0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 258;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 258 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 516;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 516 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 032;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 032 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 064;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 064 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 128;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 128 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 256;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 256 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 896 512;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 896 512 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 793 024;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 793 024 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 586 048;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 586 048 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 172 096;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 172 096 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 344 192;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 344 192 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 688 384;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 688 384 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 376 768;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 376 768 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 753 536;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 753 536 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 507 072;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 507 072 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 014 144;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 014 144 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 028 288;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 028 288 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 484 056 576;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 484 056 576 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 968 113 152;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 968 113 152 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 936 226 304;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 936 226 304 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 872 452 608;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 872 452 608 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 744 905 216;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 744 905 216 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 489 810 432;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 489 810 432 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 374 979 620 864;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 374 979 620 864 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 749 959 241 728;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 749 959 241 728 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 499 918 483 456;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 499 918 483 456 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 999 836 966 912;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 999 836 966 912 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 999 673 933 824;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 999 673 933 824 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 999 347 867 648;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 999 347 867 648 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 998 695 735 296;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 998 695 735 296 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 997 391 470 592;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 997 391 470 592 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 994 782 941 184;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 994 782 941 184 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 175 989 565 882 368;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 175 989 565 882 368 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 351 979 131 764 736;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 351 979 131 764 736 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 703 958 263 529 472;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 703 958 263 529 472 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 407 916 527 058 944;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 407 916 527 058 944 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 815 833 054 117 888;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 815 833 054 117 888 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 631 666 108 235 776;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 631 666 108 235 776 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 263 332 216 471 552;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 263 332 216 471 552 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 526 664 432 943 104;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 526 664 432 943 104 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 053 328 865 886 208;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 053 328 865 886 208 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 106 657 731 772 416;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 106 657 731 772 416 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 213 315 463 544 832;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 213 315 463 544 832 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 426 630 927 089 664;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 426 630 927 089 664 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 688 853 261 854 179 328;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 688 853 261 854 179 328 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 377 706 523 708 358 656;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 377 706 523 708 358 656 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 755 413 047 416 717 312;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 755 413 047 416 717 312 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 510 826 094 833 434 624;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 510 826 094 833 434 624 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 021 652 189 666 869 248;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 021 652 189 666 869 248 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 043 304 379 333 738 496;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 043 304 379 333 738 496 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 086 608 758 667 476 992;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 086 608 758 667 476 992 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 173 217 517 334 953 984;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 173 217 517 334 953 984 