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

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 065 28;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 065 28 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 130 56;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 130 56 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 261 12;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 261 12 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 522 24;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 522 24 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 044 48;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 044 48 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 088 96;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 088 96 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 177 92;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 177 92 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 355 84;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 355 84 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 711 68;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 711 68 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 423 36;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 423 36 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 846 72;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 846 72 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 981 693 44;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 981 693 44 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 963 386 88;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 963 386 88 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 926 773 76;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 926 773 76 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 853 547 52;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 853 547 52 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 707 095 04;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 707 095 04 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 414 190 08;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 414 190 08 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 828 380 16;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 828 380 16 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 656 760 32;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 656 760 32 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 313 520 64;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 313 520 64 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 158 627 041 28;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 158 627 041 28 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 317 254 082 56;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 317 254 082 56 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 634 508 165 12;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 634 508 165 12 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 269 016 330 24;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 269 016 330 24 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 538 032 660 48;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 538 032 660 48 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 076 065 320 96;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 076 065 320 96 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 152 130 641 92;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 152 130 641 92 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 304 261 283 84;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 304 261 283 84 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 608 522 567 68;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 608 522 567 68 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 217 045 135 36;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 217 045 135 36 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 378 434 090 270 72;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 378 434 090 270 72 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 756 868 180 541 44;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 756 868 180 541 44 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 513 736 361 082 88;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 513 736 361 082 88 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 027 472 722 165 76;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 027 472 722 165 76 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 054 945 444 331 52;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 054 945 444 331 52 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 109 890 888 663 04;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 109 890 888 663 04 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 219 781 777 326 08;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 219 781 777 326 08 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 439 563 554 652 16;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 439 563 554 652 16 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 879 127 109 304 32;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 879 127 109 304 32 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 758 254 218 608 64;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 758 254 218 608 64 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 107 516 508 437 217 28;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 107 516 508 437 217 28 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 215 033 016 874 434 56;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 215 033 016 874 434 56 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 430 066 033 748 869 12;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 430 066 033 748 869 12 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 860 132 067 497 738 24;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 860 132 067 497 738 24 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 720 264 134 995 476 48;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 720 264 134 995 476 48 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 440 528 269 990 952 96;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 440 528 269 990 952 96 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 881 056 539 981 905 92;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 881 056 539 981 905 92 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 762 113 079 963 811 84;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 762 113 079 963 811 84 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 524 226 159 927 623 68;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 524 226 159 927 623 68 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 503 048 452 319 855 247 36;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 503 048 452 319 855 247 36 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 006 096 904 639 710 494 72;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 006 096 904 639 710 494 72 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 012 193 809 279 420 989 44;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 012 193 809 279 420 989 44 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 024 387 618 558 841 978 88;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 024 387 618 558 841 978 88 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 048 775 237 117 683 957 76;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 048 775 237 117 683 957 76 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 097 550 474 235 367 915 52;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 097 550 474 235 367 915 52 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 195 100 948 470 735 831 04;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 195 100 948 470 735 831 04 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 390 201 896 941 471 662 08;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 390 201 896 941 471 662 08 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 780 403 793 882 943 324 16;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 780 403 793 882 943 324 16 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 560 807 587 765 886 648 32;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 560 807 587 765 886 648 32 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 003 121 615 175 531 773 296 64;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 003 121 615 175 531 773 296 64 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 006 243 230 351 063 546 593 28;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 006 243 230 351 063 546 593 28 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 012 486 460 702 127 093 186 56;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 012 486 460 702 127 093 186 56 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 024 972 921 404 254 186 373 12;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 024 972 921 404 254 186 373 12 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 049 945 842 808 508 372 746 24;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 049 945 842 808 508 372 746 24 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 099 891 685 617 016 745 492 48;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 099 891 685 617 016 745 492 48 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 199 783 371 234 033 490 984 96;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 199 783 371 234 033 490 984 96 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 399 566 742 468 066 981 969 92;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 399 566 742 468 066 981 969 92 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 799 133 484 936 133 963 939 84;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 799 133 484 936 133 963 939 84 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 598 266 969 872 267 927 879 68;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 598 266 969 872 267 927 879 68 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 003 196 533 939 744 535 855 759 36;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 003 196 533 939 744 535 855 759 36 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 006 393 067 879 489 071 711 518 72;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 006 393 067 879 489 071 711 518 72 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 012 786 135 758 978 143 423 037 44;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 012 786 135 758 978 143 423 037 44 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 025 572 271 517 956 286 846 074 88;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 025 572 271 517 956 286 846 074 88 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 051 144 543 035 912 573 692 149 76;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 051 144 543 035 912 573 692 149 76 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 102 289 086 071 825 147 384 299 52;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 102 289 086 071 825 147 384 299 52 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 204 578 172 143 650 294 768 599 04;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 204 578 172 143 650 294 768 599 04 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 409 156 344 287 300 589 537 198 08;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 409 156 344 287 300 589 537 198 08 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 818 312 688 574 601 179 074 396 16;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 818 312 688 574 601 179 074 396 16 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 636 625 377 149 202 358 148 792 32;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 636 625 377 149 202 358 148 792 32 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 003 273 250 754 298 404 716 297 584 64;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 003 273 250 754 298 404 716 297 584 64 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 006 546 501 508 596 809 432 595 169 28;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 006 546 501 508 596 809 432 595 169 28 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 013 093 003 017 193 618 865 190 338 56;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 013 093 003 017 193 618 865 190 338 56 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 026 186 006 034 387 237 730 380 677 12;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 026 186 006 034 387 237 730 380 677 12 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 052 372 012 068 774 475 460 761 354 24;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 052 372 012 068 774 475 460 761 354 24 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 104 744 024 137 548 950 921 522 708 48;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 104 744 024 137 548 950 921 522 708 48 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 209 488 048 275 097 901 843 045 416 96;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 209 488 048 275 097 901 843 045 416 96 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 418 976 096 550 195 803 686 090 833 92;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 418 976 096 550 195 803 686 090 833 92 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 837 952 193 100 391 607 372 181 667 84;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 837 952 193 100 391 607 372 181 667 84 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 001 675 904 386 200 783 214 744 363 335 68;
  • 90) 0.500 000 000 000 000 000 000 000 000 001 675 904 386 200 783 214 744 363 335 68 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 003 351 808 772 401 566 429 488 726 671 36;
  • 91) 0.000 000 000 000 000 000 000 000 000 003 351 808 772 401 566 429 488 726 671 36 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 006 703 617 544 803 132 858 977 453 342 72;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 64 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