0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 068 84;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 068 84 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 137 68;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 137 68 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 275 36;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 275 36 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 550 72;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 550 72 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 101 44;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 101 44 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 202 88;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 202 88 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 405 76;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 405 76 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 811 52;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 811 52 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 873 623 04;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 873 623 04 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 747 246 08;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 747 246 08 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 494 492 16;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 494 492 16 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 988 984 32;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 988 984 32 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 977 968 64;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 977 968 64 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 955 937 28;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 955 937 28 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 911 874 56;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 911 874 56 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 823 749 12;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 823 749 12 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 647 498 24;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 647 498 24 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 895 294 996 48;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 895 294 996 48 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 790 589 992 96;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 790 589 992 96 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 581 179 985 92;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 581 179 985 92 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 162 359 971 84;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 162 359 971 84 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 324 719 943 68;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 324 719 943 68 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 649 439 887 36;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 649 439 887 36 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 298 879 774 72;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 298 879 774 72 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 597 759 549 44;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 597 759 549 44 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 195 519 098 88;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 195 519 098 88 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 391 038 197 76;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 391 038 197 76 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 782 076 395 52;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 782 076 395 52 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 345 564 152 791 04;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 345 564 152 791 04 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 691 128 305 582 08;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 691 128 305 582 08 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 382 256 611 164 16;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 382 256 611 164 16 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 764 513 222 328 32;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 764 513 222 328 32 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 529 026 444 656 64;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 529 026 444 656 64 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 058 052 889 313 28;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 058 052 889 313 28 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 116 105 778 626 56;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 116 105 778 626 56 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 232 211 557 253 12;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 232 211 557 253 12 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 464 423 114 506 24;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 464 423 114 506 24 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 928 846 229 012 48;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 928 846 229 012 48 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 777 857 692 458 024 96;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 777 857 692 458 024 96 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 555 715 384 916 049 92;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 555 715 384 916 049 92 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 111 430 769 832 099 84;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 111 430 769 832 099 84 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 222 861 539 664 199 68;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 222 861 539 664 199 68 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 445 723 079 328 399 36;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 445 723 079 328 399 36 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 891 446 158 656 798 72;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 891 446 158 656 798 72 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 782 892 317 313 597 44;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 782 892 317 313 597 44 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 565 784 634 627 194 88;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 565 784 634 627 194 88 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 063 131 569 269 254 389 76;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 063 131 569 269 254 389 76 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 126 263 138 538 508 779 52;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 126 263 138 538 508 779 52 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 252 526 277 077 017 559 04;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 252 526 277 077 017 559 04 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 505 052 554 154 035 118 08;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 505 052 554 154 035 118 08 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 010 105 108 308 070 236 16;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 010 105 108 308 070 236 16 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 020 210 216 616 140 472 32;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 020 210 216 616 140 472 32 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 040 420 433 232 280 944 64;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 040 420 433 232 280 944 64 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 080 840 866 464 561 889 28;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 080 840 866 464 561 889 28 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 161 681 732 929 123 778 56;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 161 681 732 929 123 778 56 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 323 363 465 858 247 557 12;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 323 363 465 858 247 557 12 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 646 726 931 716 495 114 24;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 646 726 931 716 495 114 24 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 001 293 453 863 432 990 228 48;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 001 293 453 863 432 990 228 48 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 002 586 907 726 865 980 456 96;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 002 586 907 726 865 980 456 96 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 005 173 815 453 731 960 913 92;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 005 173 815 453 731 960 913 92 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 010 347 630 907 463 921 827 84;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 010 347 630 907 463 921 827 84 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 020 695 261 814 927 843 655 68;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 020 695 261 814 927 843 655 68 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 041 390 523 629 855 687 311 36;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 041 390 523 629 855 687 311 36 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 082 781 047 259 711 374 622 72;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 082 781 047 259 711 374 622 72 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 165 562 094 519 422 749 245 44;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 165 562 094 519 422 749 245 44 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 331 124 189 038 845 498 490 88;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 331 124 189 038 845 498 490 88 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 662 248 378 077 690 996 981 76;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 662 248 378 077 690 996 981 76 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 001 324 496 756 155 381 993 963 52;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 001 324 496 756 155 381 993 963 52 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 002 648 993 512 310 763 987 927 04;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 002 648 993 512 310 763 987 927 04 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 005 297 987 024 621 527 975 854 08;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 005 297 987 024 621 527 975 854 08 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 010 595 974 049 243 055 951 708 16;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 010 595 974 049 243 055 951 708 16 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 021 191 948 098 486 111 903 416 32;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 021 191 948 098 486 111 903 416 32 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 042 383 896 196 972 223 806 832 64;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 042 383 896 196 972 223 806 832 64 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 084 767 792 393 944 447 613 665 28;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 084 767 792 393 944 447 613 665 28 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 169 535 584 787 888 895 227 330 56;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 169 535 584 787 888 895 227 330 56 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 339 071 169 575 777 790 454 661 12;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 339 071 169 575 777 790 454 661 12 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 678 142 339 151 555 580 909 322 24;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 678 142 339 151 555 580 909 322 24 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 001 356 284 678 303 111 161 818 644 48;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 001 356 284 678 303 111 161 818 644 48 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 002 712 569 356 606 222 323 637 288 96;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 002 712 569 356 606 222 323 637 288 96 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 005 425 138 713 212 444 647 274 577 92;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 005 425 138 713 212 444 647 274 577 92 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 010 850 277 426 424 889 294 549 155 84;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 010 850 277 426 424 889 294 549 155 84 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 021 700 554 852 849 778 589 098 311 68;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 021 700 554 852 849 778 589 098 311 68 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 043 401 109 705 699 557 178 196 623 36;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 043 401 109 705 699 557 178 196 623 36 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 086 802 219 411 399 114 356 393 246 72;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 086 802 219 411 399 114 356 393 246 72 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 173 604 438 822 798 228 712 786 493 44;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 173 604 438 822 798 228 712 786 493 44 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 347 208 877 645 596 457 425 572 986 88;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 347 208 877 645 596 457 425 572 986 88 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 694 417 755 291 192 914 851 145 973 76;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 694 417 755 291 192 914 851 145 973 76 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 001 388 835 510 582 385 829 702 291 947 52;
  • 89) 0.250 000 000 000 000 000 000 000 000 001 388 835 510 582 385 829 702 291 947 52 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 002 777 671 021 164 771 659 404 583 895 04;
  • 90) 0.500 000 000 000 000 000 000 000 000 002 777 671 021 164 771 659 404 583 895 04 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 005 555 342 042 329 543 318 809 167 790 08;
  • 91) 0.000 000 000 000 000 000 000 000 000 005 555 342 042 329 543 318 809 167 790 08 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 011 110 684 084 659 086 637 618 335 580 16;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 034 42 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