Decimal to 64 Bit IEEE 754 Binary: Convert Number 3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572 to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard, From Base Ten Decimal System

Number 3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572(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;
  • 3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572 ÷ 2 = 1 570 796 326 794 896 619 231 321 691 639 751 442 098 584 699 687 552 910 487 472 286 + 0;
  • 1 570 796 326 794 896 619 231 321 691 639 751 442 098 584 699 687 552 910 487 472 286 ÷ 2 = 785 398 163 397 448 309 615 660 845 819 875 721 049 292 349 843 776 455 243 736 143 + 0;
  • 785 398 163 397 448 309 615 660 845 819 875 721 049 292 349 843 776 455 243 736 143 ÷ 2 = 392 699 081 698 724 154 807 830 422 909 937 860 524 646 174 921 888 227 621 868 071 + 1;
  • 392 699 081 698 724 154 807 830 422 909 937 860 524 646 174 921 888 227 621 868 071 ÷ 2 = 196 349 540 849 362 077 403 915 211 454 968 930 262 323 087 460 944 113 810 934 035 + 1;
  • 196 349 540 849 362 077 403 915 211 454 968 930 262 323 087 460 944 113 810 934 035 ÷ 2 = 98 174 770 424 681 038 701 957 605 727 484 465 131 161 543 730 472 056 905 467 017 + 1;
  • 98 174 770 424 681 038 701 957 605 727 484 465 131 161 543 730 472 056 905 467 017 ÷ 2 = 49 087 385 212 340 519 350 978 802 863 742 232 565 580 771 865 236 028 452 733 508 + 1;
  • 49 087 385 212 340 519 350 978 802 863 742 232 565 580 771 865 236 028 452 733 508 ÷ 2 = 24 543 692 606 170 259 675 489 401 431 871 116 282 790 385 932 618 014 226 366 754 + 0;
  • 24 543 692 606 170 259 675 489 401 431 871 116 282 790 385 932 618 014 226 366 754 ÷ 2 = 12 271 846 303 085 129 837 744 700 715 935 558 141 395 192 966 309 007 113 183 377 + 0;
  • 12 271 846 303 085 129 837 744 700 715 935 558 141 395 192 966 309 007 113 183 377 ÷ 2 = 6 135 923 151 542 564 918 872 350 357 967 779 070 697 596 483 154 503 556 591 688 + 1;
  • 6 135 923 151 542 564 918 872 350 357 967 779 070 697 596 483 154 503 556 591 688 ÷ 2 = 3 067 961 575 771 282 459 436 175 178 983 889 535 348 798 241 577 251 778 295 844 + 0;
  • 3 067 961 575 771 282 459 436 175 178 983 889 535 348 798 241 577 251 778 295 844 ÷ 2 = 1 533 980 787 885 641 229 718 087 589 491 944 767 674 399 120 788 625 889 147 922 + 0;
  • 1 533 980 787 885 641 229 718 087 589 491 944 767 674 399 120 788 625 889 147 922 ÷ 2 = 766 990 393 942 820 614 859 043 794 745 972 383 837 199 560 394 312 944 573 961 + 0;
  • 766 990 393 942 820 614 859 043 794 745 972 383 837 199 560 394 312 944 573 961 ÷ 2 = 383 495 196 971 410 307 429 521 897 372 986 191 918 599 780 197 156 472 286 980 + 1;
  • 383 495 196 971 410 307 429 521 897 372 986 191 918 599 780 197 156 472 286 980 ÷ 2 = 191 747 598 485 705 153 714 760 948 686 493 095 959 299 890 098 578 236 143 490 + 0;
  • 191 747 598 485 705 153 714 760 948 686 493 095 959 299 890 098 578 236 143 490 ÷ 2 = 95 873 799 242 852 576 857 380 474 343 246 547 979 649 945 049 289 118 071 745 + 0;
  • 95 873 799 242 852 576 857 380 474 343 246 547 979 649 945 049 289 118 071 745 ÷ 2 = 47 936 899 621 426 288 428 690 237 171 623 273 989 824 972 524 644 559 035 872 + 1;
