0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 782;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 782 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 143 564;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 143 564 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 287 128;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 287 128 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 574 256;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 574 256 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 148 512;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 148 512 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 297 024;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 297 024 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 594 048;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 594 048 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 188 096;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 188 096 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 376 192;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 376 192 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 752 384;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 752 384 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 504 768;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 504 768 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 043 009 536;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 043 009 536 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 086 019 072;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 086 019 072 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 172 038 144;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 172 038 144 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 344 076 288;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 344 076 288 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 688 152 576;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 688 152 576 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 376 305 152;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 376 305 152 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 752 610 304;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 752 610 304 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 505 220 608;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 505 220 608 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 883 010 441 216;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 883 010 441 216 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 766 020 882 432;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 766 020 882 432 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 532 041 764 864;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 532 041 764 864 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 064 083 529 728;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 064 083 529 728 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 128 167 059 456;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 128 167 059 456 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 256 334 118 912;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 256 334 118 912 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 512 668 237 824;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 512 668 237 824 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 025 336 475 648;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 025 336 475 648 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 050 672 951 296;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 050 672 951 296 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 500 101 345 902 592;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 500 101 345 902 592 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 000 202 691 805 184;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 000 202 691 805 184 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 000 405 383 610 368;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 000 405 383 610 368 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 000 810 767 220 736;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 000 810 767 220 736 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 001 621 534 441 472;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 001 621 534 441 472 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 003 243 068 882 944;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 003 243 068 882 944 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 006 486 137 765 888;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 006 486 137 765 888 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 012 972 275 531 776;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 012 972 275 531 776 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 025 944 551 063 552;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 025 944 551 063 552 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 051 889 102 127 104;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 051 889 102 127 104 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 000 103 778 204 254 208;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 000 103 778 204 254 208 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 000 207 556 408 508 416;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 000 207 556 408 508 416 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 000 415 112 817 016 832;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 000 415 112 817 016 832 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 000 830 225 634 033 664;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 000 830 225 634 033 664 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 001 660 451 268 067 328;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 001 660 451 268 067 328 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 003 320 902 536 134 656;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 003 320 902 536 134 656 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 006 641 805 072 269 312;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 006 641 805 072 269 312 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 013 283 610 144 538 624;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 013 283 610 144 538 624 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 026 567 220 289 077 248;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 026 567 220 289 077 248 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 053 134 440 578 154 496;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 053 134 440 578 154 496 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 000 106 268 881 156 308 992;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 000 106 268 881 156 308 992 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 000 212 537 762 312 617 984;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 000 212 537 762 312 617 984 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 000 425 075 524 625 235 968;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 000 425 075 524 625 235 968 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 000 850 151 049 250 471 936;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 000 850 151 049 250 471 936 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 001 700 302 098 500 943 872;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 001 700 302 098 500 943 872 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 003 400 604 197 001 887 744;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 003 400 604 197 001 887 744 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 006 801 208 394 003 775 488;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 006 801 208 394 003 775 488 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 013 602 416 788 007 550 976;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 013 602 416 788 007 550 976 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 027 204 833 576 015 101 952;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 027 204 833 576 015 101 952 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 054 409 667 152 030 203 904;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 054 409 667 152 030 203 904 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 000 108 819 334 304 060 407 808;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 000 108 819 334 304 060 407 808 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 000 217 638 668 608 120 815 616;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 000 217 638 668 608 120 815 616 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 000 435 277 337 216 241 631 232;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 000 435 277 337 216 241 631 232 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 000 870 554 674 432 483 262 464;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 000 870 554 674 432 483 262 464 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 001 741 109 348 864 966 524 928;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 001 741 109 348 864 966 524 928 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 003 482 218 697 729 933 049 856;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 003 482 218 697 729 933 049 856 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 006 964 437 395 459 866 099 712;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 006 964 437 395 459 866 099 712 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 013 928 874 790 919 732 199 424;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 013 928 874 790 919 732 199 424 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 027 857 749 581 839 464 398 848;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 027 857 749 581 839 464 398 848 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 055 715 499 163 678 928 797 696;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 055 715 499 163 678 928 797 696 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 000 111 430 998 327 357 857 595 392;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 000 111 430 998 327 357 857 595 392 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 000 222 861 996 654 715 715 190 784;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 000 222 861 996 654 715 715 190 784 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 000 445 723 993 309 431 430 381 568;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 000 445 723 993 309 431 430 381 568 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 000 891 447 986 618 862 860 763 136;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 000 891 447 986 618 862 860 763 136 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 001 782 895 973 237 725 721 526 272;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 001 782 895 973 237 725 721 526 272 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 003 565 791 946 475 451 443 052 544;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 003 565 791 946 475 451 443 052 544 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 007 131 583 892 950 902 886 105 088;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 007 131 583 892 950 902 886 105 088 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 014 263 167 785 901 805 772 210 176;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 014 263 167 785 901 805 772 210 176 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 028 526 335 571 803 611 544 420 352;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 028 526 335 571 803 611 544 420 352 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 057 052 671 143 607 223 088 840 704;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 057 052 671 143 607 223 088 840 704 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 000 114 105 342 287 214 446 177 681 408;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 000 114 105 342 287 214 446 177 681 408 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 000 228 210 684 574 428 892 355 362 816;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 000 228 210 684 574 428 892 355 362 816 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 000 456 421 369 148 857 784 710 725 632;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 000 456 421 369 148 857 784 710 725 632 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 000 912 842 738 297 715 569 421 451 264;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 000 912 842 738 297 715 569 421 451 264 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 001 825 685 476 595 431 138 842 902 528;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 001 825 685 476 595 431 138 842 902 528 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 003 651 370 953 190 862 277 685 805 056;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 003 651 370 953 190 862 277 685 805 056 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 007 302 741 906 381 724 555 371 610 112;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 007 302 741 906 381 724 555 371 610 112 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 014 605 483 812 763 449 110 743 220 224;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 014 605 483 812 763 449 110 743 220 224 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 029 210 967 625 526 898 221 486 440 448;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 029 210 967 625 526 898 221 486 440 448 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 058 421 935 251 053 796 442 972 880 896;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 058 421 935 251 053 796 442 972 880 896 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 116 843 870 502 107 592 885 945 761 792;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 116 843 870 502 107 592 885 945 761 792 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 233 687 741 004 215 185 771 891 523 584;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 233 687 741 004 215 185 771 891 523 584 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 467 375 482 008 430 371 543 783 047 168;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

6. Normalize the binary representation of the number.

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

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


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-39 + 1 023)(10) =

984(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

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

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


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

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

Exponent (11 bits) =
011 1101 1000

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


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 891 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

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 797 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 10, 2026 [October]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: October - by our visitors

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