0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
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;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 404 437 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 404 437 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 808 874 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 808 874 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 617 749 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 617 749 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 235 499 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 235 499 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 470 998 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 470 998 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 941 996 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 941 996 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 883 993 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 883 993 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 767 987 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 767 987 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 535 974 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 535 974 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 071 948 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 071 948 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 118 143 897 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 118 143 897 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 236 287 795 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 236 287 795 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 472 575 590 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 472 575 590 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 945 151 180 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 945 151 180 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 890 302 361 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 890 302 361 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 780 604 723 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 780 604 723 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 561 209 446 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 561 209 446 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 122 418 892 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 122 418 892 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 244 837 785 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 244 837 785 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 812 489 675 571 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 812 489 675 571 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 624 979 351 142 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 624 979 351 142 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 249 958 702 284 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 249 958 702 284 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 499 917 404 569 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 499 917 404 569 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 999 834 809 139 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 999 834 809 139 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 999 669 618 278 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 999 669 618 278 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 999 339 236 556 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 999 339 236 556 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 998 678 473 113 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 998 678 473 113 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 999 997 356 946 227 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 999 997 356 946 227 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 999 994 713 892 454 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 999 994 713 892 454 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 999 989 427 784 908 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 999 989 427 784 908 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 999 978 855 569 817 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 999 978 855 569 817 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 999 957 711 139 635 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 999 957 711 139 635 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 999 915 422 279 270 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 999 915 422 279 270 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 999 830 844 558 540 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 999 830 844 558 540 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 999 661 689 117 081 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 999 661 689 117 081 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 999 323 378 234 163 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 999 323 378 234 163 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 998 646 756 468 326 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 998 646 756 468 326 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 999 997 293 512 936 652 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 999 997 293 512 936 652 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 999 994 587 025 873 305 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 999 994 587 025 873 305 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 999 989 174 051 746 611 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 999 989 174 051 746 611 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 999 978 348 103 493 222 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 999 978 348 103 493 222 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 999 956 696 206 986 444 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 999 956 696 206 986 444 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 999 913 392 413 972 889 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 999 913 392 413 972 889 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 999 826 784 827 945 779 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 999 826 784 827 945 779 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 999 653 569 655 891 558 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 999 653 569 655 891 558 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 999 307 139 311 783 116 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 999 307 139 311 783 116 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 998 614 278 623 566 233 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 998 614 278 623 566 233 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 999 997 228 557 247 132 467 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 999 997 228 557 247 132 467 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 999 994 457 114 494 264 934 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 999 994 457 114 494 264 934 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 999 988 914 228 988 529 868 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 999 988 914 228 988 529 868 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 999 977 828 457 977 059 737 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 999 977 828 457 977 059 737 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 999 955 656 915 954 119 475 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 999 955 656 915 954 119 475 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 999 911 313 831 908 238 950 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 999 911 313 831 908 238 950 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 999 822 627 663 816 477 900 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 999 822 627 663 816 477 900 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 999 645 255 327 632 955 801 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 999 645 255 327 632 955 801 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 999 290 510 655 265 911 603 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 999 290 510 655 265 911 603 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 999 998 581 021 310 531 823 206 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 999 998 581 021 310 531 823 206 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 999 997 162 042 621 063 646 412 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 999 997 162 042 621 063 646 412 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 999 994 324 085 242 127 292 825 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 999 994 324 085 242 127 292 825 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 999 988 648 170 484 254 585 651 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 999 988 648 170 484 254 585 651 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 999 977 296 340 968 509 171 302 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 999 977 296 340 968 509 171 302 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 999 954 592 681 937 018 342 604 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 999 954 592 681 937 018 342 604 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 999 909 185 363 874 036 685 209 6;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 999 909 185 363 874 036 685 209 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 999 818 370 727 748 073 370 419 2;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 999 818 370 727 748 073 370 419 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 999 636 741 455 496 146 740 838 4;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 999 636 741 455 496 146 740 838 4 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 999 273 482 910 992 293 481 676 8;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 999 273 482 910 992 293 481 676 8 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 999 998 546 965 821 984 586 963 353 6;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 999 998 546 965 821 984 586 963 353 6 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 999 997 093 931 643 969 173 926 707 2;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 999 997 093 931 643 969 173 926 707 2 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 999 994 187 863 287 938 347 853 414 4;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 999 994 187 863 287 938 347 853 414 4 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 999 988 375 726 575 876 695 706 828 8;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 999 988 375 726 575 876 695 706 828 8 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 999 976 751 453 151 753 391 413 657 6;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 999 976 751 453 151 753 391 413 657 6 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 999 953 502 906 303 506 782 827 315 2;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 999 953 502 906 303 506 782 827 315 2 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 999 907 005 812 607 013 565 654 630 4;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 999 907 005 812 607 013 565 654 630 4 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 814 011 625 214 027 131 309 260 8;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 814 011 625 214 027 131 309 260 8 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 628 023 250 428 054 262 618 521 6;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 628 023 250 428 054 262 618 521 6 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 256 046 500 856 108 525 237 043 2;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 256 046 500 856 108 525 237 043 2 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 998 512 093 001 712 217 050 474 086 4;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 998 512 093 001 712 217 050 474 086 4 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 997 024 186 003 424 434 100 948 172 8;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 997 024 186 003 424 434 100 948 172 8 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 994 048 372 006 848 868 201 896 345 6;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 994 048 372 006 848 868 201 896 345 6 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 988 096 744 013 697 736 403 792 691 2;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 988 096 744 013 697 736 403 792 691 2 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 193 488 027 395 472 807 585 382 4;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 976 193 488 027 395 472 807 585 382 4 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 952 386 976 054 790 945 615 170 764 8;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 952 386 976 054 790 945 615 170 764 8 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 904 773 952 109 581 891 230 341 529 6;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 904 773 952 109 581 891 230 341 529 6 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 809 547 904 219 163 782 460 683 059 2;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 809 547 904 219 163 782 460 683 059 2 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 619 095 808 438 327 564 921 366 118 4;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 619 095 808 438 327 564 921 366 118 4 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 238 191 616 876 655 129 842 732 236 8;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 238 191 616 876 655 129 842 732 236 8 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 476 383 233 753 310 259 685 464 473 6;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 476 383 233 753 310 259 685 464 473 6 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 952 766 467 506 620 519 370 928 947 2;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 996 952 766 467 506 620 519 370 928 947 2 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 905 532 935 013 241 038 741 857 894 4;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 993 905 532 935 013 241 038 741 857 894 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 811 065 870 026 482 077 483 715 788 8;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 987 811 065 870 026 482 077 483 715 788 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 975 622 131 740 052 964 154 967 431 577 6;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


7. 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): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. 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;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. 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, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


12. 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) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 218 7 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

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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