64bit IEEE 754: Decimal ↗ Double Precision Floating Point Binary: 110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013 Convert the Number to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From a Base Ten Decimal System Number

Number 110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013(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;
  • 110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013 ÷ 2 = 55 000 005 000 505 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 006 + 1;
  • 55 000 005 000 505 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 006 ÷ 2 = 27 500 002 500 252 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 + 0;
  • 27 500 002 500 252 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 ÷ 2 = 13 750 001 250 126 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 + 1;
  • 13 750 001 250 126 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 ÷ 2 = 6 875 000 625 063 125 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 + 1;
  • 6 875 000 625 063 125 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 3 437 500 312 531 562 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 3 437 500 312 531 562 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 1 718 750 156 265 781 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 1 718 750 156 265 781 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 859 375 078 132 890 625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 859 375 078 132 890 625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 429 687 539 066 445 312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 429 687 539 066 445 312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 214 843 769 533 222 656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 214 843 769 533 222 656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 107 421 884 766 611 328 125 000 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 107 421 884 766 611 328 125 000 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 53 710 942 383 305 664 062 500 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 53 710 942 383 305 664 062 500 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 26 855 471 191 652 832 031 250 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 26 855 471 191 652 832 031 250 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 13 427 735 595 826 416 015 625 000 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 13 427 735 595 826 416 015 625 000 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 6 713 867 797 913 208 007 812 500 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 6 713 867 797 913 208 007 812 500 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 3 356 933 898 956 604 003 906 250 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 3 356 933 898 956 604 003 906 250 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 1 678 466 949 478 302 001 953 125 000 000 000 000 000 000 000 000 000 000 000 + 0;
  • 1 678 466 949 478 302 001 953 125 000 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 839 233 474 739 151 000 976 562 500 000 000 000 000 000 000 000 000 000 000 + 0;
  • 839 233 474 739 151 000 976 562 500 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 419 616 737 369 575 500 488 281 250 000 000 000 000 000 000 000 000 000 000 + 0;
  • 419 616 737 369 575 500 488 281 250 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 209 808 368 684 787 750 244 140 625 000 000 000 000 000 000 000 000 000 000 + 0;
