • 26 481 536 735 589 551 519 843 286 076 ÷ 2 = 13 240 768 367 794 775 759 921 643 038 + 0;
  • 13 240 768 367 794 775 759 921 643 038 ÷ 2 = 6 620 384 183 897 387 879 960 821 519 + 0;
  • 6 620 384 183 897 387 879 960 821 519 ÷ 2 = 3 310 192 091 948 693 939 980 410 759 + 1;
  • 3 310 192 091 948 693 939 980 410 759 ÷ 2 = 1 655 096 045 974 346 969 990 205 379 + 1;
  • 1 655 096 045 974 346 969 990 205 379 ÷ 2 = 827 548 022 987 173 484 995 102 689 + 1;
  • 827 548 022 987 173 484 995 102 689 ÷ 2 = 413 774 011 493 586 742 497 551 344 + 1;
  • 413 774 011 493 586 742 497 551 344 ÷ 2 = 206 887 005 746 793 371 248 775 672 + 0;
  • 206 887 005 746 793 371 248 775 672 ÷ 2 = 103 443 502 873 396 685 624 387 836 + 0;
  • 103 443 502 873 396 685 624 387 836 ÷ 2 = 51 721 751 436 698 342 812 193 918 + 0;
  • 51 721 751 436 698 342 812 193 918 ÷ 2 = 25 860 875 718 349 171 406 096 959 + 0;
  • 25 860 875 718 349 171 406 096 959 ÷ 2 = 12 930 437 859 174 585 703 048 479 + 1;
  • 12 930 437 859 174 585 703 048 479 ÷ 2 = 6 465 218 929 587 292 851 524 239 + 1;
  • 6 465 218 929 587 292 851 524 239 ÷ 2 = 3 232 609 464 793 646 425 762 119 + 1;
  • 3 232 609 464 793 646 425 762 119 ÷ 2 = 1 616 304 732 396 823 212 881 059 + 1;
  • 1 616 304 732 396 823 212 881 059 ÷ 2 = 808 152 366 198 411 606 440 529 + 1;
  • 808 152 366 198 411 606 440 529 ÷ 2 = 404 076 183 099 205 803 220 264 + 1;
  • 404 076 183 099 205 803 220 264 ÷ 2 = 202 038 091 549 602 901 610 132 + 0;
  • 202 038 091 549 602 901 610 132 ÷ 2 = 101 019 045 774 801 450 805 066 + 0;
  • 101 019 045 774 801 450 805 066 ÷ 2 = 50 509 522 887 400 725 402 533 + 0;
  • 50 509 522 887 400 725 402 533 ÷ 2 = 25 254 761 443 700 362 701 266 + 1;
  • 25 254 761 443 700 362 701 266 ÷ 2 = 12 627 380 721 850 181 350 633 + 0;
  • 12 627 380 721 850 181 350 633 ÷ 2 = 6 313 690 360 925 090 675 316 + 1;
  • 6 313 690 360 925 090 675 316 ÷ 2 = 3 156 845 180 462 545 337 658 + 0;
  • 3 156 845 180 462 545 337 658 ÷ 2 = 1 578 422 590 231 272 668 829 + 0;
  • 1 578 422 590 231 272 668 829 ÷ 2 = 789 211 295 115 636 334 414 + 1;
  • 789 211 295 115 636 334 414 ÷ 2 = 394 605 647 557 818 167 207 + 0;
  • 394 605 647 557 818 167 207 ÷ 2 = 197 302 823 778 909 083 603 + 1;
  • 197 302 823 778 909 083 603 ÷ 2 = 98 651 411 889 454 541 801 + 1;
  • 98 651 411 889 454 541 801 ÷ 2 = 49 325 705 944 727 270 900 + 1;
  • 49 325 705 944 727 270 900 ÷ 2 = 24 662 852 972 363 635 450 + 0;
  • 24 662 852 972 363 635 450 ÷ 2 = 12 331 426 486 181 817 725 + 0;
  • 12 331 426 486 181 817 725 ÷ 2 = 6 165 713 243 090 908 862 + 1;
  • 6 165 713 243 090 908 862 ÷ 2 = 3 082 856 621 545 454 431 + 0;
  • 3 082 856 621 545 454 431 ÷ 2 = 1 541 428 310 772 727 215 + 1;
  • 1 541 428 310 772 727 215 ÷ 2 = 770 714 155 386 363 607 + 1;
  • 770 714 155 386 363 607 ÷ 2 = 385 357 077 693 181 803 + 1;
  • 385 357 077 693 181 803 ÷ 2 = 192 678 538 846 590 901 + 1;
  • 192 678 538 846 590 901 ÷ 2 = 96 339 269 423 295 450 + 1;
  • 96 339 269 423 295 450 ÷ 2 = 48 169 634 711 647 725 + 0;
  • 48 169 634 711 647 725 ÷ 2 = 24 084 817 355 823 862 + 1;
  • 24 084 817 355 823 862 ÷ 2 = 12 042 408 677 911 931 + 0;
  • 12 042 408 677 911 931 ÷ 2 = 6 021 204 338 955 965 + 1;
  • 6 021 204 338 955 965 ÷ 2 = 3 010 602 169 477 982 + 1;
  • 3 010 602 169 477 982 ÷ 2 = 1 505 301 084 738 991 + 0;
  • 1 505 301 084 738 991 ÷ 2 = 752 650 542 369 495 + 1;
  • 752 650 542 369 495 ÷ 2 = 376 325 271 184 747 + 1;
  • 376 325 271 184 747 ÷ 2 = 188 162 635 592 373 + 1;
  • 188 162 635 592 373 ÷ 2 = 94 081 317 796 186 + 1;
  • 94 081 317 796 186 ÷ 2 = 47 040 658 898 093 + 0;
  • 47 040 658 898 093 ÷ 2 = 23 520 329 449 046 + 1;
  • 23 520 329 449 046 ÷ 2 = 11 760 164 724 523 + 0;
  • 11 760 164 724 523 ÷ 2 = 5 880 082 362 261 + 1;
  • 5 880 082 362 261 ÷ 2 = 2 940 041 181 130 + 1;
  • 2 940 041 181 130 ÷ 2 = 1 470 020 590 565 + 0;
  • 1 470 020 590 565 ÷ 2 = 735 010 295 282 + 1;
  • 735 010 295 282 ÷ 2 = 367 505 147 641 + 0;
  • 367 505 147 641 ÷ 2 = 183 752 573 820 + 1;
  • 183 752 573 820 ÷ 2 = 91 876 286 910 + 0;
  • 91 876 286 910 ÷ 2 = 45 938 143 455 + 0;
  • 45 938 143 455 ÷ 2 = 22 969 071 727 + 1;
  • 22 969 071 727 ÷ 2 = 11 484 535 863 + 1;
  • 11 484 535 863 ÷ 2 = 5 742 267 931 + 1;
  • 5 742 267 931 ÷ 2 = 2 871 133 965 + 1;
  • 2 871 133 965 ÷ 2 = 1 435 566 982 + 1;
  • 1 435 566 982 ÷ 2 = 717 783 491 + 0;
  • 717 783 491 ÷ 2 = 358 891 745 + 1;
  • 358 891 745 ÷ 2 = 179 445 872 + 1;
  • 179 445 872 ÷ 2 = 89 722 936 + 0;
  • 89 722 936 ÷ 2 = 44 861 468 + 0;
  • 44 861 468 ÷ 2 = 22 430 734 + 0;
  • 22 430 734 ÷ 2 = 11 215 367 + 0;
  • 11 215 367 ÷ 2 = 5 607 683 + 1;
  • 5 607 683 ÷ 2 = 2 803 841 + 1;
  • 2 803 841 ÷ 2 = 1 401 920 + 1;
  • 1 401 920 ÷ 2 = 700 960 + 0;
  • 700 960 ÷ 2 = 350 480 + 0;
  • 350 480 ÷ 2 = 175 240 + 0;
  • 175 240 ÷ 2 = 87 620 + 0;
  • 87 620 ÷ 2 = 43 810 + 0;
  • 43 810 ÷ 2 = 21 905 + 0;
  • 21 905 ÷ 2 = 10 952 + 1;
  • 10 952 ÷ 2 = 5 476 + 0;
  • 5 476 ÷ 2 = 2 738 + 0;
  • 2 738 ÷ 2 = 1 369 + 0;
  • 1 369 ÷ 2 = 684 + 1;
  • 684 ÷ 2 = 342 + 0;
  • 342 ÷ 2 = 171 + 0;
  • 171 ÷ 2 = 85 + 1;
  • 85 ÷ 2 = 42 + 1;
  • 42 ÷ 2 = 21 + 0;
  • 21 ÷ 2 = 10 + 1;
  • 10 ÷ 2 = 5 + 0;
  • 5 ÷ 2 = 2 + 1;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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

