0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 93;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 93 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 86;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 86 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 72;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 72 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 44;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 44 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 894 88;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 894 88 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 789 76;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 789 76 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 579 52;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 579 52 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 159 04;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 159 04 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 318 08;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 318 08 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 636 16;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 636 16 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 272 32;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 272 32 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 544 64;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 544 64 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 089 28;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 089 28 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 178 56;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 178 56 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 357 12;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 357 12 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 584 714 24;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 584 714 24 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 169 428 48;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 169 428 48 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 338 856 96;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 338 856 96 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 677 713 92;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 677 713 92 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 355 427 84;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 355 427 84 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 710 855 68;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 710 855 68 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 421 711 36;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 421 711 36 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 843 422 72;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 843 422 72 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 686 845 44;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 686 845 44 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 373 690 88;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 373 690 88 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 526 747 381 76;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 526 747 381 76 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 053 494 763 52;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 053 494 763 52 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 106 989 527 04;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 106 989 527 04 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 213 979 054 08;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 213 979 054 08 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 427 958 108 16;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 427 958 108 16 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 855 916 216 32;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 855 916 216 32 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 711 832 432 64;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 711 832 432 64 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 423 664 865 28;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 423 664 865 28 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 847 329 730 56;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 847 329 730 56 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 694 659 461 12;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 694 659 461 12 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 389 318 922 24;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 389 318 922 24 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 126 778 637 844 48;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 126 778 637 844 48 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 253 557 275 688 96;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 253 557 275 688 96 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 507 114 551 377 92;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 507 114 551 377 92 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 014 229 102 755 84;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 014 229 102 755 84 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 028 458 205 511 68;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 028 458 205 511 68 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 056 916 411 023 36;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 056 916 411 023 36 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 113 832 822 046 72;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 113 832 822 046 72 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 227 665 644 093 44;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 227 665 644 093 44 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 696 455 331 288 186 88;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 696 455 331 288 186 88 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 392 910 662 576 373 76;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 392 910 662 576 373 76 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 785 821 325 152 747 52;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 785 821 325 152 747 52 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 571 642 650 305 495 04;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 571 642 650 305 495 04 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 143 285 300 610 990 08;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 143 285 300 610 990 08 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 286 570 601 221 980 16;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 286 570 601 221 980 16 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 573 141 202 443 960 32;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 573 141 202 443 960 32 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 146 282 404 887 920 64;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 146 282 404 887 920 64 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 292 564 809 775 841 28;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 292 564 809 775 841 28 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 585 129 619 551 682 56;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 585 129 619 551 682 56 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 170 259 239 103 365 12;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 170 259 239 103 365 12 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 498 340 518 478 206 730 24;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 498 340 518 478 206 730 24 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 996 681 036 956 413 460 48;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 996 681 036 956 413 460 48 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 993 362 073 912 826 920 96;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 993 362 073 912 826 920 96 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 986 724 147 825 653 841 92;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 986 724 147 825 653 841 92 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 973 448 295 651 307 683 84;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 973 448 295 651 307 683 84 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 946 896 591 302 615 367 68;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 946 896 591 302 615 367 68 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 893 793 182 605 230 735 36;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 893 793 182 605 230 735 36 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 787 586 365 210 461 470 72;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 787 586 365 210 461 470 72 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 575 172 730 420 922 941 44;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 575 172 730 420 922 941 44 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 150 345 460 841 845 882 88;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 150 345 460 841 845 882 88 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 998 300 690 921 683 691 765 76;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 998 300 690 921 683 691 765 76 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 996 601 381 843 367 383 531 52;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 996 601 381 843 367 383 531 52 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 993 202 763 686 734 767 063 04;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 993 202 763 686 734 767 063 04 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 986 405 527 373 469 534 126 08;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 986 405 527 373 469 534 126 08 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 972 811 054 746 939 068 252 16;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 972 811 054 746 939 068 252 16 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 945 622 109 493 878 136 504 32;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 945 622 109 493 878 136 504 32 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 891 244 218 987 756 273 008 64;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 891 244 218 987 756 273 008 64 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 782 488 437 975 512 546 017 28;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 782 488 437 975 512 546 017 28 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 564 976 875 951 025 092 034 56;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 564 976 875 951 025 092 034 56 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 129 953 751 902 050 184 069 12;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 129 953 751 902 050 184 069 12 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 998 259 907 503 804 100 368 138 24;
  • 77) 0.698 547 363 281 249 999 999 999 999 998 259 907 503 804 100 368 138 24 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 996 519 815 007 608 200 736 276 48;
  • 78) 0.397 094 726 562 499 999 999 999 999 996 519 815 007 608 200 736 276 48 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 993 039 630 015 216 401 472 552 96;
  • 79) 0.794 189 453 124 999 999 999 999 999 993 039 630 015 216 401 472 552 96 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 986 079 260 030 432 802 945 105 92;
  • 80) 0.588 378 906 249 999 999 999 999 999 986 079 260 030 432 802 945 105 92 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 972 158 520 060 865 605 890 211 84;
  • 81) 0.176 757 812 499 999 999 999 999 999 972 158 520 060 865 605 890 211 84 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 944 317 040 121 731 211 780 423 68;
  • 82) 0.353 515 624 999 999 999 999 999 999 944 317 040 121 731 211 780 423 68 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 888 634 080 243 462 423 560 847 36;
  • 83) 0.707 031 249 999 999 999 999 999 999 888 634 080 243 462 423 560 847 36 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 777 268 160 486 924 847 121 694 72;
  • 84) 0.414 062 499 999 999 999 999 999 999 777 268 160 486 924 847 121 694 72 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 554 536 320 973 849 694 243 389 44;
  • 85) 0.828 124 999 999 999 999 999 999 999 554 536 320 973 849 694 243 389 44 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 109 072 641 947 699 388 486 778 88;
  • 86) 0.656 249 999 999 999 999 999 999 999 109 072 641 947 699 388 486 778 88 × 2 = 1 + 0.312 499 999 999 999 999 999 999 998 218 145 283 895 398 776 973 557 76;
  • 87) 0.312 499 999 999 999 999 999 999 998 218 145 283 895 398 776 973 557 76 × 2 = 0 + 0.624 999 999 999 999 999 999 999 996 436 290 567 790 797 553 947 115 52;
  • 88) 0.624 999 999 999 999 999 999 999 996 436 290 567 790 797 553 947 115 52 × 2 = 1 + 0.249 999 999 999 999 999 999 999 992 872 581 135 581 595 107 894 231 04;
  • 89) 0.249 999 999 999 999 999 999 999 992 872 581 135 581 595 107 894 231 04 × 2 = 0 + 0.499 999 999 999 999 999 999 999 985 745 162 271 163 190 215 788 462 08;
  • 90) 0.499 999 999 999 999 999 999 999 985 745 162 271 163 190 215 788 462 08 × 2 = 0 + 0.999 999 999 999 999 999 999 999 971 490 324 542 326 380 431 576 924 16;
  • 91) 0.999 999 999 999 999 999 999 999 971 490 324 542 326 380 431 576 924 16 × 2 = 1 + 0.999 999 999 999 999 999 999 999 942 980 649 084 652 760 863 153 848 32;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 965 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