64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093(10) converted and written in 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093 ÷ 2 = 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 505 546 + 1;
  • 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 505 546 ÷ 2 = 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 752 773 + 0;
  • 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 752 773 ÷ 2 = 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 876 386 + 1;
  • 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 876 386 ÷ 2 = 69 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 438 193 + 0;
  • 69 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 438 193 ÷ 2 = 34 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 219 096 + 1;
  • 34 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 219 096 ÷ 2 = 17 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 109 548 + 0;
  • 17 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 109 548 ÷ 2 = 8 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 774 + 0;
  • 8 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 554 774 ÷ 2 = 4 340 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 387 + 0;
  • 4 340 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 387 ÷ 2 = 2 170 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 693 + 1;
  • 2 170 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 693 ÷ 2 = 1 085 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 346 + 1;
  • 1 085 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 346 ÷ 2 = 542 534 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 173 + 0;
  • 542 534 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 222 173 ÷ 2 = 271 267 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 086 + 1;
  • 271 267 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 086 ÷ 2 = 135 633 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 543 + 0;
  • 135 633 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 543 ÷ 2 = 67 816 840 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 771 + 1;
  • 67 816 840 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 777 771 ÷ 2 = 33 908 420 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 + 1;
  • 33 908 420 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 888 885 ÷ 2 = 16 954 210 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 + 1;
  • 16 954 210 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 444 442 ÷ 2 = 8 477 105 034 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 + 0;
  • 8 477 105 034 722 222 222 222 222 222 222 222 222 222 222 222 222 222 222 221 ÷ 2 = 4 238 552 517 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 + 1;
  • 4 238 552 517 361 111 111 111 111 111 111 111 111 111 111 111 111 111 111 110 ÷ 2 = 2 119 276 258 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 + 0;