  • 47 936 899 621 426 288 428 690 237 171 623 273 989 824 972 524 644 559 035 872 ÷ 2 = 23 968 449 810 713 144 214 345 118 585 811 636 994 912 486 262 322 279 517 936 + 0;
  • 23 968 449 810 713 144 214 345 118 585 811 636 994 912 486 262 322 279 517 936 ÷ 2 = 11 984 224 905 356 572 107 172 559 292 905 818 497 456 243 131 161 139 758 968 + 0;
  • 11 984 224 905 356 572 107 172 559 292 905 818 497 456 243 131 161 139 758 968 ÷ 2 = 5 992 112 452 678 286 053 586 279 646 452 909 248 728 121 565 580 569 879 484 + 0;
  • 5 992 112 452 678 286 053 586 279 646 452 909 248 728 121 565 580 569 879 484 ÷ 2 = 2 996 056 226 339 143 026 793 139 823 226 454 624 364 060 782 790 284 939 742 + 0;
  • 2 996 056 226 339 143 026 793 139 823 226 454 624 364 060 782 790 284 939 742 ÷ 2 = 1 498 028 113 169 571 513 396 569 911 613 227 312 182 030 391 395 142 469 871 + 0;
  • 1 498 028 113 169 571 513 396 569 911 613 227 312 182 030 391 395 142 469 871 ÷ 2 = 749 014 056 584 785 756 698 284 955 806 613 656 091 015 195 697 571 234 935 + 1;
  • 749 014 056 584 785 756 698 284 955 806 613 656 091 015 195 697 571 234 935 ÷ 2 = 374 507 028 292 392 878 349 142 477 903 306 828 045 507 597 848 785 617 467 + 1;
  • 374 507 028 292 392 878 349 142 477 903 306 828 045 507 597 848 785 617 467 ÷ 2 = 187 253 514 146 196 439 174 571 238 951 653 414 022 753 798 924 392 808 733 + 1;
  • 187 253 514 146 196 439 174 571 238 951 653 414 022 753 798 924 392 808 733 ÷ 2 = 93 626 757 073 098 219 587 285 619 475 826 707 011 376 899 462 196 404 366 + 1;
  • 93 626 757 073 098 219 587 285 619 475 826 707 011 376 899 462 196 404 366 ÷ 2 = 46 813 378 536 549 109 793 642 809 737 913 353 505 688 449 731 098 202 183 + 0;
  • 46 813 378 536 549 109 793 642 809 737 913 353 505 688 449 731 098 202 183 ÷ 2 = 23 406 689 268 274 554 896 821 404 868 956 676 752 844 224 865 549 101 091 + 1;
  • 23 406 689 268 274 554 896 821 404 868 956 676 752 844 224 865 549 101 091 ÷ 2 = 11 703 344 634 137 277 448 410 702 434 478 338 376 422 112 432 774 550 545 + 1;
  • 11 703 344 634 137 277 448 410 702 434 478 338 376 422 112 432 774 550 545 ÷ 2 = 5 851 672 317 068 638 724 205 351 217 239 169 188 211 056 216 387 275 272 + 1;
  • 5 851 672 317 068 638 724 205 351 217 239 169 188 211 056 216 387 275 272 ÷ 2 = 2 925 836 158 534 319 362 102 675 608 619 584 594 105 528 108 193 637 636 + 0;
  • 2 925 836 158 534 319 362 102 675 608 619 584 594 105 528 108 193 637 636 ÷ 2 = 1 462 918 079 267 159 681 051 337 804 309 792 297 052 764 054 096 818 818 + 0;
  • 1 462 918 079 267 159 681 051 337 804 309 792 297 052 764 054 096 818 818 ÷ 2 = 731 459 039 633 579 840 525 668 902 154 896 148 526 382 027 048 409 409 + 0;
  • 731 459 039 633 579 840 525 668 902 154 896 148 526 382 027 048 409 409 ÷ 2 = 365 729 519 816 789 920 262 834 451 077 448 074 263 191 013 524 204 704 + 1;
  • 365 729 519 816 789 920 262 834 451 077 448 074 263 191 013 524 204 704 ÷ 2 = 182 864 759 908 394 960 131 417 225 538 724 037 131 595 506 762 102 352 + 0;
