0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 026;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 026 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 052;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 052 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 104;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 104 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 208;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 208 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 720 416;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 720 416 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 440 832;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 440 832 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 881 664;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 881 664 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 763 328;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 763 328 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 526 656;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 526 656 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 053 312;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 053 312 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 106 624;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 106 624 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 213 248;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 213 248 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 426 496;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 426 496 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 852 992;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 852 992 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 681 705 984;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 681 705 984 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 363 411 968;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 363 411 968 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 726 823 936;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 726 823 936 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 453 647 872;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 453 647 872 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 907 295 744;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 907 295 744 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 814 591 488;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 814 591 488 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 629 182 976;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 629 182 976 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 258 365 952;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 258 365 952 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 516 731 904;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 516 731 904 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 033 463 808;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 033 463 808 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 746 066 927 616;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 746 066 927 616 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 492 133 855 232;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 492 133 855 232 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 984 267 710 464;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 984 267 710 464 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 968 535 420 928;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 968 535 420 928 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 937 070 841 856;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 937 070 841 856 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 874 141 683 712;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 874 141 683 712 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 748 283 367 424;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 748 283 367 424 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 496 566 734 848;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 496 566 734 848 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 993 133 469 696;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 993 133 469 696 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 349 986 266 939 392;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 349 986 266 939 392 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 699 972 533 878 784;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 699 972 533 878 784 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 399 945 067 757 568;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 399 945 067 757 568 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 799 890 135 515 136;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 799 890 135 515 136 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 599 780 271 030 272;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 599 780 271 030 272 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 199 560 542 060 544;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 199 560 542 060 544 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 399 121 084 121 088;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 399 121 084 121 088 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 798 242 168 242 176;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 798 242 168 242 176 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 596 484 336 484 352;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 596 484 336 484 352 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 192 968 672 968 704;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 192 968 672 968 704 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 342 385 937 345 937 408;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 342 385 937 345 937 408 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 684 771 874 691 874 816;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 684 771 874 691 874 816 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 369 543 749 383 749 632;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 369 543 749 383 749 632 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 739 087 498 767 499 264;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 739 087 498 767 499 264 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 478 174 997 534 998 528;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 478 174 997 534 998 528 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 956 349 995 069 997 056;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 956 349 995 069 997 056 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 