  • 2 119 276 258 680 555 555 555 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 1 059 638 129 340 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 1 059 638 129 340 277 777 777 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 529 819 064 670 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 529 819 064 670 138 888 888 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 264 909 532 335 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 264 909 532 335 069 444 444 444 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 132 454 766 167 534 722 222 222 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 132 454 766 167 534 722 222 222 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 66 227 383 083 767 361 111 111 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 66 227 383 083 767 361 111 111 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 33 113 691 541 883 680 555 555 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 33 113 691 541 883 680 555 555 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 16 556 845 770 941 840 277 777 777 777 777 777 777 777 777 777 777 777 777 + 1;
  • 16 556 845 770 941 840 277 777 777 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 8 278 422 885 470 920 138 888 888 888 888 888 888 888 888 888 888 888 888 + 1;
  • 8 278 422 885 470 920 138 888 888 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 4 139 211 442 735 460 069 444 444 444 444 444 444 444 444 444 444 444 444 + 0;
  • 4 139 211 442 735 460 069 444 444 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 2 069 605 721 367 730 034 722 222 222 222 222 222 222 222 222 222 222 222 + 0;
  • 2 069 605 721 367 730 034 722 222 222 222 222 222 222 222 222 222 222 222 ÷ 2 = 1 034 802 860 683 865 017 361 111 111 111 111 111 111 111 111 111 111 111 + 0;
  • 1 034 802 860 683 865 017 361 111 111 111 111 111 111 111 111 111 111 111 ÷ 2 = 517 401 430 341 932 508 680 555 555 555 555 555 555 555 555 555 555 555 + 1;
  • 517 401 430 341 932 508 680 555 555 555 555 555 555 555 555 555 555 555 ÷ 2 = 258 700 715 170 966 254 340 277 777 777 777 777 777 777 777 777 777 777 + 1;
  • 258 700 715 170 966 254 340 277 777 777 777 777 777 777 777 777 777 777 ÷ 2 = 129 350 357 585 483 127 170 138 888 888 888 888 888 888 888 888 888 888 + 1;
  • 129 350 357 585 483 127 170 138 888 888 888 888 888 888 888 888 888 888 ÷ 2 = 64 675 178 792 741 563 585 069 444 444 444 444 444 444 444 444 444 444 + 0;
  • 64 675 178 792 741 563 585 069 444 444 444 444 444 444 444 444 444 444 ÷ 2 = 32 337 589 396 370 781 792 534 722 222 222 222 222 222 222 222 222 222 + 0;
  • 32 337 589 396 370 781 792 534 722 222 222 222 222 222 222 222 222 222 ÷ 2 = 16 168 794 698 185 390 896 267 361 111 111 111 111 111 111 111 111 111 + 0;
  • 16 168 794 698 185 390 896 267 361 111 111 111 111 111 111 111 111 111 ÷ 2 = 8 084 397 349 092 695 448 133 680 555 555 555 555 555 555 555 555 555 + 1;
  • 8 084 397 349 092 695 448 133 680 555 555 555 555 555 555 555 555 555 ÷ 2 = 4 042 198 674 546 347 724 066 840 277 777 777 777 777 777 777 777 777 + 1;
  • 4 042 198 674 546 347 724 066 840 277 777 777 777 777 777 777 777 777 ÷ 2 = 2 021 099 337 273 173 862 033 420 138 888 888 888 888 888 888 888 888 + 1;
  • 2 021 099 337 273 173 862 033 420 138 888 888 888 888 888 888 888 888 ÷ 2 = 1 010 549 668 636 586 931 016 710 069 444 444 444 444 444 444 444 444 + 0;
  • 1 010 549 668 636 586 931 016 710 069 444 444 444 444 444 444 444 444 ÷ 2 = 505 274 834 318 293 465 508 355 034 722 222 222 222 222 222 222 222 + 0;