  • 209 808 368 684 787 750 244 140 625 000 000 000 000 000 000 000 000 000 000 ÷ 2 = 104 904 184 342 393 875 122 070 312 500 000 000 000 000 000 000 000 000 000 + 0;
  • 104 904 184 342 393 875 122 070 312 500 000 000 000 000 000 000 000 000 000 ÷ 2 = 52 452 092 171 196 937 561 035 156 250 000 000 000 000 000 000 000 000 000 + 0;
  • 52 452 092 171 196 937 561 035 156 250 000 000 000 000 000 000 000 000 000 ÷ 2 = 26 226 046 085 598 468 780 517 578 125 000 000 000 000 000 000 000 000 000 + 0;
  • 26 226 046 085 598 468 780 517 578 125 000 000 000 000 000 000 000 000 000 ÷ 2 = 13 113 023 042 799 234 390 258 789 062 500 000 000 000 000 000 000 000 000 + 0;
  • 13 113 023 042 799 234 390 258 789 062 500 000 000 000 000 000 000 000 000 ÷ 2 = 6 556 511 521 399 617 195 129 394 531 250 000 000 000 000 000 000 000 000 + 0;
  • 6 556 511 521 399 617 195 129 394 531 250 000 000 000 000 000 000 000 000 ÷ 2 = 3 278 255 760 699 808 597 564 697 265 625 000 000 000 000 000 000 000 000 + 0;
  • 3 278 255 760 699 808 597 564 697 265 625 000 000 000 000 000 000 000 000 ÷ 2 = 1 639 127 880 349 904 298 782 348 632 812 500 000 000 000 000 000 000 000 + 0;
  • 1 639 127 880 349 904 298 782 348 632 812 500 000 000 000 000 000 000 000 ÷ 2 = 819 563 940 174 952 149 391 174 316 406 250 000 000 000 000 000 000 000 + 0;
  • 819 563 940 174 952 149 391 174 316 406 250 000 000 000 000 000 000 000 ÷ 2 = 409 781 970 087 476 074 695 587 158 203 125 000 000 000 000 000 000 000 + 0;
  • 409 781 970 087 476 074 695 587 158 203 125 000 000 000 000 000 000 000 ÷ 2 = 204 890 985 043 738 037 347 793 579 101 562 500 000 000 000 000 000 000 + 0;
  • 204 890 985 043 738 037 347 793 579 101 562 500 000 000 000 000 000 000 ÷ 2 = 102 445 492 521 869 018 673 896 789 550 781 250 000 000 000 000 000 000 + 0;
  • 102 445 492 521 869 018 673 896 789 550 781 250 000 000 000 000 000 000 ÷ 2 = 51 222 746 260 934 509 336 948 394 775 390 625 000 000 000 000 000 000 + 0;
  • 51 222 746 260 934 509 336 948 394 775 390 625 000 000 000 000 000 000 ÷ 2 = 25 611 373 130 467 254 668 474 197 387 695 312 500 000 000 000 000 000 + 0;
  • 25 611 373 130 467 254 668 474 197 387 695 312 500 000 000 000 000 000 ÷ 2 = 12 805 686 565 233 627 334 237 098 693 847 656 250 000 000 000 000 000 + 0;
  • 12 805 686 565 233 627 334 237 098 693 847 656 250 000 000 000 000 000 ÷ 2 = 6 402 843 282 616 813 667 118 549 346 923 828 125 000 000 000 000 000 + 0;
  • 6 402 843 282 616 813 667 118 549 346 923 828 125 000 000 000 000 000 ÷ 2 = 3 201 421 641 308 406 833 559 274 673 461 914 062 500 000 000 000 000 + 0;
  • 3 201 421 641 308 406 833 559 274 673 461 914 062 500 000 000 000 000 ÷ 2 = 1 600 710 820 654 203 416 779 637 336 730 957 031 250 000 000 000 000 + 0;
  • 1 600 710 820 654 203 416 779 637 336 730 957 031 250 000 000 000 000 ÷ 2 = 800 355 410 327 101 708 389 818 668 365 478 515 625 000 000 000 000 + 0;
  • 800 355 410 327 101 708 389 818 668 365 478 515 625 000 000 000 000 ÷ 2 = 400 177 705 163 550 854 194 909 334 182 739 257 812 500 000 000 000 + 0;
  • 400 177 705 163 550 854 194 909 334 182 739 257 812 500 000 000 000 ÷ 2 = 200 088 852 581 775 427 097 454 667 091 369 628 906 250 000 000 000 + 0;
  • 200 088 852 581 775 427 097 454 667 091 369 628 906 250 000 000 000 ÷ 2 = 100 044 426 290 887 713 548 727 333 545 684 814 453 125 000 000 000 + 0;
  • 100 044 426 290 887 713 548 727 333 545 684 814 453 125 000 000 000 ÷ 2 = 50 022 213 145 443 856 774 363 666 772 842 407 226 562 500 000 000 + 0;