  • 182 864 759 908 394 960 131 417 225 538 724 037 131 595 506 762 102 352 ÷ 2 = 91 432 379 954 197 480 065 708 612 769 362 018 565 797 753 381 051 176 + 0;
  • 91 432 379 954 197 480 065 708 612 769 362 018 565 797 753 381 051 176 ÷ 2 = 45 716 189 977 098 740 032 854 306 384 681 009 282 898 876 690 525 588 + 0;
  • 45 716 189 977 098 740 032 854 306 384 681 009 282 898 876 690 525 588 ÷ 2 = 22 858 094 988 549 370 016 427 153 192 340 504 641 449 438 345 262 794 + 0;
  • 22 858 094 988 549 370 016 427 153 192 340 504 641 449 438 345 262 794 ÷ 2 = 11 429 047 494 274 685 008 213 576 596 170 252 320 724 719 172 631 397 + 0;
  • 11 429 047 494 274 685 008 213 576 596 170 252 320 724 719 172 631 397 ÷ 2 = 5 714 523 747 137 342 504 106 788 298 085 126 160 362 359 586 315 698 + 1;
  • 5 714 523 747 137 342 504 106 788 298 085 126 160 362 359 586 315 698 ÷ 2 = 2 857 261 873 568 671 252 053 394 149 042 563 080 181 179 793 157 849 + 0;
  • 2 857 261 873 568 671 252 053 394 149 042 563 080 181 179 793 157 849 ÷ 2 = 1 428 630 936 784 335 626 026 697 074 521 281 540 090 589 896 578 924 + 1;
  • 1 428 630 936 784 335 626 026 697 074 521 281 540 090 589 896 578 924 ÷ 2 = 714 315 468 392 167 813 013 348 537 260 640 770 045 294 948 289 462 + 0;
  • 714 315 468 392 167 813 013 348 537 260 640 770 045 294 948 289 462 ÷ 2 = 357 157 734 196 083 906 506 674 268 630 320 385 022 647 474 144 731 + 0;
  • 357 157 734 196 083 906 506 674 268 630 320 385 022 647 474 144 731 ÷ 2 = 178 578 867 098 041 953 253 337 134 315 160 192 511 323 737 072 365 + 1;
  • 178 578 867 098 041 953 253 337 134 315 160 192 511 323 737 072 365 ÷ 2 = 89 289 433 549 020 976 626 668 567 157 580 096 255 661 868 536 182 + 1;
  • 89 289 433 549 020 976 626 668 567 157 580 096 255 661 868 536 182 ÷ 2 = 44 644 716 774 510 488 313 334 283 578 790 048 127 830 934 268 091 + 0;
  • 44 644 716 774 510 488 313 334 283 578 790 048 127 830 934 268 091 ÷ 2 = 22 322 358 387 255 244 156 667 141 789 395 024 063 915 467 134 045 + 1;
  • 22 322 358 387 255 244 156 667 141 789 395 024 063 915 467 134 045 ÷ 2 = 11 161 179 193 627 622 078 333 570 894 697 512 031 957 733 567 022 + 1;
  • 11 161 179 193 627 622 078 333 570 894 697 512 031 957 733 567 022 ÷ 2 = 5 580 589 596 813 811 039 166 785 447 348 756 015 978 866 783 511 + 0;
  • 5 580 589 596 813 811 039 166 785 447 348 756 015 978 866 783 511 ÷ 2 = 2 790 294 798 406 905 519 583 392 723 674 378 007 989 433 391 755 + 1;
  • 2 790 294 798 406 905 519 583 392 723 674 378 007 989 433 391 755 ÷ 2 = 1 395 147 399 203 452 759 791 696 361 837 189 003 994 716 695 877 + 1;
  • 1 395 147 399 203 452 759 791 696 361 837 189 003 994 716 695 877 ÷ 2 = 697 573 699 601 726 379 895 848 180 918 594 501 997 358 347 938 + 1;
  • 697 573 699 601 726 379 895 848 180 918 594 501 997 358 347 938 ÷ 2 = 348 786 849 800 863 189 947 924 090 459 297 250 998 679 173 969 + 0;
  • 348 786 849 800 863 189 947 924 090 459 297 250 998 679 173 969 ÷ 2 = 174 393 424 900 431 594 973 962 045 229 648 625 499 339 586 984 + 1;
  • 174 393 424 900 431 594 973 962 045 229 648 625 499 339 586 984 ÷ 2 = 87 196 712 450 215 797 486 981 022 614 824 312 749 669 793 492 + 0;