  • 505 274 834 318 293 465 508 355 034 722 222 222 222 222 222 222 222 ÷ 2 = 252 637 417 159 146 732 754 177 517 361 111 111 111 111 111 111 111 + 0;
  • 252 637 417 159 146 732 754 177 517 361 111 111 111 111 111 111 111 ÷ 2 = 126 318 708 579 573 366 377 088 758 680 555 555 555 555 555 555 555 + 1;
  • 126 318 708 579 573 366 377 088 758 680 555 555 555 555 555 555 555 ÷ 2 = 63 159 354 289 786 683 188 544 379 340 277 777 777 777 777 777 777 + 1;
  • 63 159 354 289 786 683 188 544 379 340 277 777 777 777 777 777 777 ÷ 2 = 31 579 677 144 893 341 594 272 189 670 138 888 888 888 888 888 888 + 1;
  • 31 579 677 144 893 341 594 272 189 670 138 888 888 888 888 888 888 ÷ 2 = 15 789 838 572 446 670 797 136 094 835 069 444 444 444 444 444 444 + 0;
  • 15 789 838 572 446 670 797 136 094 835 069 444 444 444 444 444 444 ÷ 2 = 7 894 919 286 223 335 398 568 047 417 534 722 222 222 222 222 222 + 0;
  • 7 894 919 286 223 335 398 568 047 417 534 722 222 222 222 222 222 ÷ 2 = 3 947 459 643 111 667 699 284 023 708 767 361 111 111 111 111 111 + 0;
  • 3 947 459 643 111 667 699 284 023 708 767 361 111 111 111 111 111 ÷ 2 = 1 973 729 821 555 833 849 642 011 854 383 680 555 555 555 555 555 + 1;
  • 1 973 729 821 555 833 849 642 011 854 383 680 555 555 555 555 555 ÷ 2 = 986 864 910 777 916 924 821 005 927 191 840 277 777 777 777 777 + 1;
  • 986 864 910 777 916 924 821 005 927 191 840 277 777 777 777 777 ÷ 2 = 493 432 455 388 958 462 410 502 963 595 920 138 888 888 888 888 + 1;
  • 493 432 455 388 958 462 410 502 963 595 920 138 888 888 888 888 ÷ 2 = 246 716 227 694 479 231 205 251 481 797 960 069 444 444 444 444 + 0;
  • 246 716 227 694 479 231 205 251 481 797 960 069 444 444 444 444 ÷ 2 = 123 358 113 847 239 615 602 625 740 898 980 034 722 222 222 222 + 0;
  • 123 358 113 847 239 615 602 625 740 898 980 034 722 222 222 222 ÷ 2 = 61 679 056 923 619 807 801 312 870 449 490 017 361 111 111 111 + 0;
  • 61 679 056 923 619 807 801 312 870 449 490 017 361 111 111 111 ÷ 2 = 30 839 528 461 809 903 900 656 435 224 745 008 680 555 555 555 + 1;
  • 30 839 528 461 809 903 900 656 435 224 745 008 680 555 555 555 ÷ 2 = 15 419 764 230 904 951 950 328 217 612 372 504 340 277 777 777 + 1;
  • 15 419 764 230 904 951 950 328 217 612 372 504 340 277 777 777 ÷ 2 = 7 709 882 115 452 475 975 164 108 806 186 252 170 138 888 888 + 1;
  • 7 709 882 115 452 475 975 164 108 806 186 252 170 138 888 888 ÷ 2 = 3 854 941 057 726 237 987 582 054 403 093 126 085 069 444 444 + 0;
  • 3 854 941 057 726 237 987 582 054 403 093 126 085 069 444 444 ÷ 2 = 1 927 470 528 863 118 993 791 027 201 546 563 042 534 722 222 + 0;
  • 1 927 470 528 863 118 993 791 027 201 546 563 042 534 722 222 ÷ 2 = 963 735 264 431 559 496 895 513 600 773 281 521 267 361 111 + 0;
  • 963 735 264 431 559 496 895 513 600 773 281 521 267 361 111 ÷ 2 = 481 867 632 215 779 748 447 756 800 386 640 760 633 680 555 + 1;
  • 481 867 632 215 779 748 447 756 800 386 640 760 633 680 555 ÷ 2 = 240 933 816 107 889 874 223 878 400 193 320 380 316 840 277 + 1;
  • 240 933 816 107 889 874 223 878 400 193 320 380 316 840 277 ÷ 2 = 120 466 908 053 944 937 111 939 200 096 660 190 158 420 138 + 1;
  • 120 466 908 053 944 937 111 939 200 096 660 190 158 420 138 ÷ 2 = 60 233 454 026 972 468 555 969 600 048 330 095 079 210 069 + 0;