  • 50 022 213 145 443 856 774 363 666 772 842 407 226 562 500 000 000 ÷ 2 = 25 011 106 572 721 928 387 181 833 386 421 203 613 281 250 000 000 + 0;
  • 25 011 106 572 721 928 387 181 833 386 421 203 613 281 250 000 000 ÷ 2 = 12 505 553 286 360 964 193 590 916 693 210 601 806 640 625 000 000 + 0;
  • 12 505 553 286 360 964 193 590 916 693 210 601 806 640 625 000 000 ÷ 2 = 6 252 776 643 180 482 096 795 458 346 605 300 903 320 312 500 000 + 0;
  • 6 252 776 643 180 482 096 795 458 346 605 300 903 320 312 500 000 ÷ 2 = 3 126 388 321 590 241 048 397 729 173 302 650 451 660 156 250 000 + 0;
  • 3 126 388 321 590 241 048 397 729 173 302 650 451 660 156 250 000 ÷ 2 = 1 563 194 160 795 120 524 198 864 586 651 325 225 830 078 125 000 + 0;
  • 1 563 194 160 795 120 524 198 864 586 651 325 225 830 078 125 000 ÷ 2 = 781 597 080 397 560 262 099 432 293 325 662 612 915 039 062 500 + 0;
  • 781 597 080 397 560 262 099 432 293 325 662 612 915 039 062 500 ÷ 2 = 390 798 540 198 780 131 049 716 146 662 831 306 457 519 531 250 + 0;
  • 390 798 540 198 780 131 049 716 146 662 831 306 457 519 531 250 ÷ 2 = 195 399 270 099 390 065 524 858 073 331 415 653 228 759 765 625 + 0;
  • 195 399 270 099 390 065 524 858 073 331 415 653 228 759 765 625 ÷ 2 = 97 699 635 049 695 032 762 429 036 665 707 826 614 379 882 812 + 1;
  • 97 699 635 049 695 032 762 429 036 665 707 826 614 379 882 812 ÷ 2 = 48 849 817 524 847 516 381 214 518 332 853 913 307 189 941 406 + 0;
  • 48 849 817 524 847 516 381 214 518 332 853 913 307 189 941 406 ÷ 2 = 24 424 908 762 423 758 190 607 259 166 426 956 653 594 970 703 + 0;
  • 24 424 908 762 423 758 190 607 259 166 426 956 653 594 970 703 ÷ 2 = 12 212 454 381 211 879 095 303 629 583 213 478 326 797 485 351 + 1;
  • 12 212 454 381 211 879 095 303 629 583 213 478 326 797 485 351 ÷ 2 = 6 106 227 190 605 939 547 651 814 791 606 739 163 398 742 675 + 1;
  • 6 106 227 190 605 939 547 651 814 791 606 739 163 398 742 675 ÷ 2 = 3 053 113 595 302 969 773 825 907 395 803 369 581 699 371 337 + 1;
  • 3 053 113 595 302 969 773 825 907 395 803 369 581 699 371 337 ÷ 2 = 1 526 556 797 651 484 886 912 953 697 901 684 790 849 685 668 + 1;
  • 1 526 556 797 651 484 886 912 953 697 901 684 790 849 685 668 ÷ 2 = 763 278 398 825 742 443 456 476 848 950 842 395 424 842 834 + 0;
  • 763 278 398 825 742 443 456 476 848 950 842 395 424 842 834 ÷ 2 = 381 639 199 412 871 221 728 238 424 475 421 197 712 421 417 + 0;
  • 381 639 199 412 871 221 728 238 424 475 421 197 712 421 417 ÷ 2 = 190 819 599 706 435 610 864 119 212 237 710 598 856 210 708 + 1;
  • 190 819 599 706 435 610 864 119 212 237 710 598 856 210 708 ÷ 2 = 95 409 799 853 217 805 432 059 606 118 855 299 428 105 354 + 0;
  • 95 409 799 853 217 805 432 059 606 118 855 299 428 105 354 ÷ 2 = 47 704 899 926 608 902 716 029 803 059 427 649 714 052 677 + 0;
  • 47 704 899 926 608 902 716 029 803 059 427 649 714 052 677 ÷ 2 = 23 852 449 963 304 451 358 014 901 529 713 824 857 026 338 + 1;
  • 23 852 449 963 304 451 358 014 901 529 713 824 857 026 338 ÷ 2 = 11 926 224 981 652 225 679 007 450 764 856 912 428 513 169 + 0;
  • 11 926 224 981 652 225 679 007 450 764 856 912 428 513 169 ÷ 2 = 5 963 112 490 826 112 839 503 725 382 428 456 214 256 584 + 1;
  • 5 963 112 490 826 112 839 503 725 382 428 456 214 256 584 ÷ 2 = 2 981 556 245 413 056 419 751 862 691 214 228 107 128 292 + 0;