  • 87 196 712 450 215 797 486 981 022 614 824 312 749 669 793 492 ÷ 2 = 43 598 356 225 107 898 743 490 511 307 412 156 374 834 896 746 + 0;
  • 43 598 356 225 107 898 743 490 511 307 412 156 374 834 896 746 ÷ 2 = 21 799 178 112 553 949 371 745 255 653 706 078 187 417 448 373 + 0;
  • 21 799 178 112 553 949 371 745 255 653 706 078 187 417 448 373 ÷ 2 = 10 899 589 056 276 974 685 872 627 826 853 039 093 708 724 186 + 1;
  • 10 899 589 056 276 974 685 872 627 826 853 039 093 708 724 186 ÷ 2 = 5 449 794 528 138 487 342 936 313 913 426 519 546 854 362 093 + 0;
  • 5 449 794 528 138 487 342 936 313 913 426 519 546 854 362 093 ÷ 2 = 2 724 897 264 069 243 671 468 156 956 713 259 773 427 181 046 + 1;
  • 2 724 897 264 069 243 671 468 156 956 713 259 773 427 181 046 ÷ 2 = 1 362 448 632 034 621 835 734 078 478 356 629 886 713 590 523 + 0;
  • 1 362 448 632 034 621 835 734 078 478 356 629 886 713 590 523 ÷ 2 = 681 224 316 017 310 917 867 039 239 178 314 943 356 795 261 + 1;
  • 681 224 316 017 310 917 867 039 239 178 314 943 356 795 261 ÷ 2 = 340 612 158 008 655 458 933 519 619 589 157 471 678 397 630 + 1;
  • 340 612 158 008 655 458 933 519 619 589 157 471 678 397 630 ÷ 2 = 170 306 079 004 327 729 466 759 809 794 578 735 839 198 815 + 0;
  • 170 306 079 004 327 729 466 759 809 794 578 735 839 198 815 ÷ 2 = 85 153 039 502 163 864 733 379 904 897 289 367 919 599 407 + 1;
  • 85 153 039 502 163 864 733 379 904 897 289 367 919 599 407 ÷ 2 = 42 576 519 751 081 932 366 689 952 448 644 683 959 799 703 + 1;
  • 42 576 519 751 081 932 366 689 952 448 644 683 959 799 703 ÷ 2 = 21 288 259 875 540 966 183 344 976 224 322 341 979 899 851 + 1;
  • 21 288 259 875 540 966 183 344 976 224 322 341 979 899 851 ÷ 2 = 10 644 129 937 770 483 091 672 488 112 161 170 989 949 925 + 1;
  • 10 644 129 937 770 483 091 672 488 112 161 170 989 949 925 ÷ 2 = 5 322 064 968 885 241 545 836 244 056 080 585 494 974 962 + 1;
  • 5 322 064 968 885 241 545 836 244 056 080 585 494 974 962 ÷ 2 = 2 661 032 484 442 620 772 918 122 028 040 292 747 487 481 + 0;
  • 2 661 032 484 442 620 772 918 122 028 040 292 747 487 481 ÷ 2 = 1 330 516 242 221 310 386 459 061 014 020 146 373 743 740 + 1;
  • 1 330 516 242 221 310 386 459 061 014 020 146 373 743 740 ÷ 2 = 665 258 121 110 655 193 229 530 507 010 073 186 871 870 + 0;
  • 665 258 121 110 655 193 229 530 507 010 073 186 871 870 ÷ 2 = 332 629 060 555 327 596 614 765 253 505 036 593 435 935 + 0;
  • 332 629 060 555 327 596 614 765 253 505 036 593 435 935 ÷ 2 = 166 314 530 277 663 798 307 382 626 752 518 296 717 967 + 1;
  • 166 314 530 277 663 798 307 382 626 752 518 296 717 967 ÷ 2 = 83 157 265 138 831 899 153 691 313 376 259 148 358 983 + 1;
  • 83 157 265 138 831 899 153 691 313 376 259 148 358 983 ÷ 2 = 41 578 632 569 415 949 576 845 656 688 129 574 179 491 + 1;
  • 41 578 632 569 415 949 576 845 656 688 129 574 179 491 ÷ 2 = 20 789 316 284 707 974 788 422 828 344 064 787 089 745 + 1;
  • 20 789 316 284 707 974 788 422 828 344 064 787 089 745 ÷ 2 = 10 394 658 142 353 987 394 211 414 172 032 393 544 872 + 1;
