0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 086;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 086 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 172;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 172 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 344;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 344 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 688;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 688 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 721 376;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 721 376 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 442 752;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 442 752 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 885 504;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 885 504 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 771 008;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 771 008 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 542 016;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 542 016 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 084 032;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 084 032 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 168 064;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 168 064 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 336 128;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 336 128 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 672 256;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 672 256 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 344 512;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 344 512 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 682 689 024;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 682 689 024 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 365 378 048;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 365 378 048 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 730 756 096;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 730 756 096 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 461 512 192;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 461 512 192 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 923 024 384;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 923 024 384 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 846 048 768;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 846 048 768 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 692 097 536;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 692 097 536 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 384 195 072;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 384 195 072 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 768 390 144;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 768 390 144 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 536 780 288;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 536 780 288 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 747 073 560 576;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 747 073 560 576 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 494 147 121 152;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 494 147 121 152 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 988 294 242 304;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 988 294 242 304 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 976 588 484 608;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 976 588 484 608 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 953 176 969 216;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 953 176 969 216 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 906 353 938 432;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 906 353 938 432 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 812 707 876 864;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 812 707 876 864 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 625 415 753 728;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 625 415 753 728 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 175 250 831 507 456;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 175 250 831 507 456 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 350 501 663 014 912;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 350 501 663 014 912 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 701 003 326 029 824;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 701 003 326 029 824 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 402 006 652 059 648;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 402 006 652 059 648 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 804 013 304 119 296;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 804 013 304 119 296 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 608 026 608 238 592;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 608 026 608 238 592 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 216 053 216 477 184;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 216 053 216 477 184 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 432 106 432 954 368;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 432 106 432 954 368 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 864 212 865 908 736;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 864 212 865 908 736 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 728 425 731 817 472;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 728 425 731 817 472 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 456 851 463 634 944;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 456 851 463 634 944 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 342 913 702 927 269 888;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 342 913 702 927 269 888 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 685 827 405 854 539 776;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 685 827 405 854 539 776 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 371 654 811 709 079 552;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 371 654 811 709 079 552 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 743 309 623 418 159 104;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 743 309 623 418 159 104 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 486 619 246 836 318 208;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 486 619 246 836 318 208 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 973 238 493 672 636 416;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 973 238 493 672 636 416 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 