  • 2 981 556 245 413 056 419 751 862 691 214 228 107 128 292 ÷ 2 = 1 490 778 122 706 528 209 875 931 345 607 114 053 564 146 + 0;
  • 1 490 778 122 706 528 209 875 931 345 607 114 053 564 146 ÷ 2 = 745 389 061 353 264 104 937 965 672 803 557 026 782 073 + 0;
  • 745 389 061 353 264 104 937 965 672 803 557 026 782 073 ÷ 2 = 372 694 530 676 632 052 468 982 836 401 778 513 391 036 + 1;
  • 372 694 530 676 632 052 468 982 836 401 778 513 391 036 ÷ 2 = 186 347 265 338 316 026 234 491 418 200 889 256 695 518 + 0;
  • 186 347 265 338 316 026 234 491 418 200 889 256 695 518 ÷ 2 = 93 173 632 669 158 013 117 245 709 100 444 628 347 759 + 0;
  • 93 173 632 669 158 013 117 245 709 100 444 628 347 759 ÷ 2 = 46 586 816 334 579 006 558 622 854 550 222 314 173 879 + 1;
  • 46 586 816 334 579 006 558 622 854 550 222 314 173 879 ÷ 2 = 23 293 408 167 289 503 279 311 427 275 111 157 086 939 + 1;
  • 23 293 408 167 289 503 279 311 427 275 111 157 086 939 ÷ 2 = 11 646 704 083 644 751 639 655 713 637 555 578 543 469 + 1;
  • 11 646 704 083 644 751 639 655 713 637 555 578 543 469 ÷ 2 = 5 823 352 041 822 375 819 827 856 818 777 789 271 734 + 1;
  • 5 823 352 041 822 375 819 827 856 818 777 789 271 734 ÷ 2 = 2 911 676 020 911 187 909 913 928 409 388 894 635 867 + 0;
  • 2 911 676 020 911 187 909 913 928 409 388 894 635 867 ÷ 2 = 1 455 838 010 455 593 954 956 964 204 694 447 317 933 + 1;
  • 1 455 838 010 455 593 954 956 964 204 694 447 317 933 ÷ 2 = 727 919 005 227 796 977 478 482 102 347 223 658 966 + 1;
  • 727 919 005 227 796 977 478 482 102 347 223 658 966 ÷ 2 = 363 959 502 613 898 488 739 241 051 173 611 829 483 + 0;
  • 363 959 502 613 898 488 739 241 051 173 611 829 483 ÷ 2 = 181 979 751 306 949 244 369 620 525 586 805 914 741 + 1;
  • 181 979 751 306 949 244 369 620 525 586 805 914 741 ÷ 2 = 90 989 875 653 474 622 184 810 262 793 402 957 370 + 1;
  • 90 989 875 653 474 622 184 810 262 793 402 957 370 ÷ 2 = 45 494 937 826 737 311 092 405 131 396 701 478 685 + 0;
  • 45 494 937 826 737 311 092 405 131 396 701 478 685 ÷ 2 = 22 747 468 913 368 655 546 202 565 698 350 739 342 + 1;
  • 22 747 468 913 368 655 546 202 565 698 350 739 342 ÷ 2 = 11 373 734 456 684 327 773 101 282 849 175 369 671 + 0;
  • 11 373 734 456 684 327 773 101 282 849 175 369 671 ÷ 2 = 5 686 867 228 342 163 886 550 641 424 587 684 835 + 1;
  • 5 686 867 228 342 163 886 550 641 424 587 684 835 ÷ 2 = 2 843 433 614 171 081 943 275 320 712 293 842 417 + 1;
  • 2 843 433 614 171 081 943 275 320 712 293 842 417 ÷ 2 = 1 421 716 807 085 540 971 637 660 356 146 921 208 + 1;
  • 1 421 716 807 085 540 971 637 660 356 146 921 208 ÷ 2 = 710 858 403 542 770 485 818 830 178 073 460 604 + 0;
  • 710 858 403 542 770 485 818 830 178 073 460 604 ÷ 2 = 355 429 201 771 385 242 909 415 089 036 730 302 + 0;
  • 355 429 201 771 385 242 909 415 089 036 730 302 ÷ 2 = 177 714 600 885 692 621 454 707 544 518 365 151 + 0;
  • 177 714 600 885 692 621 454 707 544 518 365 151 ÷ 2 = 88 857 300 442 846 310 727 353 772 259 182 575 + 1;
  • 88 857 300 442 846 310 727 353 772 259 182 575 ÷ 2 = 44 428 650 221 423 155 363 676 886 129 591 287 + 1;
  • 44 428 650 221 423 155 363 676 886 129 591 287 ÷ 2 = 22 214 325 110 711 577 681 838 443 064 795 643 + 1;
  • 22 214 325 110 711 577 681 838 443 064 795 643 ÷ 2 = 11 107 162 555 355 788 840 919 221 532 397 821 + 1;
  • 11 107 162 555 355 788 840 919 221 532 397 821 ÷ 2 = 5 553 581 277 677 894 420 459 610 766 198 910 + 1;
  • 5 553 581 277 677 894 420 459 610 766 198 910 ÷ 2 = 2 776 790 638 838 947 210 229 805 383 099 455 + 0;
