0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 577 08;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 577 08 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 154 16;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 154 16 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 308 32;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 308 32 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 616 64;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 616 64 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 233 28;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 233 28 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 466 56;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 466 56 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 933 12;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 933 12 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 866 24;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 866 24 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 131 732 48;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 131 732 48 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 263 464 96;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 263 464 96 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 526 929 92;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 526 929 92 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 053 859 84;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 053 859 84 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 107 719 68;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 107 719 68 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 215 439 36;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 215 439 36 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 430 878 72;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 430 878 72 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 861 757 44;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 861 757 44 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 723 514 88;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 723 514 88 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 971 447 029 76;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 971 447 029 76 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 942 894 059 52;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 942 894 059 52 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 885 788 119 04;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 885 788 119 04 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 771 576 238 08;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 771 576 238 08 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 543 152 476 16;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 543 152 476 16 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 086 304 952 32;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 086 304 952 32 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 172 609 904 64;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 172 609 904 64 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 345 219 809 28;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 345 219 809 28 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 690 439 618 56;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 690 439 618 56 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 380 879 237 12;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 380 879 237 12 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 761 758 474 24;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 761 758 474 24 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 501 523 516 948 48;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 501 523 516 948 48 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 003 047 033 896 96;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 003 047 033 896 96 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 006 094 067 793 92;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 006 094 067 793 92 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 012 188 135 587 84;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 012 188 135 587 84 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 024 376 271 175 68;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 024 376 271 175 68 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 048 752 542 351 36;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 048 752 542 351 36 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 097 505 084 702 72;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 097 505 084 702 72 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 195 010 169 405 44;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 195 010 169 405 44 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 390 020 338 810 88;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 390 020 338 810 88 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 780 040 677 621 76;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 780 040 677 621 76 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 001 560 081 355 243 52;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 001 560 081 355 243 52 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 003 120 162 710 487 04;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 003 120 162 710 487 04 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 006 240 325 420 974 08;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 006 240 325 420 974 08 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 012 480 650 841 948 16;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 012 480 650 841 948 16 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 024 961 301 683 896 32;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 024 961 301 683 896 32 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 049 922 603 367 792 64;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 049 922 603 367 792 64 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 099 845 206 735 585 28;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 099 845 206 735 585 28 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 199 690 413 471 170 56;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 199 690 413 471 170 56 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 399 380 826 942 341 12;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 399 380 826 942 341 12 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 798 761 653 884 682 24;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 798 761 653 884 682 24 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 001 597 523 307 769 364 48;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 001 597 523 307 769 364 48 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 003 195 046 615 538 728 96;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 003 195 046 615 538 728 96 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 006 390 093 231 077 457 92;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 006 390 093 231 077 457 92 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 012 780 186 462 154 915 84;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 012 780 186 462 154 915 84 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 025 560 372 924 309 831 68;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 025 560 372 924 309 831 68 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 051 120 745 848 619 663 36;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 051 120 745 848 619 663 36 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 102 241 491 697 239 326 72;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 102 241 491 697 239 326 72 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 204 482 983 394 478 653 44;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 204 482 983 394 478 653 44 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 408 965 966 788 957 306 88;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 408 965 966 788 957 306 88 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 817 931 933 577 914 613 76;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 817 931 933 577 914 613 76 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 001 635 863 867 155 829 227 52;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 001 635 863 867 155 829 227 52 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 003 271 727 734 311 658 455 04;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 003 271 727 734 311 658 455 04 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 006 543 455 468 623 316 910 08;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 006 543 455 468 623 316 910 08 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 013 086 910 937 246 633 820 16;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 013 086 910 937 246 633 820 16 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 026 173 821 874 493 267 640 32;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 026 173 821 874 493 267 640 32 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 052 347 643 748 986 535 280 64;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 052 347 643 748 986 535 280 64 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 104 695 287 497 973 070 561 28;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 104 695 287 497 973 070 561 28 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 209 390 574 995 946 141 122 56;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 209 390 574 995 946 141 122 56 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 418 781 149 991 892 282 245 12;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 418 781 149 991 892 282 245 12 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 837 562 299 983 784 564 490 24;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 837 562 299 983 784 564 490 24 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 001 675 124 599 967 569 128 980 48;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 001 675 124 599 967 569 128 980 48 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 003 350 249 199 935 138 257 960 96;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 003 350 249 199 935 138 257 960 96 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 006 700 498 399 870 276 515 921 92;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 006 700 498 399 870 276 515 921 92 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 013 400 996 799 740 553 031 843 84;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 013 400 996 799 740 553 031 843 84 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 026 801 993 599 481 106 063 687 68;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 026 801 993 599 481 106 063 687 68 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 053 603 987 198 962 212 127 375 36;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 053 603 987 198 962 212 127 375 36 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 107 207 974 397 924 424 254 750 72;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 107 207 974 397 924 424 254 750 72 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 214 415 948 795 848 848 509 501 44;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 214 415 948 795 848 848 509 501 44 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 428 831 897 591 697 697 019 002 88;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 428 831 897 591 697 697 019 002 88 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 857 663 795 183 395 394 038 005 76;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 857 663 795 183 395 394 038 005 76 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 001 715 327 590 366 790 788 076 011 52;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 001 715 327 590 366 790 788 076 011 52 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 003 430 655 180 733 581 576 152 023 04;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 003 430 655 180 733 581 576 152 023 04 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 006 861 310 361 467 163 152 304 046 08;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 006 861 310 361 467 163 152 304 046 08 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 013 722 620 722 934 326 304 608 092 16;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 013 722 620 722 934 326 304 608 092 16 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 027 445 241 445 868 652 609 216 184 32;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 027 445 241 445 868 652 609 216 184 32 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 054 890 482 891 737 305 218 432 368 64;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 054 890 482 891 737 305 218 432 368 64 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 109 780 965 783 474 610 436 864 737 28;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 109 780 965 783 474 610 436 864 737 28 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 219 561 931 566 949 220 873 729 474 56;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 219 561 931 566 949 220 873 729 474 56 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 439 123 863 133 898 441 747 458 949 12;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 439 123 863 133 898 441 747 458 949 12 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 878 247 726 267 796 883 494 917 898 24;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 878 247 726 267 796 883 494 917 898 24 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 001 756 495 452 535 593 766 989 835 796 48;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 001 756 495 452 535 593 766 989 835 796 48 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 512 990 905 071 187 533 979 671 592 96;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 003 512 990 905 071 187 533 979 671 592 96 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 007 025 981 810 142 375 067 959 343 185 92;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(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 1010


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 1010 =

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


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 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 788 54 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 1010

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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