1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057(10) =


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


3. Normalize the binary representation of the number.

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


1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057(10) =


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


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


1.0101 0110 0100 0100 0000 1110 0001 1011 1110 0101 0110 1011 1101 1010 1111 1010 0111 0100 1010 0011 1111 0000 1111 0011 0000 0011 0101 0011 1001 0111 0001 1011 1000 1001 1010 1101 0011 1010 0101 0111 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 1(2) × 2209


4. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)


Exponent (unadjusted): 209


Mantissa (not normalized):
1.0101 0110 0100 0100 0000 1110 0001 1011 1110 0101 0110 1011 1101 1010 1111 1010 0111 0100 1010 0011 1111 0000 1111 0011 0000 0011 0101 0011 1001 0111 0001 1011 1000 1001 1010 1101 0011 1010 0101 0111 0000 0100 0000 0000 0000 0000 0000 0000 0000 0000 0001 1100 1


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


209 + 2(11-1) - 1 =


(209 + 1 023)(10) =


1 232(10)


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

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 232 ÷ 2 = 616 + 0;
  • 616 ÷ 2 = 308 + 0;
  • 308 ÷ 2 = 154 + 0;
  • 154 ÷ 2 = 77 + 0;
  • 77 ÷ 2 = 38 + 1;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =


1232(10) =


100 1101 0000(2)


8. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.


b) Adjust its length to 52 bits, by removing the excess bits, from the right (if any of the excess bits is set on 1, we are losing precision...).


Mantissa (normalized) =


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


0101 0110 0100 0100 0000 1110 0001 1011 1110 0101 0110 1011 1101


9. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)


Exponent (11 bits) =
100 1101 0000


Mantissa (52 bits) =
0101 0110 0100 0100 0000 1110 0001 1011 1110 0101 0110 1011 1101


The base ten decimal number 1 100 000 000 011 010 101 010 000 000 000 000 000 000 000 000 000 000 000 000 000 057 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 1101 0000 - 0101 0110 0100 0100 0000 1110 0001 1011 1110 0101 0110 1011 1101

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