  • 2 776 790 638 838 947 210 229 805 383 099 455 ÷ 2 = 1 388 395 319 419 473 605 114 902 691 549 727 + 1;
  • 1 388 395 319 419 473 605 114 902 691 549 727 ÷ 2 = 694 197 659 709 736 802 557 451 345 774 863 + 1;
  • 694 197 659 709 736 802 557 451 345 774 863 ÷ 2 = 347 098 829 854 868 401 278 725 672 887 431 + 1;
  • 347 098 829 854 868 401 278 725 672 887 431 ÷ 2 = 173 549 414 927 434 200 639 362 836 443 715 + 1;
  • 173 549 414 927 434 200 639 362 836 443 715 ÷ 2 = 86 774 707 463 717 100 319 681 418 221 857 + 1;
  • 86 774 707 463 717 100 319 681 418 221 857 ÷ 2 = 43 387 353 731 858 550 159 840 709 110 928 + 1;
  • 43 387 353 731 858 550 159 840 709 110 928 ÷ 2 = 21 693 676 865 929 275 079 920 354 555 464 + 0;
  • 21 693 676 865 929 275 079 920 354 555 464 ÷ 2 = 10 846 838 432 964 637 539 960 177 277 732 + 0;
  • 10 846 838 432 964 637 539 960 177 277 732 ÷ 2 = 5 423 419 216 482 318 769 980 088 638 866 + 0;
  • 5 423 419 216 482 318 769 980 088 638 866 ÷ 2 = 2 711 709 608 241 159 384 990 044 319 433 + 0;
  • 2 711 709 608 241 159 384 990 044 319 433 ÷ 2 = 1 355 854 804 120 579 692 495 022 159 716 + 1;
  • 1 355 854 804 120 579 692 495 022 159 716 ÷ 2 = 677 927 402 060 289 846 247 511 079 858 + 0;
  • 677 927 402 060 289 846 247 511 079 858 ÷ 2 = 338 963 701 030 144 923 123 755 539 929 + 0;
  • 338 963 701 030 144 923 123 755 539 929 ÷ 2 = 169 481 850 515 072 461 561 877 769 964 + 1;
  • 169 481 850 515 072 461 561 877 769 964 ÷ 2 = 84 740 925 257 536 230 780 938 884 982 + 0;
  • 84 740 925 257 536 230 780 938 884 982 ÷ 2 = 42 370 462 628 768 115 390 469 442 491 + 0;
  • 42 370 462 628 768 115 390 469 442 491 ÷ 2 = 21 185 231 314 384 057 695 234 721 245 + 1;
  • 21 185 231 314 384 057 695 234 721 245 ÷ 2 = 10 592 615 657 192 028 847 617 360 622 + 1;
  • 10 592 615 657 192 028 847 617 360 622 ÷ 2 = 5 296 307 828 596 014 423 808 680 311 + 0;
  • 5 296 307 828 596 014 423 808 680 311 ÷ 2 = 2 648 153 914 298 007 211 904 340 155 + 1;
  • 2 648 153 914 298 007 211 904 340 155 ÷ 2 = 1 324 076 957 149 003 605 952 170 077 + 1;
  • 1 324 076 957 149 003 605 952 170 077 ÷ 2 = 662 038 478 574 501 802 976 085 038 + 1;
  • 662 038 478 574 501 802 976 085 038 ÷ 2 = 331 019 239 287 250 901 488 042 519 + 0;
  • 331 019 239 287 250 901 488 042 519 ÷ 2 = 165 509 619 643 625 450 744 021 259 + 1;
  • 165 509 619 643 625 450 744 021 259 ÷ 2 = 82 754 809 821 812 725 372 010 629 + 1;
  • 82 754 809 821 812 725 372 010 629 ÷ 2 = 41 377 404 910 906 362 686 005 314 + 1;
  • 41 377 404 910 906 362 686 005 314 ÷ 2 = 20 688 702 455 453 181 343 002 657 + 0;
  • 20 688 702 455 453 181 343 002 657 ÷ 2 = 10 344 351 227 726 590 671 501 328 + 1;
  • 10 344 351 227 726 590 671 501 328 ÷ 2 = 5 172 175 613 863 295 335 750 664 + 0;
  • 5 172 175 613 863 295 335 750 664 ÷ 2 = 2 586 087 806 931 647 667 875 332 + 0;
  • 2 586 087 806 931 647 667 875 332 ÷ 2 = 1 293 043 903 465 823 833 937 666 + 0;
  • 1 293 043 903 465 823 833 937 666 ÷ 2 = 646 521 951 732 911 916 968 833 + 0;
  • 646 521 951 732 911 916 968 833 ÷ 2 = 323 260 975 866 455 958 484 416 + 1;
  • 323 260 975 866 455 958 484 416 ÷ 2 = 161 630 487 933 227 979 242 208 + 0;
  • 161 630 487 933 227 979 242 208 ÷ 2 = 80 815 243 966 613 989 621 104 + 0;
  • 80 815 243 966 613 989 621 104 ÷ 2 = 40 407 621 983 306 994 810 552 + 0;
  • 40 407 621 983 306 994 810 552 ÷ 2 = 20 203 810 991 653 497 405 276 + 0;