  • 10 394 658 142 353 987 394 211 414 172 032 393 544 872 ÷ 2 = 5 197 329 071 176 993 697 105 707 086 016 196 772 436 + 0;
  • 5 197 329 071 176 993 697 105 707 086 016 196 772 436 ÷ 2 = 2 598 664 535 588 496 848 552 853 543 008 098 386 218 + 0;
  • 2 598 664 535 588 496 848 552 853 543 008 098 386 218 ÷ 2 = 1 299 332 267 794 248 424 276 426 771 504 049 193 109 + 0;
  • 1 299 332 267 794 248 424 276 426 771 504 049 193 109 ÷ 2 = 649 666 133 897 124 212 138 213 385 752 024 596 554 + 1;
  • 649 666 133 897 124 212 138 213 385 752 024 596 554 ÷ 2 = 324 833 066 948 562 106 069 106 692 876 012 298 277 + 0;
  • 324 833 066 948 562 106 069 106 692 876 012 298 277 ÷ 2 = 162 416 533 474 281 053 034 553 346 438 006 149 138 + 1;
  • 162 416 533 474 281 053 034 553 346 438 006 149 138 ÷ 2 = 81 208 266 737 140 526 517 276 673 219 003 074 569 + 0;
  • 81 208 266 737 140 526 517 276 673 219 003 074 569 ÷ 2 = 40 604 133 368 570 263 258 638 336 609 501 537 284 + 1;
  • 40 604 133 368 570 263 258 638 336 609 501 537 284 ÷ 2 = 20 302 066 684 285 131 629 319 168 304 750 768 642 + 0;
  • 20 302 066 684 285 131 629 319 168 304 750 768 642 ÷ 2 = 10 151 033 342 142 565 814 659 584 152 375 384 321 + 0;
  • 10 151 033 342 142 565 814 659 584 152 375 384 321 ÷ 2 = 5 075 516 671 071 282 907 329 792 076 187 692 160 + 1;
  • 5 075 516 671 071 282 907 329 792 076 187 692 160 ÷ 2 = 2 537 758 335 535 641 453 664 896 038 093 846 080 + 0;
  • 2 537 758 335 535 641 453 664 896 038 093 846 080 ÷ 2 = 1 268 879 167 767 820 726 832 448 019 046 923 040 + 0;
  • 1 268 879 167 767 820 726 832 448 019 046 923 040 ÷ 2 = 634 439 583 883 910 363 416 224 009 523 461 520 + 0;
  • 634 439 583 883 910 363 416 224 009 523 461 520 ÷ 2 = 317 219 791 941 955 181 708 112 004 761 730 760 + 0;
  • 317 219 791 941 955 181 708 112 004 761 730 760 ÷ 2 = 158 609 895 970 977 590 854 056 002 380 865 380 + 0;
  • 158 609 895 970 977 590 854 056 002 380 865 380 ÷ 2 = 79 304 947 985 488 795 427 028 001 190 432 690 + 0;
  • 79 304 947 985 488 795 427 028 001 190 432 690 ÷ 2 = 39 652 473 992 744 397 713 514 000 595 216 345 + 0;
  • 39 652 473 992 744 397 713 514 000 595 216 345 ÷ 2 = 19 826 236 996 372 198 856 757 000 297 608 172 + 1;
  • 19 826 236 996 372 198 856 757 000 297 608 172 ÷ 2 = 9 913 118 498 186 099 428 378 500 148 804 086 + 0;
  • 9 913 118 498 186 099 428 378 500 148 804 086 ÷ 2 = 4 956 559 249 093 049 714 189 250 074 402 043 + 0;
  • 4 956 559 249 093 049 714 189 250 074 402 043 ÷ 2 = 2 478 279 624 546 524 857 094 625 037 201 021 + 1;
  • 2 478 279 624 546 524 857 094 625 037 201 021 ÷ 2 = 1 239 139 812 273 262 428 547 312 518 600 510 + 1;
  • 1 239 139 812 273 262 428 547 312 518 600 510 ÷ 2 = 619 569 906 136 631 214 273 656 259 300 255 + 0;
  • 619 569 906 136 631 214 273 656 259 300 255 ÷ 2 = 309 784 953 068 315 607 136 828 129 650 127 + 1;
  • 309 784 953 068 315 607 136 828 129 650 127 ÷ 2 = 154 892 476 534 157 803 568 414 064 825 063 + 1;
  • 154 892 476 534 157 803 568 414 064 825 063 ÷ 2 = 77 446 238 267 078 901 784 207 032 412 531 + 1;
  • 77 446 238 267 078 901 784 207 032 412 531 ÷ 2 = 38 723 119 133 539 450 892 103 516 206 265 + 1;
