0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 606;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 606 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 212;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 212 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 424;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 424 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 356 848;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 356 848 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 713 696;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 713 696 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 427 392;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 427 392 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 854 784;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 854 784 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 709 568;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 709 568 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 419 136;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 419 136 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 838 272;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 838 272 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 676 544;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 676 544 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 353 088;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 353 088 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 706 176;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 706 176 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 412 352;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 412 352 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 674 824 704;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 674 824 704 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 349 649 408;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 349 649 408 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 699 298 816;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 699 298 816 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 398 597 632;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 398 597 632 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 797 195 264;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 797 195 264 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 594 390 528;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 594 390 528 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 188 781 056;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 188 781 056 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 377 562 112;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 377 562 112 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 755 124 224;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 755 124 224 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 510 248 448;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 510 248 448 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 739 020 496 896;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 739 020 496 896 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 478 040 993 792;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 478 040 993 792 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 956 081 987 584;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 956 081 987 584 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 912 163 975 168;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 912 163 975 168 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 824 327 950 336;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 824 327 950 336 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 648 655 900 672;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 648 655 900 672 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 297 311 801 344;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 297 311 801 344 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 086 594 623 602 688;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 086 594 623 602 688 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 173 189 247 205 376;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 173 189 247 205 376 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 346 378 494 410 752;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 346 378 494 410 752 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 692 756 988 821 504;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 692 756 988 821 504 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 385 513 977 643 008;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 385 513 977 643 008 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 771 027 955 286 016;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 771 027 955 286 016 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 542 055 910 572 032;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 542 055 910 572 032 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 084 111 821 144 064;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 084 111 821 144 064 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 168 223 642 288 128;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 168 223 642 288 128 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 336 447 284 576 256;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 336 447 284 576 256 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 584 672 894 569 152 512;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 584 672 894 569 152 512 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 169 345 789 138 305 024;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 169 345 789 138 305 024 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 338 691 578 276 610 048;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 338 691 578 276 610 048 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 677 383 156 553 220 096;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 677 383 156 553 220 096 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 354 766 313 106 440 192;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 354 766 313 106 440 192 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 709 532 626 212 880 384;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 709 532 626 212 880 384 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 419 065 252 425 760 768;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 419 065 252 425 760 768 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 838 130 504 851 521 536;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 838 130 504 851 521 536 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 676 261 009 703 043 072;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 676 261 009 703 043 072 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 352 522 019 406 086 144;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 352 522 019 406 086 144 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 486 705 044 038 812 172 288;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 486 705 044 038 812 172 288 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 973 410 088 077 624 344 