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 346 435 034 669 907 968;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 346 435 034 669 907 968 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 692 870 069 339 815 936;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 692 870 069 339 815 936 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 385 740 138 679 631 872;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 385 740 138 679 631 872 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 810 771 480 277 359 263 744;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 810 771 480 277 359 263 744 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 621 542 960 554 718 527 488;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 621 542 960 554 718 527 488 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 243 085 921 109 437 054 976;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 243 085 921 109 437 054 976 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 486 171 842 218 874 109 952;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 486 171 842 218 874 109 952 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 972 343 684 437 748 219 904;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 972 343 684 437 748 219 904 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 944 687 368 875 496 439 808;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 944 687 368 875 496 439 808 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 889 374 737 750 992 879 616;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 889 374 737 750 992 879 616 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 778 749 475 501 985 759 232;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 778 749 475 501 985 759 232 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 557 498 951 003 971 518 464;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 557 498 951 003 971 518 464 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 114 997 902 007 943 036 928;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 114 997 902 007 943 036 928 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 998 229 995 804 015 886 073 856;
  • 67) 0.831 736 862 659 454 345 703 124 999 998 229 995 804 015 886 073 856 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 996 459 991 608 031 772 147 712;
  • 68) 0.663 473 725 318 908 691 406 249 999 996 459 991 608 031 772 147 712 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 992 919 983 216 063 544 295 424;
  • 69) 0.326 947 450 637 817 382 812 499 999 992 919 983 216 063 544 295 424 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 985 839 966 432 127 088 590 848;
  • 70) 0.653 894 901 275 634 765 624 999 999 985 839 966 432 127 088 590 848 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 971 679 932 864 254 177 181 696;
  • 71) 0.307 789 802 551 269 531 249 999 999 971 679 932 864 254 177 181 696 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 943 359 865 728 508 354 363 392;
  • 72) 0.615 579 605 102 539 062 499 999 999 943 359 865 728 508 354 363 392 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 886 719 731 457 016 708 726 784;
  • 73) 0.231 159 210 205 078 124 999 999 999 886 719 731 457 016 708 726 784 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 773 439 462 914 033 417 453 568;
  • 74) 0.462 318 420 410 156 249 999 999 999 773 439 462 914 033 417 453 568 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 546 878 925 828 066 834 907 136;
  • 75) 0.924 636 840 820 312 499 999 999 999 546 878 925 828 066 834 907 136 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 093 757 851 656 133 669 814 272;
  • 76) 0.849 273 681 640 624 999 999 999 999 093 757 851 656 133 669 814 272 × 2 = 1 + 0.698 547 363 281 249 999 999 999 998 187 515 703 312 267 339 628 544;
  • 77) 0.698 547 363 281 249 999 999 999 998 187 515 703 312 267 339 628 544 × 2 = 1 + 0.397 094 726 562 499 999 999 999 996 375 031 406 624 534 679 257 088;
  • 78) 0.397 094 726 562 499 999 999 999 996 375 031 406 624 534 679 257 088 × 2 = 0 + 0.794 189 453 124 999 999 999 999 992 750 062 813 249 069 358 514 176;
  • 79) 0.794 189 453 124 999 999 999 999 992 750 062 813 249 069 358 514 176 × 2 = 1 + 0.588 378 906 249 999 999 999 999 985 500 125 626 498 138 717 028 352;
  • 80) 0.588 378 906 249 999 999 999 999 985 500 125 626 498 138 717 028 352 × 2 = 1 + 0.176 757 812 499 999 999 999 999 971 000 251 252 996 277 434 056 704;
  • 81) 0.176 757 812 499 999 999 999 999 971 000 251 252 996 277 434 056 704 × 2 = 0 + 0.353 515 624 999 999 999 999 999 942 000 502 505 992 554 868 113 408;
  • 82) 0.353 515 624 999 999 999 999 999 942 000 502 505 992 554 868 113 408 × 2 = 0 + 0.707 031 249 999 999 999 999 999 884 001 005 011 985 109 736 226 816;
  • 83) 0.707 031 249 999 999 999 999 999 884 001 005 011 985 109 736 226 816 × 2 = 1 + 0.414 062 499 999 999 999 999 999 768 002 010 023 970 219 472 453 632;
  • 84) 0.414 062 499 999 999 999 999 999 768 002 010 023 970 219 472 453 632 × 2 = 0 + 0.828 124 999 999 999 999 999 999 536 004 020 047 940 438 944 907 264;
  • 85) 0.828 124 999 999 999 999 999 999 536 004 020 047 940 438 944 907 264 × 2 = 1 + 0.656 249 999 999 999 999 999 999 072 008 040 095 880 877 889 814 528;
  • 86) 0.656 249 999 999 999 999 999 999 072 008 040 095 880 877 889 814 528 × 2 = 1 + 0.312 499 999 999 999 999 999 998 144 016 080 191 761 755 779 629 056;
  • 87) 0.312 499 999 999 999 999 999 998 144 016 080 191 761 755 779 629 056 × 2 = 0 + 0.624 999 999 999 999 999 999 996 288 032 160 383 523 511 559 258 112;
  • 88) 0.624 999 999 999 999 999 999 996 288 032 160 383 523 511 559 258 112 × 2 = 1 + 0.249 999 999 999 999 999 999 992 576 064 320 767 047 023 118 516 224;
  • 89) 0.249 999 999 999 999 999 999 992 576 064 320 767 047 023 118 516 224 × 2 = 0 + 0.499 999 999 999 999 999 999 985 152 128 641 534 094 046 237 032 448;
  • 90) 0.499 999 999 999 999 999 999 985 152 128 641 534 094 046 237 032 448 × 2 = 0 + 0.999 999 999 999 999 999 999 970 304 257 283 068 188 092 474 064 896;
  • 91) 0.999 999 999 999 999 999 999 970 304 257 283 068 188 092 474 064 896 × 2 = 1 + 0.999 999 999 999 999 999 999 940 608 514 566 136 376 184 948 129 792;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129(10) =

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

6. Normalize the binary representation of the number.

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129(10) =

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

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

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


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-39 + 1 023)(10) =

984(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

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

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


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

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

Exponent (11 bits) =
011 1101 1000

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


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 129 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