  • 38 723 119 133 539 450 892 103 516 206 265 ÷ 2 = 19 361 559 566 769 725 446 051 758 103 132 + 1;
  • 19 361 559 566 769 725 446 051 758 103 132 ÷ 2 = 9 680 779 783 384 862 723 025 879 051 566 + 0;
  • 9 680 779 783 384 862 723 025 879 051 566 ÷ 2 = 4 840 389 891 692 431 361 512 939 525 783 + 0;
  • 4 840 389 891 692 431 361 512 939 525 783 ÷ 2 = 2 420 194 945 846 215 680 756 469 762 891 + 1;
  • 2 420 194 945 846 215 680 756 469 762 891 ÷ 2 = 1 210 097 472 923 107 840 378 234 881 445 + 1;
  • 1 210 097 472 923 107 840 378 234 881 445 ÷ 2 = 605 048 736 461 553 920 189 117 440 722 + 1;
  • 605 048 736 461 553 920 189 117 440 722 ÷ 2 = 302 524 368 230 776 960 094 558 720 361 + 0;
  • 302 524 368 230 776 960 094 558 720 361 ÷ 2 = 151 262 184 115 388 480 047 279 360 180 + 1;
  • 151 262 184 115 388 480 047 279 360 180 ÷ 2 = 75 631 092 057 694 240 023 639 680 090 + 0;
  • 75 631 092 057 694 240 023 639 680 090 ÷ 2 = 37 815 546 028 847 120 011 819 840 045 + 0;
  • 37 815 546 028 847 120 011 819 840 045 ÷ 2 = 18 907 773 014 423 560 005 909 920 022 + 1;
  • 18 907 773 014 423 560 005 909 920 022 ÷ 2 = 9 453 886 507 211 780 002 954 960 011 + 0;
  • 9 453 886 507 211 780 002 954 960 011 ÷ 2 = 4 726 943 253 605 890 001 477 480 005 + 1;
  • 4 726 943 253 605 890 001 477 480 005 ÷ 2 = 2 363 471 626 802 945 000 738 740 002 + 1;
  • 2 363 471 626 802 945 000 738 740 002 ÷ 2 = 1 181 735 813 401 472 500 369 370 001 + 0;
  • 1 181 735 813 401 472 500 369 370 001 ÷ 2 = 590 867 906 700 736 250 184 685 000 + 1;
  • 590 867 906 700 736 250 184 685 000 ÷ 2 = 295 433 953 350 368 125 092 342 500 + 0;
  • 295 433 953 350 368 125 092 342 500 ÷ 2 = 147 716 976 675 184 062 546 171 250 + 0;
  • 147 716 976 675 184 062 546 171 250 ÷ 2 = 73 858 488 337 592 031 273 085 625 + 0;
  • 73 858 488 337 592 031 273 085 625 ÷ 2 = 36 929 244 168 796 015 636 542 812 + 1;
  • 36 929 244 168 796 015 636 542 812 ÷ 2 = 18 464 622 084 398 007 818 271 406 + 0;
  • 18 464 622 084 398 007 818 271 406 ÷ 2 = 9 232 311 042 199 003 909 135 703 + 0;
  • 9 232 311 042 199 003 909 135 703 ÷ 2 = 4 616 155 521 099 501 954 567 851 + 1;
  • 4 616 155 521 099 501 954 567 851 ÷ 2 = 2 308 077 760 549 750 977 283 925 + 1;
  • 2 308 077 760 549 750 977 283 925 ÷ 2 = 1 154 038 880 274 875 488 641 962 + 1;
  • 1 154 038 880 274 875 488 641 962 ÷ 2 = 577 019 440 137 437 744 320 981 + 0;
  • 577 019 440 137 437 744 320 981 ÷ 2 = 288 509 720 068 718 872 160 490 + 1;
  • 288 509 720 068 718 872 160 490 ÷ 2 = 144 254 860 034 359 436 080 245 + 0;
  • 144 254 860 034 359 436 080 245 ÷ 2 = 72 127 430 017 179 718 040 122 + 1;
  • 72 127 430 017 179 718 040 122 ÷ 2 = 36 063 715 008 589 859 020 061 + 0;
  • 36 063 715 008 589 859 020 061 ÷ 2 = 18 031 857 504 294 929 510 030 + 1;
  • 18 031 857 504 294 929 510 030 ÷ 2 = 9 015 928 752 147 464 755 015 + 0;
  • 9 015 928 752 147 464 755 015 ÷ 2 = 4 507 964 376 073 732 377 507 + 1;
  • 4 507 964 376 073 732 377 507 ÷ 2 = 2 253 982 188 036 866 188 753 + 1;
  • 2 253 982 188 036 866 188 753 ÷ 2 = 1 126 991 094 018 433 094 376 + 1;
  • 1 126 991 094 018 433 094 376 ÷ 2 = 563 495 547 009 216 547 188 + 0;