  • 60 233 454 026 972 468 555 969 600 048 330 095 079 210 069 ÷ 2 = 30 116 727 013 486 234 277 984 800 024 165 047 539 605 034 + 1;
  • 30 116 727 013 486 234 277 984 800 024 165 047 539 605 034 ÷ 2 = 15 058 363 506 743 117 138 992 400 012 082 523 769 802 517 + 0;
  • 15 058 363 506 743 117 138 992 400 012 082 523 769 802 517 ÷ 2 = 7 529 181 753 371 558 569 496 200 006 041 261 884 901 258 + 1;
  • 7 529 181 753 371 558 569 496 200 006 041 261 884 901 258 ÷ 2 = 3 764 590 876 685 779 284 748 100 003 020 630 942 450 629 + 0;
  • 3 764 590 876 685 779 284 748 100 003 020 630 942 450 629 ÷ 2 = 1 882 295 438 342 889 642 374 050 001 510 315 471 225 314 + 1;
  • 1 882 295 438 342 889 642 374 050 001 510 315 471 225 314 ÷ 2 = 941 147 719 171 444 821 187 025 000 755 157 735 612 657 + 0;
  • 941 147 719 171 444 821 187 025 000 755 157 735 612 657 ÷ 2 = 470 573 859 585 722 410 593 512 500 377 578 867 806 328 + 1;
  • 470 573 859 585 722 410 593 512 500 377 578 867 806 328 ÷ 2 = 235 286 929 792 861 205 296 756 250 188 789 433 903 164 + 0;
  • 235 286 929 792 861 205 296 756 250 188 789 433 903 164 ÷ 2 = 117 643 464 896 430 602 648 378 125 094 394 716 951 582 + 0;
  • 117 643 464 896 430 602 648 378 125 094 394 716 951 582 ÷ 2 = 58 821 732 448 215 301 324 189 062 547 197 358 475 791 + 0;
  • 58 821 732 448 215 301 324 189 062 547 197 358 475 791 ÷ 2 = 29 410 866 224 107 650 662 094 531 273 598 679 237 895 + 1;
  • 29 410 866 224 107 650 662 094 531 273 598 679 237 895 ÷ 2 = 14 705 433 112 053 825 331 047 265 636 799 339 618 947 + 1;
  • 14 705 433 112 053 825 331 047 265 636 799 339 618 947 ÷ 2 = 7 352 716 556 026 912 665 523 632 818 399 669 809 473 + 1;
  • 7 352 716 556 026 912 665 523 632 818 399 669 809 473 ÷ 2 = 3 676 358 278 013 456 332 761 816 409 199 834 904 736 + 1;
  • 3 676 358 278 013 456 332 761 816 409 199 834 904 736 ÷ 2 = 1 838 179 139 006 728 166 380 908 204 599 917 452 368 + 0;
  • 1 838 179 139 006 728 166 380 908 204 599 917 452 368 ÷ 2 = 919 089 569 503 364 083 190 454 102 299 958 726 184 + 0;
  • 919 089 569 503 364 083 190 454 102 299 958 726 184 ÷ 2 = 459 544 784 751 682 041 595 227 051 149 979 363 092 + 0;
  • 459 544 784 751 682 041 595 227 051 149 979 363 092 ÷ 2 = 229 772 392 375 841 020 797 613 525 574 989 681 546 + 0;
  • 229 772 392 375 841 020 797 613 525 574 989 681 546 ÷ 2 = 114 886 196 187 920 510 398 806 762 787 494 840 773 + 0;
  • 114 886 196 187 920 510 398 806 762 787 494 840 773 ÷ 2 = 57 443 098 093 960 255 199 403 381 393 747 420 386 + 1;
  • 57 443 098 093 960 255 199 403 381 393 747 420 386 ÷ 2 = 28 721 549 046 980 127 599 701 690 696 873 710 193 + 0;
  • 28 721 549 046 980 127 599 701 690 696 873 710 193 ÷ 2 = 14 360 774 523 490 063 799 850 845 348 436 855 096 + 1;
  • 14 360 774 523 490 063 799 850 845 348 436 855 096 ÷ 2 = 7 180 387 261 745 031 899 925 422 674 218 427 548 + 0;
  • 7 180 387 261 745 031 899 925 422 674 218 427 548 ÷ 2 = 3 590 193 630 872 515 949 962 711 337 109 213 774 + 0;
  • 3 590 193 630 872 515 949 962 711 337 109 213 774 ÷ 2 = 1 795 096 815 436 257 974 981 355 668 554 606 887 + 0;
  • 1 795 096 815 436 257 974 981 355 668 554 606 887 ÷ 2 = 897 548 407 718 128 987 490 677 834 277 303 443 + 1;
  • 897 548 407 718 128 987 490 677 834 277 303 443 ÷ 2 = 448 774 203 859 064 493 745 338 917 138 651 721 + 1;
