0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 041 08;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 041 08 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 082 16;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 082 16 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 164 32;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 164 32 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 328 64;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 328 64 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 657 28;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 657 28 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 314 56;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 314 56 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 629 12;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 629 12 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 258 24;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 258 24 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 516 48;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 516 48 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 485 032 96;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 485 032 96 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 970 065 92;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 970 065 92 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 940 131 84;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 940 131 84 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 880 263 68;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 880 263 68 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 760 527 36;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 760 527 36 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 521 054 72;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 521 054 72 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 055 042 109 44;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 055 042 109 44 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 110 084 218 88;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 110 084 218 88 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 220 168 437 76;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 220 168 437 76 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 440 336 875 52;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 440 336 875 52 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 880 673 751 04;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 880 673 751 04 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 761 347 502 08;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 761 347 502 08 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 522 695 004 16;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 522 695 004 16 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 045 390 008 32;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 045 390 008 32 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 090 780 016 64;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 090 780 016 64 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 181 560 033 28;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 181 560 033 28 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 363 120 066 56;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 363 120 066 56 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 726 240 133 12;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 726 240 133 12 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 452 480 266 24;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 452 480 266 24 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 904 960 532 48;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 904 960 532 48 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 809 921 064 96;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 809 921 064 96 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 251 619 842 129 92;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 251 619 842 129 92 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 503 239 684 259 84;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 503 239 684 259 84 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 006 479 368 519 68;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 006 479 368 519 68 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 012 958 737 039 36;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 012 958 737 039 36 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 025 917 474 078 72;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 025 917 474 078 72 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 051 834 948 157 44;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 051 834 948 157 44 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 103 669 896 314 88;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 103 669 896 314 88 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 207 339 792 629 76;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 207 339 792 629 76 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 414 679 585 259 52;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 414 679 585 259 52 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 829 359 170 519 04;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 829 359 170 519 04 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 001 658 718 341 038 08;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 001 658 718 341 038 08 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 003 317 436 682 076 16;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 003 317 436 682 076 16 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 006 634 873 364 152 32;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 006 634 873 364 152 32 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 013 269 746 728 304 64;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 013 269 746 728 304 64 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 026 539 493 456 609 28;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 026 539 493 456 609 28 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 053 078 986 913 218 56;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 053 078 986 913 218 56 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 106 157 973 826 437 12;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 106 157 973 826 437 12 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 212 315 947 652 874 24;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 212 315 947 652 874 24 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 424 631 895 305 748 48;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 424 631 895 305 748 48 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 849 263 790 611 496 96;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 849 263 790 611 496 96 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 001 698 527 581 222 993 92;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 001 698 527 581 222 993 92 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 003 397 055 162 445 987 84;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 003 397 055 162 445 987 84 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 006 794 110 324 891 975 68;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 006 794 110 324 891 975 68 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 013 588 220 649 783 951 36;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 013 588 220 649 783 951 36 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 027 176 441 299 567 902 72;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 027 176 441 299 567 902 72 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 054 352 882 599 135 805 44;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 054 352 882 599 135 805 44 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 108 705 765 198 271 610 88;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 108 705 765 198 271 610 88 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 217 411 530 396 543 221 76;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 217 411 530 396 543 221 76 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 434 823 060 793 086 443 52;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 434 823 060 793 086 443 52 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 869 646 121 586 172 887 04;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 869 646 121 586 172 887 04 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 001 739 292 243 172 345 774 08;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 001 739 292 243 172 345 774 08 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 003 478 584 486 344 691 548 16;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 003 478 584 486 344 691 548 16 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 006 957 168 972 689 383 096 32;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 006 957 168 972 689 383 096 32 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 013 914 337 945 378 766 192 64;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 013 914 337 945 378 766 192 64 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 027 828 675 890 757 532 385 28;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 027 828 675 890 757 532 385 28 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 055 657 351 781 515 064 770 56;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 055 657 351 781 515 064 770 56 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 111 314 703 563 030 129 541 12;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 111 314 703 563 030 129 541 12 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 222 629 407 126 060 259 082 24;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 222 629 407 126 060 259 082 24 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 445 258 814 252 120 518 164 48;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 445 258 814 252 120 518 164 48 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 890 517 628 504 241 036 328 96;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 890 517 628 504 241 036 328 96 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 001 781 035 257 008 482 072 657 92;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 001 781 035 257 008 482 072 657 92 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 003 562 070 514 016 964 145 315 84;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 003 562 070 514 016 964 145 315 84 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 007 124 141 028 033 928 290 631 68;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 007 124 141 028 033 928 290 631 68 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 014 248 282 056 067 856 581 263 36;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 014 248 282 056 067 856 581 263 36 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 028 496 564 112 135 713 162 526 72;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 028 496 564 112 135 713 162 526 72 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 056 993 128 224 271 426 325 053 44;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 056 993 128 224 271 426 325 053 44 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 113 986 256 448 542 852 650 106 88;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 113 986 256 448 542 852 650 106 88 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 227 972 512 897 085 705 300 213 76;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 227 972 512 897 085 705 300 213 76 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 455 945 025 794 171 410 600 427 52;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 455 945 025 794 171 410 600 427 52 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 911 890 051 588 342 821 200 855 04;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 911 890 051 588 342 821 200 855 04 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 001 823 780 103 176 685 642 401 710 08;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 001 823 780 103 176 685 642 401 710 08 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 003 647 560 206 353 371 284 803 420 16;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 003 647 560 206 353 371 284 803 420 16 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 007 295 120 412 706 742 569 606 840 32;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 007 295 120 412 706 742 569 606 840 32 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 014 590 240 825 413 485 139 213 680 64;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 014 590 240 825 413 485 139 213 680 64 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 029 180 481 650 826 970 278 427 361 28;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 029 180 481 650 826 970 278 427 361 28 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 058 360 963 301 653 940 556 854 722 56;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 058 360 963 301 653 940 556 854 722 56 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 116 721 926 603 307 881 113 709 445 12;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 116 721 926 603 307 881 113 709 445 12 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 233 443 853 206 615 762 227 418 890 24;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 233 443 853 206 615 762 227 418 890 24 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 466 887 706 413 231 524 454 837 780 48;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 466 887 706 413 231 524 454 837 780 48 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 933 775 412 826 463 048 909 675 560 96;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 933 775 412 826 463 048 909 675 560 96 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 001 867 550 825 652 926 097 819 351 121 92;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 020 54 converted to 64 bit double precision IEEE 754 binary floating point representation:

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

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

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

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100