  • 20 203 810 991 653 497 405 276 ÷ 2 = 10 101 905 495 826 748 702 638 + 0;
  • 10 101 905 495 826 748 702 638 ÷ 2 = 5 050 952 747 913 374 351 319 + 0;
  • 5 050 952 747 913 374 351 319 ÷ 2 = 2 525 476 373 956 687 175 659 + 1;
  • 2 525 476 373 956 687 175 659 ÷ 2 = 1 262 738 186 978 343 587 829 + 1;
  • 1 262 738 186 978 343 587 829 ÷ 2 = 631 369 093 489 171 793 914 + 1;
  • 631 369 093 489 171 793 914 ÷ 2 = 315 684 546 744 585 896 957 + 0;
  • 315 684 546 744 585 896 957 ÷ 2 = 157 842 273 372 292 948 478 + 1;
  • 157 842 273 372 292 948 478 ÷ 2 = 78 921 136 686 146 474 239 + 0;
  • 78 921 136 686 146 474 239 ÷ 2 = 39 460 568 343 073 237 119 + 1;
  • 39 460 568 343 073 237 119 ÷ 2 = 19 730 284 171 536 618 559 + 1;
  • 19 730 284 171 536 618 559 ÷ 2 = 9 865 142 085 768 309 279 + 1;
  • 9 865 142 085 768 309 279 ÷ 2 = 4 932 571 042 884 154 639 + 1;
  • 4 932 571 042 884 154 639 ÷ 2 = 2 466 285 521 442 077 319 + 1;
  • 2 466 285 521 442 077 319 ÷ 2 = 1 233 142 760 721 038 659 + 1;
  • 1 233 142 760 721 038 659 ÷ 2 = 616 571 380 360 519 329 + 1;
  • 616 571 380 360 519 329 ÷ 2 = 308 285 690 180 259 664 + 1;
  • 308 285 690 180 259 664 ÷ 2 = 154 142 845 090 129 832 + 0;
  • 154 142 845 090 129 832 ÷ 2 = 77 071 422 545 064 916 + 0;
  • 77 071 422 545 064 916 ÷ 2 = 38 535 711 272 532 458 + 0;
  • 38 535 711 272 532 458 ÷ 2 = 19 267 855 636 266 229 + 0;
  • 19 267 855 636 266 229 ÷ 2 = 9 633 927 818 133 114 + 1;
  • 9 633 927 818 133 114 ÷ 2 = 4 816 963 909 066 557 + 0;
  • 4 816 963 909 066 557 ÷ 2 = 2 408 481 954 533 278 + 1;
  • 2 408 481 954 533 278 ÷ 2 = 1 204 240 977 266 639 + 0;
  • 1 204 240 977 266 639 ÷ 2 = 602 120 488 633 319 + 1;
  • 602 120 488 633 319 ÷ 2 = 301 060 244 316 659 + 1;
  • 301 060 244 316 659 ÷ 2 = 150 530 122 158 329 + 1;
  • 150 530 122 158 329 ÷ 2 = 75 265 061 079 164 + 1;
  • 75 265 061 079 164 ÷ 2 = 37 632 530 539 582 + 0;
  • 37 632 530 539 582 ÷ 2 = 18 816 265 269 791 + 0;
  • 18 816 265 269 791 ÷ 2 = 9 408 132 634 895 + 1;
  • 9 408 132 634 895 ÷ 2 = 4 704 066 317 447 + 1;
  • 4 704 066 317 447 ÷ 2 = 2 352 033 158 723 + 1;
  • 2 352 033 158 723 ÷ 2 = 1 176 016 579 361 + 1;
  • 1 176 016 579 361 ÷ 2 = 588 008 289 680 + 1;
  • 588 008 289 680 ÷ 2 = 294 004 144 840 + 0;
  • 294 004 144 840 ÷ 2 = 147 002 072 420 + 0;
  • 147 002 072 420 ÷ 2 = 73 501 036 210 + 0;
  • 73 501 036 210 ÷ 2 = 36 750 518 105 + 0;
  • 36 750 518 105 ÷ 2 = 18 375 259 052 + 1;
  • 18 375 259 052 ÷ 2 = 9 187 629 526 + 0;
  • 9 187 629 526 ÷ 2 = 4 593 814 763 + 0;
  • 4 593 814 763 ÷ 2 = 2 296 907 381 + 1;
  • 2 296 907 381 ÷ 2 = 1 148 453 690 + 1;
  • 1 148 453 690 ÷ 2 = 574 226 845 + 0;
  • 574 226 845 ÷ 2 = 287 113 422 + 1;
  • 287 113 422 ÷ 2 = 143 556 711 + 0;
  • 143 556 711 ÷ 2 = 71 778 355 + 1;
  • 71 778 355 ÷ 2 = 35 889 177 + 1;
  • 35 889 177 ÷ 2 = 17 944 588 + 1;
  • 17 944 588 ÷ 2 = 8 972 294 + 0;
  • 8 972 294 ÷ 2 = 4 486 147 + 0;
  • 4 486 147 ÷ 2 = 2 243 073 + 1;
  • 2 243 073 ÷ 2 = 1 121 536 + 1;
  • 1 121 536 ÷ 2 = 560 768 + 0;
  • 560 768 ÷ 2 = 280 384 + 0;
  • 280 384 ÷ 2 = 140 192 + 0;
  • 140 192 ÷ 2 = 70 096 + 0;
  • 70 096 ÷ 2 = 35 048 + 0;
  • 35 048 ÷ 2 = 17 524 + 0;
  • 17 524 ÷ 2 = 8 762 + 0;
  • 8 762 ÷ 2 = 4 381 + 0;
  • 4 381 ÷ 2 = 2 190 + 1;
  • 2 190 ÷ 2 = 1 095 + 0;
  • 1 095 ÷ 2 = 547 + 1;
  • 547 ÷ 2 = 273 + 1;
  • 273 ÷ 2 = 136 + 1;
  • 136 ÷ 2 = 68 + 0;
  • 68 ÷ 2 = 34 + 0;
  • 34 ÷ 2 = 17 + 0;
  • 17 ÷ 2 = 8 + 1;
  • 8 ÷ 2 = 4 + 0;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013(10) =