  • 448 774 203 859 064 493 745 338 917 138 651 721 ÷ 2 = 224 387 101 929 532 246 872 669 458 569 325 860 + 1;
  • 224 387 101 929 532 246 872 669 458 569 325 860 ÷ 2 = 112 193 550 964 766 123 436 334 729 284 662 930 + 0;
  • 112 193 550 964 766 123 436 334 729 284 662 930 ÷ 2 = 56 096 775 482 383 061 718 167 364 642 331 465 + 0;
  • 56 096 775 482 383 061 718 167 364 642 331 465 ÷ 2 = 28 048 387 741 191 530 859 083 682 321 165 732 + 1;
  • 28 048 387 741 191 530 859 083 682 321 165 732 ÷ 2 = 14 024 193 870 595 765 429 541 841 160 582 866 + 0;
  • 14 024 193 870 595 765 429 541 841 160 582 866 ÷ 2 = 7 012 096 935 297 882 714 770 920 580 291 433 + 0;
  • 7 012 096 935 297 882 714 770 920 580 291 433 ÷ 2 = 3 506 048 467 648 941 357 385 460 290 145 716 + 1;
  • 3 506 048 467 648 941 357 385 460 290 145 716 ÷ 2 = 1 753 024 233 824 470 678 692 730 145 072 858 + 0;
  • 1 753 024 233 824 470 678 692 730 145 072 858 ÷ 2 = 876 512 116 912 235 339 346 365 072 536 429 + 0;
  • 876 512 116 912 235 339 346 365 072 536 429 ÷ 2 = 438 256 058 456 117 669 673 182 536 268 214 + 1;
  • 438 256 058 456 117 669 673 182 536 268 214 ÷ 2 = 219 128 029 228 058 834 836 591 268 134 107 + 0;
  • 219 128 029 228 058 834 836 591 268 134 107 ÷ 2 = 109 564 014 614 029 417 418 295 634 067 053 + 1;
  • 109 564 014 614 029 417 418 295 634 067 053 ÷ 2 = 54 782 007 307 014 708 709 147 817 033 526 + 1;
  • 54 782 007 307 014 708 709 147 817 033 526 ÷ 2 = 27 391 003 653 507 354 354 573 908 516 763 + 0;
  • 27 391 003 653 507 354 354 573 908 516 763 ÷ 2 = 13 695 501 826 753 677 177 286 954 258 381 + 1;
  • 13 695 501 826 753 677 177 286 954 258 381 ÷ 2 = 6 847 750 913 376 838 588 643 477 129 190 + 1;
  • 6 847 750 913 376 838 588 643 477 129 190 ÷ 2 = 3 423 875 456 688 419 294 321 738 564 595 + 0;
  • 3 423 875 456 688 419 294 321 738 564 595 ÷ 2 = 1 711 937 728 344 209 647 160 869 282 297 + 1;
  • 1 711 937 728 344 209 647 160 869 282 297 ÷ 2 = 855 968 864 172 104 823 580 434 641 148 + 1;
  • 855 968 864 172 104 823 580 434 641 148 ÷ 2 = 427 984 432 086 052 411 790 217 320 574 + 0;
  • 427 984 432 086 052 411 790 217 320 574 ÷ 2 = 213 992 216 043 026 205 895 108 660 287 + 0;
  • 213 992 216 043 026 205 895 108 660 287 ÷ 2 = 106 996 108 021 513 102 947 554 330 143 + 1;
  • 106 996 108 021 513 102 947 554 330 143 ÷ 2 = 53 498 054 010 756 551 473 777 165 071 + 1;
  • 53 498 054 010 756 551 473 777 165 071 ÷ 2 = 26 749 027 005 378 275 736 888 582 535 + 1;
  • 26 749 027 005 378 275 736 888 582 535 ÷ 2 = 13 374 513 502 689 137 868 444 291 267 + 1;
  • 13 374 513 502 689 137 868 444 291 267 ÷ 2 = 6 687 256 751 344 568 934 222 145 633 + 1;
  • 6 687 256 751 344 568 934 222 145 633 ÷ 2 = 3 343 628 375 672 284 467 111 072 816 + 1;
  • 3 343 628 375 672 284 467 111 072 816 ÷ 2 = 1 671 814 187 836 142 233 555 536 408 + 0;
  • 1 671 814 187 836 142 233 555 536 408 ÷ 2 = 835 907 093 918 071 116 777 768 204 + 0;
  • 835 907 093 918 071 116 777 768 204 ÷ 2 = 417 953 546 959 035 558 388 884 102 + 0;
  • 417 953 546 959 035 558 388 884 102 ÷ 2 = 208 976 773 479 517 779 194 442 051 + 0;
  • 208 976 773 479 517 779 194 442 051 ÷ 2 = 104 488 386 739 758 889 597 221 025 + 1;
