0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 43;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 43 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 86;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 86 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 72;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 72 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 44;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 44 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 88;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 88 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 453 76;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 453 76 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 907 52;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 907 52 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 815 04;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 815 04 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 630 08;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 630 08 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 260 16;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 260 16 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 520 32;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 520 32 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 040 64;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 461 040 64 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 922 081 28;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 922 081 28 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 844 162 56;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 844 162 56 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 688 325 12;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 688 325 12 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 376 650 24;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 376 650 24 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 753 300 48;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 753 300 48 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 506 600 96;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 506 600 96 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 013 201 92;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 013 201 92 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 026 403 84;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 026 403 84 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 052 807 68;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 052 807 68 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 105 615 36;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 105 615 36 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 211 230 72;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 211 230 72 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 422 461 44;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 422 461 44 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 844 922 88;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 844 922 88 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 505 689 845 76;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 505 689 845 76 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 011 379 691 52;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 011 379 691 52 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 022 759 383 04;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 022 759 383 04 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 045 518 766 08;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 045 518 766 08 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 091 037 532 16;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 091 037 532 16 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 182 075 064 32;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 182 075 064 32 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 364 150 128 64;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 364 150 128 64 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 728 300 257 28;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 728 300 257 28 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 353 456 600 514 56;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 353 456 600 514 56 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 706 913 201 029 12;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 706 913 201 029 12 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 413 826 402 058 24;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 413 826 402 058 24 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 827 652 804 116 48;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 827 652 804 116 48 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 655 305 608 232 96;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 655 305 608 232 96 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 310 611 216 465 92;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 310 611 216 465 92 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 621 222 432 931 84;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 621 222 432 931 84 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 242 444 865 863 68;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 242 444 865 863 68 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 484 889 731 727 36;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 484 889 731 727 36 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 969 779 463 454 72;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 969 779 463 454 72 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 939 558 926 909 44;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 939 558 926 909 44 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 691 879 117 853 818 88;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 691 879 117 853 818 88 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 383 758 235 707 637 76;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 383 758 235 707 637 76 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 767 516 471 415 275 52;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 767 516 471 415 275 52 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 535 032 942 830 551 04;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 535 032 942 830 551 04 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 070 065 885 661 102 08;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 070 065 885 661 102 08 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 140 131 771 322 204 16;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 140 131 771 322 204 16 