  • 563 495 547 009 216 547 188 ÷ 2 = 281 747 773 504 608 273 594 + 0;
  • 281 747 773 504 608 273 594 ÷ 2 = 140 873 886 752 304 136 797 + 0;
  • 140 873 886 752 304 136 797 ÷ 2 = 70 436 943 376 152 068 398 + 1;
  • 70 436 943 376 152 068 398 ÷ 2 = 35 218 471 688 076 034 199 + 0;
  • 35 218 471 688 076 034 199 ÷ 2 = 17 609 235 844 038 017 099 + 1;
  • 17 609 235 844 038 017 099 ÷ 2 = 8 804 617 922 019 008 549 + 1;
  • 8 804 617 922 019 008 549 ÷ 2 = 4 402 308 961 009 504 274 + 1;
  • 4 402 308 961 009 504 274 ÷ 2 = 2 201 154 480 504 752 137 + 0;
  • 2 201 154 480 504 752 137 ÷ 2 = 1 100 577 240 252 376 068 + 1;
  • 1 100 577 240 252 376 068 ÷ 2 = 550 288 620 126 188 034 + 0;
  • 550 288 620 126 188 034 ÷ 2 = 275 144 310 063 094 017 + 0;
  • 275 144 310 063 094 017 ÷ 2 = 137 572 155 031 547 008 + 1;
  • 137 572 155 031 547 008 ÷ 2 = 68 786 077 515 773 504 + 0;
  • 68 786 077 515 773 504 ÷ 2 = 34 393 038 757 886 752 + 0;
  • 34 393 038 757 886 752 ÷ 2 = 17 196 519 378 943 376 + 0;
  • 17 196 519 378 943 376 ÷ 2 = 8 598 259 689 471 688 + 0;
  • 8 598 259 689 471 688 ÷ 2 = 4 299 129 844 735 844 + 0;
  • 4 299 129 844 735 844 ÷ 2 = 2 149 564 922 367 922 + 0;
  • 2 149 564 922 367 922 ÷ 2 = 1 074 782 461 183 961 + 0;
  • 1 074 782 461 183 961 ÷ 2 = 537 391 230 591 980 + 1;
  • 537 391 230 591 980 ÷ 2 = 268 695 615 295 990 + 0;
  • 268 695 615 295 990 ÷ 2 = 134 347 807 647 995 + 0;
  • 134 347 807 647 995 ÷ 2 = 67 173 903 823 997 + 1;
  • 67 173 903 823 997 ÷ 2 = 33 586 951 911 998 + 1;
  • 33 586 951 911 998 ÷ 2 = 16 793 475 955 999 + 0;
  • 16 793 475 955 999 ÷ 2 = 8 396 737 977 999 + 1;
  • 8 396 737 977 999 ÷ 2 = 4 198 368 988 999 + 1;
  • 4 198 368 988 999 ÷ 2 = 2 099 184 494 499 + 1;
  • 2 099 184 494 499 ÷ 2 = 1 049 592 247 249 + 1;
  • 1 049 592 247 249 ÷ 2 = 524 796 123 624 + 1;
  • 524 796 123 624 ÷ 2 = 262 398 061 812 + 0;
  • 262 398 061 812 ÷ 2 = 131 199 030 906 + 0;
  • 131 199 030 906 ÷ 2 = 65 599 515 453 + 0;
  • 65 599 515 453 ÷ 2 = 32 799 757 726 + 1;
  • 32 799 757 726 ÷ 2 = 16 399 878 863 + 0;
  • 16 399 878 863 ÷ 2 = 8 199 939 431 + 1;
  • 8 199 939 431 ÷ 2 = 4 099 969 715 + 1;
  • 4 099 969 715 ÷ 2 = 2 049 984 857 + 1;
  • 2 049 984 857 ÷ 2 = 1 024 992 428 + 1;
  • 1 024 992 428 ÷ 2 = 512 496 214 + 0;
  • 512 496 214 ÷ 2 = 256 248 107 + 0;
  • 256 248 107 ÷ 2 = 128 124 053 + 1;
  • 128 124 053 ÷ 2 = 64 062 026 + 1;
  • 64 062 026 ÷ 2 = 32 031 013 + 0;
  • 32 031 013 ÷ 2 = 16 015 506 + 1;
  • 16 015 506 ÷ 2 = 8 007 753 + 0;
  • 8 007 753 ÷ 2 = 4 003 876 + 1;
  • 4 003 876 ÷ 2 = 2 001 938 + 0;
  • 2 001 938 ÷ 2 = 1 000 969 + 0;
  • 1 000 969 ÷ 2 = 500 484 + 1;
  • 500 484 ÷ 2 = 250 242 + 0;
  • 250 242 ÷ 2 = 125 121 + 0;
  • 125 121 ÷ 2 = 62 560 + 1;
  • 62 560 ÷ 2 = 31 280 + 0;
  • 31 280 ÷ 2 = 15 640 + 0;
  • 15 640 ÷ 2 = 7 820 + 0;
  • 7 820 ÷ 2 = 3 910 + 0;
  • 3 910 ÷ 2 = 1 955 + 0;
  • 1 955 ÷ 2 = 977 + 1;
  • 977 ÷ 2 = 488 + 1;
  • 488 ÷ 2 = 244 + 0;
  • 244 ÷ 2 = 122 + 0;
  • 122 ÷ 2 = 61 + 0;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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