  • 104 488 386 739 758 889 597 221 025 ÷ 2 = 52 244 193 369 879 444 798 610 512 + 1;
  • 52 244 193 369 879 444 798 610 512 ÷ 2 = 26 122 096 684 939 722 399 305 256 + 0;
  • 26 122 096 684 939 722 399 305 256 ÷ 2 = 13 061 048 342 469 861 199 652 628 + 0;
  • 13 061 048 342 469 861 199 652 628 ÷ 2 = 6 530 524 171 234 930 599 826 314 + 0;
  • 6 530 524 171 234 930 599 826 314 ÷ 2 = 3 265 262 085 617 465 299 913 157 + 0;
  • 3 265 262 085 617 465 299 913 157 ÷ 2 = 1 632 631 042 808 732 649 956 578 + 1;
  • 1 632 631 042 808 732 649 956 578 ÷ 2 = 816 315 521 404 366 324 978 289 + 0;
  • 816 315 521 404 366 324 978 289 ÷ 2 = 408 157 760 702 183 162 489 144 + 1;
  • 408 157 760 702 183 162 489 144 ÷ 2 = 204 078 880 351 091 581 244 572 + 0;
  • 204 078 880 351 091 581 244 572 ÷ 2 = 102 039 440 175 545 790 622 286 + 0;
  • 102 039 440 175 545 790 622 286 ÷ 2 = 51 019 720 087 772 895 311 143 + 0;
  • 51 019 720 087 772 895 311 143 ÷ 2 = 25 509 860 043 886 447 655 571 + 1;
  • 25 509 860 043 886 447 655 571 ÷ 2 = 12 754 930 021 943 223 827 785 + 1;
  • 12 754 930 021 943 223 827 785 ÷ 2 = 6 377 465 010 971 611 913 892 + 1;
  • 6 377 465 010 971 611 913 892 ÷ 2 = 3 188 732 505 485 805 956 946 + 0;
  • 3 188 732 505 485 805 956 946 ÷ 2 = 1 594 366 252 742 902 978 473 + 0;
  • 1 594 366 252 742 902 978 473 ÷ 2 = 797 183 126 371 451 489 236 + 1;
  • 797 183 126 371 451 489 236 ÷ 2 = 398 591 563 185 725 744 618 + 0;
  • 398 591 563 185 725 744 618 ÷ 2 = 199 295 781 592 862 872 309 + 0;
  • 199 295 781 592 862 872 309 ÷ 2 = 99 647 890 796 431 436 154 + 1;
  • 99 647 890 796 431 436 154 ÷ 2 = 49 823 945 398 215 718 077 + 0;
  • 49 823 945 398 215 718 077 ÷ 2 = 24 911 972 699 107 859 038 + 1;
  • 24 911 972 699 107 859 038 ÷ 2 = 12 455 986 349 553 929 519 + 0;
  • 12 455 986 349 553 929 519 ÷ 2 = 6 227 993 174 776 964 759 + 1;
  • 6 227 993 174 776 964 759 ÷ 2 = 3 113 996 587 388 482 379 + 1;
  • 3 113 996 587 388 482 379 ÷ 2 = 1 556 998 293 694 241 189 + 1;
  • 1 556 998 293 694 241 189 ÷ 2 = 778 499 146 847 120 594 + 1;
  • 778 499 146 847 120 594 ÷ 2 = 389 249 573 423 560 297 + 0;
  • 389 249 573 423 560 297 ÷ 2 = 194 624 786 711 780 148 + 1;
  • 194 624 786 711 780 148 ÷ 2 = 97 312 393 355 890 074 + 0;
  • 97 312 393 355 890 074 ÷ 2 = 48 656 196 677 945 037 + 0;
  • 48 656 196 677 945 037 ÷ 2 = 24 328 098 338 972 518 + 1;
  • 24 328 098 338 972 518 ÷ 2 = 12 164 049 169 486 259 + 0;
  • 12 164 049 169 486 259 ÷ 2 = 6 082 024 584 743 129 + 1;
  • 6 082 024 584 743 129 ÷ 2 = 3 041 012 292 371 564 + 1;
  • 3 041 012 292 371 564 ÷ 2 = 1 520 506 146 185 782 + 0;
  • 1 520 506 146 185 782 ÷ 2 = 760 253 073 092 891 + 0;
  • 760 253 073 092 891 ÷ 2 = 380 126 536 546 445 + 1;
  • 380 126 536 546 445 ÷ 2 = 190 063 268 273 222 + 1;
  • 190 063 268 273 222 ÷ 2 = 95 031 634 136 611 + 0;
  • 95 031 634 136 611 ÷ 2 = 47 515 817 068 305 + 1;
  • 47 515 817 068 305 ÷ 2 = 23 757 908 534 152 + 1;
  • 23 757 908 534 152 ÷ 2 = 11 878 954 267 076 + 0;
  • 11 878 954 267 076 ÷ 2 = 5 939 477 133 538 + 0;
  • 5 939 477 133 538 ÷ 2 = 2 969 738 566 769 + 0;
  • 2 969 738 566 769 ÷ 2 = 1 484 869 283 384 + 1;
  • 1 484 869 283 384 ÷ 2 = 742 434 641 692 + 0;
  • 742 434 641 692 ÷ 2 = 371 217 320 846 + 0;
  • 371 217 320 846 ÷ 2 = 185 608 660 423 + 0;