946 476 987 345 272 832;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 946 476 987 345 272 832 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 892 953 974 690 545 664;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 892 953 974 690 545 664 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 785 907 949 381 091 328;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 785 907 949 381 091 328 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 571 815 898 762 182 656;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 571 815 898 762 182 656 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 951 143 631 797 524 365 312;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 951 143 631 797 524 365 312 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 902 287 263 595 048 730 624;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 902 287 263 595 048 730 624 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 804 574 527 190 097 461 248;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 804 574 527 190 097 461 248 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 609 149 054 380 194 922 496;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 609 149 054 380 194 922 496 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 218 298 108 760 389 844 992;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 218 298 108 760 389 844 992 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 436 596 217 520 779 689 984;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 436 596 217 520 779 689 984 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 873 192 435 041 559 379 968;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 873 192 435 041 559 379 968 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 746 384 870 083 118 759 936;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 746 384 870 083 118 759 936 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 492 769 740 166 237 519 872;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 492 769 740 166 237 519 872 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 985 539 480 332 475 039 744;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 985 539 480 332 475 039 744 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 997 971 078 960 664 950 079 488;
  • 65) 0.707 934 215 664 863 586 425 781 249 997 971 078 960 664 950 079 488 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 995 942 157 921 329 900 158 976;
  • 66) 0.415 868 431 329 727 172 851 562 499 995 942 157 921 329 900 158 976 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 991 884 315 842 659 800 317 952;
  • 67) 0.831 736 862 659 454 345 703 124 999 991 884 315 842 659 800 317 952 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 983 768 631 685 319 600 635 904;
  • 68) 0.663 473 725 318 908 691 406 249 999 983 768 631 685 319 600 635 904 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 967 537 263 370 639 201 271 808;
  • 69) 0.326 947 450 637 817 382 812 499 999 967 537 263 370 639 201 271 808 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 935 074 526 741 278 402 543 616;
  • 70) 0.653 894 901 275 634 765 624 999 999 935 074 526 741 278 402 543 616 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 870 149 053 482 556 805 087 232;
  • 71) 0.307 789 802 551 269 531 249 999 999 870 149 053 482 556 805 087 232 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 740 298 106 965 113 610 174 464;
  • 72) 0.615 579 605 102 539 062 499 999 999 740 298 106 965 113 610 174 464 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 480 596 213 930 227 220 348 928;
  • 73) 0.231 159 210 205 078 124 999 999 999 480 596 213 930 227 220 348 928 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 961 192 427 860 454 440 697 856;
  • 74) 0.462 318 420 410 156 249 999 999 998 961 192 427 860 454 440 697 856 × 2 = 0 + 0.924 636 840 820 312 499 999 999 997 922 384 855 720 908 881 395 712;
  • 75) 0.924 636 840 820 312 499 999 999 997 922 384 855 720 908 881 395 712 × 2 = 1 + 0.849 273 681 640 624 999 999 999 995 844 769 711 441 817 762 791 424;
  • 76) 0.849 273 681 640 624 999 999 999 995 844 769 711 441 817 762 791 424 × 2 = 1 + 0.698 547 363 281 249 999 999 999 991 689 539 422 883 635 525 582 848;
  • 77) 0.698 547 363 281 249 999 999 999 991 689 539 422 883 635 525 582 848 × 2 = 1 + 0.397 094 726 562 499 999 999 999 983 379 078 845 767 271 051 165 696;
  • 78) 0.397 094 726 562 499 999 999 999 983 379 078 845 767 271 051 165 696 × 2 = 0 + 0.794 189 453 124 999 999 999 999 966 758 157 691 534 542 102 331 392;
  • 79) 0.794 189 453 124 999 999 999 999 966 758 157 691 534 542 102 331 392 × 2 = 1 + 0.588 378 906 249 999 999 999 999 933 516 315 383 069 084 204 662 784;
  • 80) 0.588 378 906 249 999 999 999 999 933 516 315 383 069 084 204 662 784 × 2 = 1 + 0.176 757 812 499 999 999 999 999 867 032 630 766 138 168 409 325 568;
  • 81) 0.176 757 812 499 999 999 999 999 867 032 630 766 138 168 409 325 568 × 2 = 0 + 0.353 515 624 999 999 999 999 999 734 065 261 532 276 336 818 651 136;
  • 82) 0.353 515 624 999 999 999 999 999 734 065 261 532 276 336 818 651 136 × 2 = 0 + 0.707 031 249 999 999 999 999 999 468 130 523 064 552 673 637 302 272;
  • 83) 0.707 031 249 999 999 999 999 999 468 130 523 064 552 673 637 302 272 × 2 = 1 + 0.414 062 499 999 999 999 999 998 936 261 046 129 105 347 274 604 544;
  • 84) 0.414 062 499 999 999 999 999 998 936 261 046 129 105 347 274 604 544 × 2 = 0 + 0.828 124 999 999 999 999 999 997 872 522 092 258 210 694 549 209 088;
  • 85) 0.828 124 999 999 999 999 999 997 872 522 092 258 210 694 549 209 088 × 2 = 1 + 0.656 249 999 999 999 999 999 995 745 044 184 516 421 389 098 418 176;
  • 86) 0.656 249 999 999 999 999 999 995 745 044 184 516 421 389 098 418 176 × 2 = 1 + 0.312 499 999 999 999 999 999 991 490 088 369 032 842 778 196 836 352;
  • 87) 0.312 499 999 999 999 999 999 991 490 088 369 032 842 778 196 836 352 × 2 = 0 + 0.624 999 999 999 999 999 999 982 980 176 738 065 685 556 393 672 704;
  • 88) 0.624 999 999 999 999 999 999 982 980 176 738 065 685 556 393 672 704 × 2 = 1 + 0.249 999 999 999 999 999 999 965 960 353 476 131 371 112 787 345 408;
  • 89) 0.249 999 999 999 999 999 999 965 960 353 476 131 371 112 787 345 408 × 2 = 0 + 0.499 999 999 999 999 999 999 931 920 706 952 262 742 225 574 690 816;
  • 90) 0.499 999 999 999 999 999 999 931 920 706 952 262 742 225 574 690 816 × 2 = 0 + 0.999 999 999 999 999 999 999 863 841 413 904 525 484 451 149 381 632;
  • 91) 0.999 999 999 999 999 999 999 863 841 413 904 525 484 451 149 381 632 × 2 = 1 + 0.999 999 999 999 999 999 999 727 682 827 809 050 968 902 298 763 264;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 043 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