0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 863 7;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 863 7 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 727 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 727 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 454 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 454 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 909 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 909 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 819 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 819 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 638 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 638 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 276 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 276 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 553 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 553 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 405 107 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 405 107 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 810 214 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 810 214 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 620 428 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 620 428 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 240 857 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 240 857 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 481 715 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 481 715 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 963 430 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 963 430 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 926 860 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 926 860 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 853 721 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 853 721 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 707 443 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 707 443 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 414 886 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 414 886 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 829 772 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 829 772 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 429 659 545 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 429 659 545 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 859 319 091 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 859 319 091 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 718 638 182 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 718 638 182 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 437 276 364 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 437 276 364 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 874 552 729 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 874 552 729 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 749 105 459 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 749 105 459 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 498 210 918 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 498 210 918 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 996 421 836 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 996 421 836 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 992 843 673 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 992 843 673 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 985 687 347 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 985 687 347 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 815 971 374 694 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 815 971 374 694 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 631 942 749 388 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 631 942 749 388 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 263 885 498 777 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 263 885 498 777 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 527 770 997 555 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 527 770 997 555 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 055 541 995 110 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 055 541 995 110 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 111 083 990 220 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 111 083 990 220 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 222 167 980 441 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 222 167 980 441 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 444 335 960 883 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 444 335 960 883 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 888 671 921 766 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 888 671 921 766 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 777 343 843 532 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 777 343 843 532 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 251 554 687 687 065 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 251 554 687 687 065 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 503 109 375 374 131 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 503 109 375 374 131 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 006 218 750 748 262 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 006 218 750 748 262 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 012 437 501 496 524 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 012 437 501 496 524 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 024 875 002 993 049 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 024 875 002 993 049 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 049 750 005 986 099 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 049 750 005 986 099 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 099 500 011 972 198 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 099 500 011 972 198 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 199 000 023 944 396 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 199 000 023 944 396 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 398 000 047 888 793 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 398 000 047 888 793 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 796 000 095 777 587 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 796 000 095 777 587 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 001 592 000 191 555 174 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 001 592 000 191 555 174 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 003 184 000 383 110 348 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 003 184 000 383 110 348 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 006 368 000 766 220 697 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 006 368 000 766 220 697 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 012 736 001 532 441 395 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 012 736 001 532 441 395 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 025 472 003 064 882 790 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 025 472 003 064 882 790 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 050 944 006 129 765 580 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 050 944 006 129 765 580 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 101 888 012 259 531 161 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 101 888 012 259 531 161 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 203 776 024 519 062 323 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 203 776 024 519 062 323 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 407 552 049 038 124 646 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 407 552 049 038 124 646 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 815 104 098 076 249 292 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 815 104 098 076 249 292 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 001 630 208 196 152 498 585 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 001 630 208 196 152 498 585 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 003 260 416 392 304 997 171 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 003 260 416 392 304 997 171 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 006 520 832 784 609 994 342 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 006 520 832 784 609 994 342 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 013 041 665 569 219 988 684 8;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 013 041 665 569 219 988 684 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 026 083 331 138 439 977 369 6;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 026 083 331 138 439 977 369 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 052 166 662 276 879 954 739 2;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 052 166 662 276 879 954 739 2 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 104 333 324 553 759 909 478 4;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 104 333 324 553 759 909 478 4 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 208 666 649 107 519 818 956 8;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 208 666 649 107 519 818 956 8 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 417 333 298 215 039 637 913 6;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 417 333 298 215 039 637 913 6 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 834 666 596 430 079 275 827 2;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 834 666 596 430 079 275 827 2 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 001 669 333 192 860 158 551 654 4;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 001 669 333 192 860 158 551 654 4 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 003 338 666 385 720 317 103 308 8;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 003 338 666 385 720 317 103 308 8 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 006 677 332 771 440 634 206 617 6;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 006 677 332 771 440 634 206 617 6 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 013 354 665 542 881 268 413 235 2;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 013 354 665 542 881 268 413 235 2 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 026 709 331 085 762 536 826 470 4;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 026 709 331 085 762 536 826 470 4 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 053 418 662 171 525 073 652 940 8;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 053 418 662 171 525 073 652 940 8 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 106 837 324 343 050 147 305 881 6;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 106 837 324 343 050 147 305 881 6 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 213 674 648 686 100 294 611 763 2;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 213 674 648 686 100 294 611 763 2 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 427 349 297 372 200 589 223 526 4;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 427 349 297 372 200 589 223 526 4 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 854 698 594 744 401 178 447 052 8;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 854 698 594 744 401 178 447 052 8 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 001 709 397 189 488 802 356 894 105 6;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 001 709 397 189 488 802 356 894 105 6 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 003 418 794 378 977 604 713 788 211 2;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 003 418 794 378 977 604 713 788 211 2 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 006 837 588 757 955 209 427 576 422 4;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 006 837 588 757 955 209 427 576 422 4 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 013 675 177 515 910 418 855 152 844 8;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 013 675 177 515 910 418 855 152 844 8 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 027 350 355 031 820 837 710 305 689 6;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 027 350 355 031 820 837 710 305 689 6 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 054 700 710 063 641 675 420 611 379 2;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 054 700 710 063 641 675 420 611 379 2 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 109 401 420 127 283 350 841 222 758 4;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 109 401 420 127 283 350 841 222 758 4 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 218 802 840 254 566 701 682 445 516 8;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 218 802 840 254 566 701 682 445 516 8 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 437 605 680 509 133 403 364 891 033 6;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 437 605 680 509 133 403 364 891 033 6 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 875 211 361 018 266 806 729 782 067 2;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 875 211 361 018 266 806 729 782 067 2 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 001 750 422 722 036 533 613 459 564 134 4;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 001 750 422 722 036 533 613 459 564 134 4 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 003 500 845 444 073 067 226 919 128 268 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 85 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 431 14 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: The Art of Strategic Ambiguity. NotebookLM YouTube Video of 6.59 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