  • 185 608 660 423 ÷ 2 = 92 804 330 211 + 1;
  • 92 804 330 211 ÷ 2 = 46 402 165 105 + 1;
  • 46 402 165 105 ÷ 2 = 23 201 082 552 + 1;
  • 23 201 082 552 ÷ 2 = 11 600 541 276 + 0;
  • 11 600 541 276 ÷ 2 = 5 800 270 638 + 0;
  • 5 800 270 638 ÷ 2 = 2 900 135 319 + 0;
  • 2 900 135 319 ÷ 2 = 1 450 067 659 + 1;
  • 1 450 067 659 ÷ 2 = 725 033 829 + 1;
  • 725 033 829 ÷ 2 = 362 516 914 + 1;
  • 362 516 914 ÷ 2 = 181 258 457 + 0;
  • 181 258 457 ÷ 2 = 90 629 228 + 1;
  • 90 629 228 ÷ 2 = 45 314 614 + 0;
  • 45 314 614 ÷ 2 = 22 657 307 + 0;
  • 22 657 307 ÷ 2 = 11 328 653 + 1;
  • 11 328 653 ÷ 2 = 5 664 326 + 1;
  • 5 664 326 ÷ 2 = 2 832 163 + 0;
  • 2 832 163 ÷ 2 = 1 416 081 + 1;
  • 1 416 081 ÷ 2 = 708 040 + 1;
  • 708 040 ÷ 2 = 354 020 + 0;
  • 354 020 ÷ 2 = 177 010 + 0;
  • 177 010 ÷ 2 = 88 505 + 0;
  • 88 505 ÷ 2 = 44 252 + 1;
  • 44 252 ÷ 2 = 22 126 + 0;
  • 22 126 ÷ 2 = 11 063 + 0;
  • 11 063 ÷ 2 = 5 531 + 1;
  • 5 531 ÷ 2 = 2 765 + 1;
  • 2 765 ÷ 2 = 1 382 + 1;
  • 1 382 ÷ 2 = 691 + 0;
  • 691 ÷ 2 = 345 + 1;
  • 345 ÷ 2 = 172 + 1;
  • 172 ÷ 2 = 86 + 0;
  • 86 ÷ 2 = 43 + 0;
  • 43 ÷ 2 = 21 + 1;
  • 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.


1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093(10) =

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


1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093(10) =

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

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

1.0101 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001 1010 0101 1110 1010 0100 1110 0010 1000 0110 0001 1111 1001 1011 0110 1001 0010 0111 0001 0100 0001 1110 0010 1010 1011 1000 1110 0011 1000 1110 0011 1000 1110 0011 1000 1101 0111 0101 1000 1010 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 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001 1010 0101 1110 1010 0100 1110 0010 1000 0110 0001 1111 1001 1011 0110 1001 0010 0111 0001 0100 0001 1110 0010 1010 1011 1000 1110 0011 1000 1110 0011 1000 1110 0011 1000 1101 0111 0101 1000 1010 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 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001 1 0100 1011 1101 0100 1001 1100 0101 0000 1100 0011 1111 0011 0110 1101 0010 0100 1110 0010 1000 0011 1100 0101 0101 0111 0001 1100 0111 0001 1100 0111 0001 1100 0111 0001 1010 1110 1011 0001 0101 =

0101 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001


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 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001


The base ten decimal number 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 093 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1101 0000 - 0101 1001 1011 1001 0001 1011 0010 1110 0011 1000 1000 1101 1001

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

» Number 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 092 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

» Number 1 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 111 011 094 converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point representation = ?

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

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

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100