576;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 973 410 088 077 624 344 576 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 946 820 176 155 248 689 152;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 946 820 176 155 248 689 152 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 893 640 352 310 497 378 304;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 893 640 352 310 497 378 304 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 787 280 704 620 994 756 608;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 787 280 704 620 994 756 608 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 574 561 409 241 989 513 216;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 574 561 409 241 989 513 216 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 149 122 818 483 979 026 432;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 149 122 818 483 979 026 432 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 298 245 636 967 958 052 864;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 298 245 636 967 958 052 864 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 596 491 273 935 916 105 728;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 596 491 273 935 916 105 728 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 192 982 547 871 832 211 456;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 192 982 547 871 832 211 456 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 385 965 095 743 664 422 912;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 385 965 095 743 664 422 912 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 996 771 930 191 487 328 845 824;
  • 64) 0.353 967 107 832 431 793 212 890 624 996 771 930 191 487 328 845 824 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 993 543 860 382 974 657 691 648;
  • 65) 0.707 934 215 664 863 586 425 781 249 993 543 860 382 974 657 691 648 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 987 087 720 765 949 315 383 296;
  • 66) 0.415 868 431 329 727 172 851 562 499 987 087 720 765 949 315 383 296 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 974 175 441 531 898 630 766 592;
  • 67) 0.831 736 862 659 454 345 703 124 999 974 175 441 531 898 630 766 592 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 948 350 883 063 797 261 533 184;
  • 68) 0.663 473 725 318 908 691 406 249 999 948 350 883 063 797 261 533 184 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 896 701 766 127 594 523 066 368;
  • 69) 0.326 947 450 637 817 382 812 499 999 896 701 766 127 594 523 066 368 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 793 403 532 255 189 046 132 736;
  • 70) 0.653 894 901 275 634 765 624 999 999 793 403 532 255 189 046 132 736 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 586 807 064 510 378 092 265 472;
  • 71) 0.307 789 802 551 269 531 249 999 999 586 807 064 510 378 092 265 472 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 173 614 129 020 756 184 530 944;
  • 72) 0.615 579 605 102 539 062 499 999 999 173 614 129 020 756 184 530 944 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 347 228 258 041 512 369 061 888;
  • 73) 0.231 159 210 205 078 124 999 999 998 347 228 258 041 512 369 061 888 × 2 = 0 + 0.462 318 420 410 156 249 999 999 996 694 456 516 083 024 738 123 776;
  • 74) 0.462 318 420 410 156 249 999 999 996 694 456 516 083 024 738 123 776 × 2 = 0 + 0.924 636 840 820 312 499 999 999 993 388 913 032 166 049 476 247 552;
  • 75) 0.924 636 840 820 312 499 999 999 993 388 913 032 166 049 476 247 552 × 2 = 1 + 0.849 273 681 640 624 999 999 999 986 777 826 064 332 098 952 495 104;
  • 76) 0.849 273 681 640 624 999 999 999 986 777 826 064 332 098 952 495 104 × 2 = 1 + 0.698 547 363 281 249 999 999 999 973 555 652 128 664 197 904 990 208;
  • 77) 0.698 547 363 281 249 999 999 999 973 555 652 128 664 197 904 990 208 × 2 = 1 + 0.397 094 726 562 499 999 999 999 947 111 304 257 328 395 809 980 416;
  • 78) 0.397 094 726 562 499 999 999 999 947 111 304 257 328 395 809 980 416 × 2 = 0 + 0.794 189 453 124 999 999 999 999 894 222 608 514 656 791 619 960 832;
  • 79) 0.794 189 453 124 999 999 999 999 894 222 608 514 656 791 619 960 832 × 2 = 1 + 0.588 378 906 249 999 999 999 999 788 445 217 029 313 583 239 921 664;
  • 80) 0.588 378 906 249 999 999 999 999 788 445 217 029 313 583 239 921 664 × 2 = 1 + 0.176 757 812 499 999 999 999 999 576 890 434 058 627 166 479 843 328;
  • 81) 0.176 757 812 499 999 999 999 999 576 890 434 058 627 166 479 843 328 × 2 = 0 + 0.353 515 624 999 999 999 999 999 153 780 868 117 254 332 959 686 656;
  • 82) 0.353 515 624 999 999 999 999 999 153 780 868 117 254 332 959 686 656 × 2 = 0 + 0.707 031 249 999 999 999 999 998 307 561 736 234 508 665 919 373 312;
  • 83) 0.707 031 249 999 999 999 999 998 307 561 736 234 508 665 919 373 312 × 2 = 1 + 0.414 062 499 999 999 999 999 996 615 123 472 469 017 331 838 746 624;
  • 84) 0.414 062 499 999 999 999 999 996 615 123 472 469 017 331 838 746 624 × 2 = 0 + 0.828 124 999 999 999 999 999 993 230 246 944 938 034 663 677 493 248;
  • 85) 0.828 124 999 999 999 999 999 993 230 246 944 938 034 663 677 493 248 × 2 = 1 + 0.656 249 999 999 999 999 999 986 460 493 889 876 069 327 354 986 496;
  • 86) 0.656 249 999 999 999 999 999 986 460 493 889 876 069 327 354 986 496 × 2 = 1 + 0.312 499 999 999 999 999 999 972 920 987 779 752 138 654 709 972 992;
  • 87) 0.312 499 999 999 999 999 999 972 920 987 779 752 138 654 709 972 992 × 2 = 0 + 0.624 999 999 999 999 999 999 945 841 975 559 504 277 309 419 945 984;
  • 88) 0.624 999 999 999 999 999 999 945 841 975 559 504 277 309 419 945 984 × 2 = 1 + 0.249 999 999 999 999 999 999 891 683 951 119 008 554 618 839 891 968;
  • 89) 0.249 999 999 999 999 999 999 891 683 951 119 008 554 618 839 891 968 × 2 = 0 + 0.499 999 999 999 999 999 999 783 367 902 238 017 109 237 679 783 936;
  • 90) 0.499 999 999 999 999 999 999 783 367 902 238 017 109 237 679 783 936 × 2 = 0 + 0.999 999 999 999 999 999 999 566 735 804 476 034 218 475 359 567 872;
  • 91) 0.999 999 999 999 999 999 999 566 735 804 476 034 218 475 359 567 872 × 2 = 1 + 0.999 999 999 999 999 999 999 133 471 608 952 068 436 950 719 135 744;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803(10) =

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

6. Normalize the binary representation of the number.

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803(10) =

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

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

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


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-39 + 1 023)(10) =

984(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

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

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


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

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

Exponent (11 bits) =
011 1101 1000

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


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 803 converted to 64 bit double precision IEEE 754 binary floating point representation:

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

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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