0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 868 32;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 868 32 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 736 64;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 736 64 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 473 28;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 473 28 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 946 56;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 946 56 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 893 12;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 893 12 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 786 24;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 786 24 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 572 48;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 572 48 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 663 144 96;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 663 144 96 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 326 289 92;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 326 289 92 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 652 579 84;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 652 579 84 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 305 159 68;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 305 159 68 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 610 319 36;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 610 319 36 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 220 638 72;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 220 638 72 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 441 277 44;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 441 277 44 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 882 554 88;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 882 554 88 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 765 109 76;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 765 109 76 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 530 219 52;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 530 219 52 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 119 060 439 04;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 119 060 439 04 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 238 120 878 08;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 238 120 878 08 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 476 241 756 16;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 476 241 756 16 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 952 483 512 32;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 952 483 512 32 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 904 967 024 64;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 904 967 024 64 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 809 934 049 28;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 809 934 049 28 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 619 868 098 56;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 619 868 098 56 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 239 736 197 12;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 239 736 197 12 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 479 472 394 24;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 479 472 394 24 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 958 944 788 48;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 958 944 788 48 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 917 889 576 96;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 917 889 576 96 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 155 835 779 153 92;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 155 835 779 153 92 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 311 671 558 307 84;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 311 671 558 307 84 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 623 343 116 615 68;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 623 343 116 615 68 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 246 686 233 231 36;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 246 686 233 231 36 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 493 372 466 462 72;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 493 372 466 462 72 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 986 744 932 925 44;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 986 744 932 925 44 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 973 489 865 850 88;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 973 489 865 850 88 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 946 979 731 701 76;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 946 979 731 701 76 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 893 959 463 403 52;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 893 959 463 403 52 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 787 918 926 807 04;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 787 918 926 807 04 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 391 575 837 853 614 08;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 391 575 837 853 614 08 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 783 151 675 707 228 16;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 783 151 675 707 228 16 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 566 303 351 414 456 32;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 566 303 351 414 456 32 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 132 606 702 828 912 64;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 132 606 702 828 912 64 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 265 213 405 657 825 28;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 265 213 405 657 825 28 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 530 426 811 315 650 56;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 530 426 811 315 650 56 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 060 853 622 631 301 12;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 060 853 622 631 301 12 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 121 707 245 262 602 24;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 121 707 245 262 602 24 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 243 414 490 525 204 48;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 243 414 490 525 204 48 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 486 828 981 050 408 96;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 486 828 981 050 408 96 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 973 657 962 100 817 92;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 973 657 962 100 817 92 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 001 947 315 924 201 635 84;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 001 947 315 924 201 635 84 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 003 894 631 848 403 271 68;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 003 894 631 848 403 271 68 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 007 789 263 696 806 543 36;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 007 789 263 696 806 543 36 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 015 578 527 393 613 086 72;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 015 578 527 393 613 086 72 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 031 157 054 787 226 173 44;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 031 157 054 787 226 173 44 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 062 314 109 574 452 346 88;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 062 314 109 574 452 346 88 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 124 628 219 148 904 693 76;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 124 628 219 148 904 693 76 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 249 256 438 297 809 387 52;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 249 256 438 297 809 387 52 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 498 512 876 595 618 775 04;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 498 512 876 595 618 775 04 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 997 025 753 191 237 550 08;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 997 025 753 191 237 550 08 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 001 994 051 506 382 475 100 16;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 001 994 051 506 382 475 100 16 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 003 988 103 012 764 950 200 32;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 003 988 103 012 764 950 200 32 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 007 976 206 025 529 900 400 64;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 007 976 206 025 529 900 400 64 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 015 952 412 051 059 800 801 28;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 015 952 412 051 059 800 801 28 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 031 904 824 102 119 601 602 56;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 031 904 824 102 119 601 602 56 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 063 809 648 204 239 203 205 12;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 063 809 648 204 239 203 205 12 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 127 619 296 408 478 406 410 24;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 127 619 296 408 478 406 410 24 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 255 238 592 816 956 812 820 48;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 255 238 592 816 956 812 820 48 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 510 477 185 633 913 625 640 96;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 510 477 185 633 913 625 640 96 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 001 020 954 371 267 827 251 281 92;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 001 020 954 371 267 827 251 281 92 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 002 041 908 742 535 654 502 563 84;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 002 041 908 742 535 654 502 563 84 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 004 083 817 485 071 309 005 127 68;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 004 083 817 485 071 309 005 127 68 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 008 167 634 970 142 618 010 255 36;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 008 167 634 970 142 618 010 255 36 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 016 335 269 940 285 236 020 510 72;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 016 335 269 940 285 236 020 510 72 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 032 670 539 880 570 472 041 021 44;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 032 670 539 880 570 472 041 021 44 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 065 341 079 761 140 944 082 042 88;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 065 341 079 761 140 944 082 042 88 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 130 682 159 522 281 888 164 085 76;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 130 682 159 522 281 888 164 085 76 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 261 364 319 044 563 776 328 171 52;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 261 364 319 044 563 776 328 171 52 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 522 728 638 089 127 552 656 343 04;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 522 728 638 089 127 552 656 343 04 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 001 045 457 276 178 255 105 312 686 08;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 001 045 457 276 178 255 105 312 686 08 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 002 090 914 552 356 510 210 625 372 16;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 002 090 914 552 356 510 210 625 372 16 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 004 181 829 104 713 020 421 250 744 32;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 004 181 829 104 713 020 421 250 744 32 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 008 363 658 209 426 040 842 501 488 64;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 008 363 658 209 426 040 842 501 488 64 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 016 727 316 418 852 081 685 002 977 28;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 016 727 316 418 852 081 685 002 977 28 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 033 454 632 837 704 163 370 005 954 56;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 033 454 632 837 704 163 370 005 954 56 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 066 909 265 675 408 326 740 011 909 12;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 066 909 265 675 408 326 740 011 909 12 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 133 818 531 350 816 653 480 023 818 24;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 133 818 531 350 816 653 480 023 818 24 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 267 637 062 701 633 306 960 047 636 48;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 267 637 062 701 633 306 960 047 636 48 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 535 274 125 403 266 613 920 095 272 96;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 535 274 125 403 266 613 920 095 272 96 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 001 070 548 250 806 533 227 840 190 545 92;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 001 070 548 250 806 533 227 840 190 545 92 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 002 141 096 501 613 066 455 680 381 091 84;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 002 141 096 501 613 066 455 680 381 091 84 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 004 282 193 003 226 132 911 360 762 183 68;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 16 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