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


3. Normalize the binary representation of the number.

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


110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013(10) =


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


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


1.0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 1101 0100 0011 1111 1101 0111 0000 0010 0001 0111 0111 0110 0100 1000 0111 1110 1111 1000 1110 1011 0110 1111 0010 0010 1001 0011 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 01(2) × 2206


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

Sign 0 (a positive number)


Exponent (unadjusted): 206


Mantissa (not normalized):
1.0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 1101 0100 0011 1111 1101 0111 0000 0010 0001 0111 0111 0110 0100 1000 0111 1110 1111 1000 1110 1011 0110 1111 0010 0010 1001 0011 1100 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 01


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


206 + 2(11-1) - 1 =


(206 + 1 023)(10) =


1 229(10)


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

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 1 229 ÷ 2 = 614 + 1;
  • 614 ÷ 2 = 307 + 0;
  • 307 ÷ 2 = 153 + 1;
  • 153 ÷ 2 = 76 + 1;
  • 76 ÷ 2 = 38 + 0;
  • 38 ÷ 2 = 19 + 0;
  • 19 ÷ 2 = 9 + 1;
  • 9 ÷ 2 = 4 + 1;
  • 4 ÷ 2 = 2 + 0;
  • 2 ÷ 2 = 1 + 0;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =


1229(10) =


100 1100 1101(2)


8. Normalize the mantissa.

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


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


Mantissa (normalized) =


1. 0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 1101 01 0000 1111 1111 0101 1100 0000 1000 0101 1101 1101 1001 0010 0001 1111 1011 1110 0011 1010 1101 1011 1100 1000 1010 0100 1111 0010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1101 =


0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 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 1100 1101


Mantissa (52 bits) =
0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 1101


The base ten decimal number 110 000 010 001 010 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 013 converted and written in 64 bit double precision IEEE 754 binary floating point representation:
0 - 100 1100 1101 - 0001 0001 1101 0000 0000 1100 1110 1011 0010 0001 1111 0011 1101

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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