0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 814 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 814 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 629 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 629 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 258 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 258 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 516 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 516 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 033 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 033 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 546 067 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 546 067 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 092 134 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 092 134 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 184 268 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 184 268 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 368 537 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 368 537 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 737 075 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 737 075 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 474 150 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 474 150 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 234 948 300 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 234 948 300 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 469 896 601 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 469 896 601 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 939 793 203 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 939 793 203 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 879 586 406 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 879 586 406 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 759 172 812 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 759 172 812 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 518 345 625 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 518 345 625 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 036 691 251 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 036 691 251 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 073 382 502 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 073 382 502 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 146 765 004 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 146 765 004 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 293 530 009 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 293 530 009 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 587 060 019 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 587 060 019 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 174 120 038 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 174 120 038 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 348 240 076 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 348 240 076 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 700 696 480 153 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 700 696 480 153 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 401 392 960 307 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 401 392 960 307 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 802 785 920 614 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 802 785 920 614 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 605 571 841 228 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 605 571 841 228 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 211 143 682 457 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 211 143 682 457 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 422 287 364 915 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 422 287 364 915 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 844 574 729 830 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 844 574 729 830 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 689 149 459 660 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 689 149 459 660 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 378 298 919 321 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 378 298 919 321 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 030 756 597 838 643 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 030 756 597 838 643 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 061 513 195 677 286 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 061 513 195 677 286 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 123 026 391 354 572 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 123 026 391 354 572 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 246 052 782 709 145 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 246 052 782 709 145 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 492 105 565 418 291 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 492 105 565 418 291 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 984 211 130 836 582 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 984 211 130 836 582 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 968 422 261 673 164 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 968 422 261 673 164 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 936 844 523 346 329 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 936 844 523 346 329 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 873 689 046 692 659 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 873 689 046 692 659 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 747 378 093 385 318 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 747 378 093 385 318 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 494 756 186 770 636 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 494 756 186 770 636 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 998 989 512 373 541 273 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 998 989 512 373 541 273 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 997 979 024 747 082 547 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 997 979 024 747 082 547 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 995 958 049 494 165 094 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 995 958 049 494 165 094 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 991 916 098 988 330 188 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 991 916 098 988 330 188 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 983 832 197 976 660 377 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 983 832 197 976 660 377 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 967 664 395 953 320 755 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 967 664 395 953 320 755 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 935 328 791 906 641 510 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 935 328 791 906 641 510 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 870 657 583 813 283 020 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 870 657 583 813 283 020 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 741 315 167 626 566 041 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 741 315 167 626 566 041 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 482 630 335 253 132 083 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 482 630 335 253 132 083 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 998 965 260 670 506 264 166 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 998 965 260 670 506 264 166 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 997 930 521 341 012 528 332 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 997 930 521 341 012 528 332 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 995 861 042 682 025 056 665 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 995 861 042 682 025 056 665 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 991 722 085 364 050 113 331 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 991 722 085 364 050 113 331 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 983 444 170 728 100 226 662 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 983 444 170 728 100 226 662 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 966 888 341 456 200 453 324 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 966 888 341 456 200 453 324 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 933 776 682 912 400 906 649 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 933 776 682 912 400 906 649 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 867 553 365 824 801 813 299 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 867 553 365 824 801 813 299 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 735 106 731 649 603 626 598 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 735 106 731 649 603 626 598 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 470 213 463 299 207 253 196 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 470 213 463 299 207 253 196 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 998 940 426 926 598 414 506 393 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 998 940 426 926 598 414 506 393 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 997 880 853 853 196 829 012 787 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 997 880 853 853 196 829 012 787 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 995 761 707 706 393 658 025 574 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 995 761 707 706 393 658 025 574 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 991 523 415 412 787 316 051 148 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 991 523 415 412 787 316 051 148 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 983 046 830 825 574 632 102 297 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 983 046 830 825 574 632 102 297 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 966 093 661 651 149 264 204 595 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 966 093 661 651 149 264 204 595 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 932 187 323 302 298 528 409 190 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 932 187 323 302 298 528 409 190 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 864 374 646 604 597 056 818 380 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 864 374 646 604 597 056 818 380 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 728 749 293 209 194 113 636 761 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 728 749 293 209 194 113 636 761 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 457 498 586 418 388 227 273 523 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 457 498 586 418 388 227 273 523 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 998 914 997 172 836 776 454 547 046 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 998 914 997 172 836 776 454 547 046 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 997 829 994 345 673 552 909 094 092 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 997 829 994 345 673 552 909 094 092 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 995 659 988 691 347 105 818 188 185 6;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 995 659 988 691 347 105 818 188 185 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 991 319 977 382 694 211 636 376 371 2;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 991 319 977 382 694 211 636 376 371 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 982 639 954 765 388 423 272 752 742 4;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 982 639 954 765 388 423 272 752 742 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 965 279 909 530 776 846 545 505 484 8;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 965 279 909 530 776 846 545 505 484 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 930 559 819 061 553 693 091 010 969 6;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 930 559 819 061 553 693 091 010 969 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 861 119 638 123 107 386 182 021 939 2;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 861 119 638 123 107 386 182 021 939 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 722 239 276 246 214 772 364 043 878 4;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 722 239 276 246 214 772 364 043 878 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 444 478 552 492 429 544 728 087 756 8;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 444 478 552 492 429 544 728 087 756 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 998 888 957 104 984 859 089 456 175 513 6;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 998 888 957 104 984 859 089 456 175 513 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 997 777 914 209 969 718 178 912 351 027 2;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 997 777 914 209 969 718 178 912 351 027 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 995 555 828 419 939 436 357 824 702 054 4;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 995 555 828 419 939 436 357 824 702 054 4 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 991 111 656 839 878 872 715 649 404 108 8;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 991 111 656 839 878 872 715 649 404 108 8 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 982 223 313 679 757 745 431 298 808 217 6;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 982 223 313 679 757 745 431 298 808 217 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 964 446 627 359 515 490 862 597 616 435 2;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 964 446 627 359 515 490 862 597 616 435 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 928 893 254 719 030 981 725 195 232 870 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 407 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 400 1 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: Reputation: Bridge Between Company's Actions & Market Value. NotebookLM YouTube Video of 6.55 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