912 699 990 139 994 112;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 912 699 990 139 994 112 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 825 399 980 279 988 224;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 825 399 980 279 988 224 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 650 799 960 559 976 448;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 650 799 960 559 976 448 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 301 599 921 119 952 896;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 301 599 921 119 952 896 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 950 603 199 842 239 905 792;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 950 603 199 842 239 905 792 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 901 206 399 684 479 811 584;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 901 206 399 684 479 811 584 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 802 412 799 368 959 623 168;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 802 412 799 368 959 623 168 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 604 825 598 737 919 246 336;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 604 825 598 737 919 246 336 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 209 651 197 475 838 492 672;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 209 651 197 475 838 492 672 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 419 302 394 951 676 985 344;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 419 302 394 951 676 985 344 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 838 604 789 903 353 970 688;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 838 604 789 903 353 970 688 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 677 209 579 806 707 941 376;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 677 209 579 806 707 941 376 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 354 419 159 613 415 882 752;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 354 419 159 613 415 882 752 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 708 838 319 226 831 765 504;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 708 838 319 226 831 765 504 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 997 417 676 638 453 663 531 008;
  • 65) 0.707 934 215 664 863 586 425 781 249 997 417 676 638 453 663 531 008 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 994 835 353 276 907 327 062 016;
  • 66) 0.415 868 431 329 727 172 851 562 499 994 835 353 276 907 327 062 016 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 989 670 706 553 814 654 124 032;
  • 67) 0.831 736 862 659 454 345 703 124 999 989 670 706 553 814 654 124 032 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 979 341 413 107 629 308 248 064;
  • 68) 0.663 473 725 318 908 691 406 249 999 979 341 413 107 629 308 248 064 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 958 682 826 215 258 616 496 128;
  • 69) 0.326 947 450 637 817 382 812 499 999 958 682 826 215 258 616 496 128 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 917 365 652 430 517 232 992 256;
  • 70) 0.653 894 901 275 634 765 624 999 999 917 365 652 430 517 232 992 256 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 834 731 304 861 034 465 984 512;
  • 71) 0.307 789 802 551 269 531 249 999 999 834 731 304 861 034 465 984 512 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 669 462 609 722 068 931 969 024;
  • 72) 0.615 579 605 102 539 062 499 999 999 669 462 609 722 068 931 969 024 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 338 925 219 444 137 863 938 048;
  • 73) 0.231 159 210 205 078 124 999 999 999 338 925 219 444 137 863 938 048 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 677 850 438 888 275 727 876 096;
  • 74) 0.462 318 420 410 156 249 999 999 998 677 850 438 888 275 727 876 096 × 2 = 0 + 0.924 636 840 820 312 499 999 999 997 355 700 877 776 551 455 752 192;
  • 75) 0.924 636 840 820 312 499 999 999 997 355 700 877 776 551 455 752 192 × 2 = 1 + 0.849 273 681 640 624 999 999 999 994 711 401 755 553 102 911 504 384;
  • 76) 0.849 273 681 640 624 999 999 999 994 711 401 755 553 102 911 504 384 × 2 = 1 + 0.698 547 363 281 249 999 999 999 989 422 803 511 106 205 823 008 768;
  • 77) 0.698 547 363 281 249 999 999 999 989 422 803 511 106 205 823 008 768 × 2 = 1 + 0.397 094 726 562 499 999 999 999 978 845 607 022 212 411 646 017 536;
  • 78) 0.397 094 726 562 499 999 999 999 978 845 607 022 212 411 646 017 536 × 2 = 0 + 0.794 189 453 124 999 999 999 999 957 691 214 044 424 823 292 035 072;
  • 79) 0.794 189 453 124 999 999 999 999 957 691 214 044 424 823 292 035 072 × 2 = 1 + 0.588 378 906 249 999 999 999 999 915 382 428 088 849 646 584 070 144;
  • 80) 0.588 378 906 249 999 999 999 999 915 382 428 088 849 646 584 070 144 × 2 = 1 + 0.176 757 812 499 999 999 999 999 830 764 856 177 699 293 168 140 288;
  • 81) 0.176 757 812 499 999 999 999 999 830 764 856 177 699 293 168 140 288 × 2 = 0 + 0.353 515 624 999 999 999 999 999 661 529 712 355 398 586 336 280 576;
  • 82) 0.353 515 624 999 999 999 999 999 661 529 712 355 398 586 336 280 576 × 2 = 0 + 0.707 031 249 999 999 999 999 999 323 059 424 710 797 172 672 561 152;
  • 83) 0.707 031 249 999 999 999 999 999 323 059 424 710 797 172 672 561 152 × 2 = 1 + 0.414 062 499 999 999 999 999 998 646 118 849 421 594 345 345 122 304;
  • 84) 0.414 062 499 999 999 999 999 998 646 118 849 421 594 345 345 122 304 × 2 = 0 + 0.828 124 999 999 999 999 999 997 292 237 698 843 188 690 690 244 608;
  • 85) 0.828 124 999 999 999 999 999 997 292 237 698 843 188 690 690 244 608 × 2 = 1 + 0.656 249 999 999 999 999 999 994 584 475 397 686 377 381 380 489 216;
  • 86) 0.656 249 999 999 999 999 999 994 584 475 397 686 377 381 380 489 216 × 2 = 1 + 0.312 499 999 999 999 999 999 989 168 950 795 372 754 762 760 978 432;
  • 87) 0.312 499 999 999 999 999 999 989 168 950 795 372 754 762 760 978 432 × 2 = 0 + 0.624 999 999 999 999 999 999 978 337 901 590 745 509 525 521 956 864;
  • 88) 0.624 999 999 999 999 999 999 978 337 901 590 745 509 525 521 956 864 × 2 = 1 + 0.249 999 999 999 999 999 999 956 675 803 181 491 019 051 043 913 728;
  • 89) 0.249 999 999 999 999 999 999 956 675 803 181 491 019 051 043 913 728 × 2 = 0 + 0.499 999 999 999 999 999 999 913 351 606 362 982 038 102 087 827 456;
  • 90) 0.499 999 999 999 999 999 999 913 351 606 362 982 038 102 087 827 456 × 2 = 0 + 0.999 999 999 999 999 999 999 826 703 212 725 964 076 204 175 654 912;
  • 91) 0.999 999 999 999 999 999 999 826 703 212 725 964 076 204 175 654 912 × 2 = 1 + 0.999 999 999 999 999 999 999 653 406 425 451 928 152 408 351 309 824;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 013 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