3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572(10) =


111 1010 0011 0000 0100 1001 0101 1001 1110 1000 1111 1011 0010 0000 0010 0101 1101 0001 1101 0101 0111 0010 0010 1101 0010 1110 0111 1101 1001 0000 0001 0010 1010 0011 1110 0101 1111 0110 1010 0010 1110 1101 1001 0100 0001 0001 1101 1110 0000 1001 0001 0011 1100(2)


3. Normalize the binary representation of the number.

Shift the decimal mark 210 positions to the left, so that only one non zero digit remains to the left of it:


3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572(10) =


111 1010 0011 0000 0100 1001 0101 1001 1110 1000 1111 1011 0010 0000 0010 0101 1101 0001 1101 0101 0111 0010 0010 1101 0010 1110 0111 1101 1001 0000 0001 0010 1010 0011 1110 0101 1111 0110 1010 0010 1110 1101 1001 0100 0001 0001 1101 1110 0000 1001 0001 0011 1100(2) =


111 1010 0011 0000 0100 1001 0101 1001 1110 1000 1111 1011 0010 0000 0010 0101 1101 0001 1101 0101 0111 0010 0010 1101 0010 1110 0111 1101 1001 0000 0001 0010 1010 0011 1110 0101 1111 0110 1010 0010 1110 1101 1001 0100 0001 0001 1101 1110 0000 1001 0001 0011 1100(2) × 20 =


1.1110 1000 1100 0001 0010 0101 0110 0111 1010 0011 1110 1100 1000 0000 1001 0111 0100 0111 0101 0101 1100 1000 1011 0100 1011 1001 1111 0110 0100 0000 0100 1010 1000 1111 1001 0111 1101 1010 1000 1011 1011 0110 0101 0000 0100 0111 0111 1000 0010 0100 0100 1111 00(2) × 2210


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): 210


Mantissa (not normalized):
1.1110 1000 1100 0001 0010 0101 0110 0111 1010 0011 1110 1100 1000 0000 1001 0111 0100 0111 0101 0101 1100 1000 1011 0100 1011 1001 1111 0110 0100 0000 0100 1010 1000 1111 1001 0111 1101 1010 1000 1011 1011 0110 0101 0000 0100 0111 0111 1000 0010 0100 0100 1111 00


5. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =


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


210 + 2(11-1) - 1 =


(210 + 1 023)(10) =


1 233(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;
  • 1 233 ÷ 2 = 616 + 1;
  • 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) =


1233(10) =


100 1101 0001(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. 1110 1000 1100 0001 0010 0101 0110 0111 1010 0011 1110 1100 1000 00 0010 0101 1101 0001 1101 0101 0111 0010 0010 1101 0010 1110 0111 1101 1001 0000 0001 0010 1010 0011 1110 0101 1111 0110 1010 0010 1110 1101 1001 0100 0001 0001 1101 1110 0000 1001 0001 0011 1100 =


1110 1000 1100 0001 0010 0101 0110 0111 1010 0011 1110 1100 1000


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 0001


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


The base ten decimal number 3 141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 572 converted and written in 64 bit double precision IEEE 754 binary floating point representation:

0 - 100 1101 0001 - 1110 1000 1100 0001 0010 0101 0110 0111 1010 0011 1110 1100 1000

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