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 280 263 542 644 408 32;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 280 263 542 644 408 32 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 560 527 085 288 816 64;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 560 527 085 288 816 64 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 977 121 054 170 577 633 28;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 977 121 054 170 577 633 28 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 954 242 108 341 155 266 56;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 954 242 108 341 155 266 56 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 908 484 216 682 310 533 12;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 908 484 216 682 310 533 12 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 816 968 433 364 621 066 24;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 816 968 433 364 621 066 24 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 633 936 866 729 242 132 48;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 633 936 866 729 242 132 48 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 267 873 733 458 484 264 96;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 267 873 733 458 484 264 96 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 535 747 466 916 968 529 92;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 535 747 466 916 968 529 92 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 071 494 933 833 937 059 84;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 071 494 933 833 937 059 84 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 142 989 867 667 874 119 68;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 142 989 867 667 874 119 68 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 285 979 735 335 748 239 36;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 285 979 735 335 748 239 36 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 571 959 470 671 496 478 72;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 571 959 470 671 496 478 72 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 001 143 918 941 342 992 957 44;
  • 65) 0.707 934 215 664 863 586 425 781 250 001 143 918 941 342 992 957 44 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 002 287 837 882 685 985 914 88;
  • 66) 0.415 868 431 329 727 172 851 562 500 002 287 837 882 685 985 914 88 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 004 575 675 765 371 971 829 76;
  • 67) 0.831 736 862 659 454 345 703 125 000 004 575 675 765 371 971 829 76 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 009 151 351 530 743 943 659 52;
  • 68) 0.663 473 725 318 908 691 406 250 000 009 151 351 530 743 943 659 52 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 018 302 703 061 487 887 319 04;
  • 69) 0.326 947 450 637 817 382 812 500 000 018 302 703 061 487 887 319 04 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 036 605 406 122 975 774 638 08;
  • 70) 0.653 894 901 275 634 765 625 000 000 036 605 406 122 975 774 638 08 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 073 210 812 245 951 549 276 16;
  • 71) 0.307 789 802 551 269 531 250 000 000 073 210 812 245 951 549 276 16 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 146 421 624 491 903 098 552 32;
  • 72) 0.615 579 605 102 539 062 500 000 000 146 421 624 491 903 098 552 32 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 292 843 248 983 806 197 104 64;
  • 73) 0.231 159 210 205 078 125 000 000 000 292 843 248 983 806 197 104 64 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 585 686 497 967 612 394 209 28;
  • 74) 0.462 318 420 410 156 250 000 000 000 585 686 497 967 612 394 209 28 × 2 = 0 + 0.924 636 840 820 312 500 000 000 001 171 372 995 935 224 788 418 56;
  • 75) 0.924 636 840 820 312 500 000 000 001 171 372 995 935 224 788 418 56 × 2 = 1 + 0.849 273 681 640 625 000 000 000 002 342 745 991 870 449 576 837 12;
  • 76) 0.849 273 681 640 625 000 000 000 002 342 745 991 870 449 576 837 12 × 2 = 1 + 0.698 547 363 281 250 000 000 000 004 685 491 983 740 899 153 674 24;
  • 77) 0.698 547 363 281 250 000 000 000 004 685 491 983 740 899 153 674 24 × 2 = 1 + 0.397 094 726 562 500 000 000 000 009 370 983 967 481 798 307 348 48;
  • 78) 0.397 094 726 562 500 000 000 000 009 370 983 967 481 798 307 348 48 × 2 = 0 + 0.794 189 453 125 000 000 000 000 018 741 967 934 963 596 614 696 96;
  • 79) 0.794 189 453 125 000 000 000 000 018 741 967 934 963 596 614 696 96 × 2 = 1 + 0.588 378 906 250 000 000 000 000 037 483 935 869 927 193 229 393 92;
  • 80) 0.588 378 906 250 000 000 000 000 037 483 935 869 927 193 229 393 92 × 2 = 1 + 0.176 757 812 500 000 000 000 000 074 967 871 739 854 386 458 787 84;
  • 81) 0.176 757 812 500 000 000 000 000 074 967 871 739 854 386 458 787 84 × 2 = 0 + 0.353 515 625 000 000 000 000 000 149 935 743 479 708 772 917 575 68;
  • 82) 0.353 515 625 000 000 000 000 000 149 935 743 479 708 772 917 575 68 × 2 = 0 + 0.707 031 250 000 000 000 000 000 299 871 486 959 417 545 835 151 36;
  • 83) 0.707 031 250 000 000 000 000 000 299 871 486 959 417 545 835 151 36 × 2 = 1 + 0.414 062 500 000 000 000 000 000 599 742 973 918 835 091 670 302 72;
  • 84) 0.414 062 500 000 000 000 000 000 599 742 973 918 835 091 670 302 72 × 2 = 0 + 0.828 125 000 000 000 000 000 001 199 485 947 837 670 183 340 605 44;
  • 85) 0.828 125 000 000 000 000 000 001 199 485 947 837 670 183 340 605 44 × 2 = 1 + 0.656 250 000 000 000 000 000 002 398 971 895 675 340 366 681 210 88;
  • 86) 0.656 250 000 000 000 000 000 002 398 971 895 675 340 366 681 210 88 × 2 = 1 + 0.312 500 000 000 000 000 000 004 797 943 791 350 680 733 362 421 76;
  • 87) 0.312 500 000 000 000 000 000 004 797 943 791 350 680 733 362 421 76 × 2 = 0 + 0.625 000 000 000 000 000 000 009 595 887 582 701 361 466 724 843 52;
  • 88) 0.625 000 000 000 000 000 000 009 595 887 582 701 361 466 724 843 52 × 2 = 1 + 0.250 000 000 000 000 000 000 019 191 775 165 402 722 933 449 687 04;
  • 89) 0.250 000 000 000 000 000 000 019 191 775 165 402 722 933 449 687 04 × 2 = 0 + 0.500 000 000 000 000 000 000 038 383 550 330 805 445 866 899 374 08;
  • 90) 0.500 000 000 000 000 000 000 038 383 550 330 805 445 866 899 374 08 × 2 = 1 + 0.000 000 000 000 000 000 000 076 767 100 661 610 891 733 798 748 16;
  • 91) 0.000 000 000 000 000 000 000 076 767 100 661 610 891 733 798 748 16 × 2 = 0 + 0.000 000 000 000 000 000 000 153 534 201 323 221 783 467 597 496 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 215 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 123 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Why Do Colleagues From Other Teams Have Better Results? Gemini Notebook YouTube Video of 11.31 minutes

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