0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 176;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 176 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 352;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 352 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 704;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 704 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 361 408;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 361 408 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 722 816;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 722 816 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 445 632;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 445 632 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 891 264;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 891 264 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 782 528;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 782 528 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 565 056;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 565 056 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 130 112;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 130 112 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 260 224;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 260 224 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 520 448;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 520 448 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 040 896;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 040 896 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 081 792;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 081 792 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 684 163 584;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 684 163 584 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 368 327 168;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 368 327 168 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 736 654 336;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 736 654 336 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 473 308 672;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 473 308 672 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 946 617 344;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 946 617 344 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 893 234 688;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 893 234 688 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 786 469 376;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 786 469 376 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 572 938 752;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 572 938 752 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 145 877 504;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 145 877 504 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 374 291 755 008;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 374 291 755 008 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 748 583 510 016;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 748 583 510 016 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 497 167 020 032;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 497 167 020 032 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 994 334 040 064;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 994 334 040 064 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 988 668 080 128;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 988 668 080 128 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 977 336 160 256;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 977 336 160 256 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 954 672 320 512;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 954 672 320 512 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 909 344 641 024;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 909 344 641 024 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 818 689 282 048;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 818 689 282 048 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 175 637 378 564 096;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 175 637 378 564 096 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 351 274 757 128 192;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 351 274 757 128 192 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 702 549 514 256 384;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 702 549 514 256 384 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 405 099 028 512 768;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 405 099 028 512 768 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 810 198 057 025 536;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 810 198 057 025 536 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 620 396 114 051 072;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 620 396 114 051 072 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 240 792 228 102 144;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 240 792 228 102 144 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 481 584 456 204 288;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 481 584 456 204 288 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 963 168 912 408 576;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 963 168 912 408 576 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 926 337 824 817 152;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 926 337 824 817 152 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 852 675 649 634 304;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 852 675 649 634 304 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 343 705 351 299 268 608;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 343 705 351 299 268 608 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 687 410 702 598 537 216;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 687 410 702 598 537 216 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 374 821 405 197 074 432;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 374 821 405 197 074 432 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 749 642 810 394 148 864;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 749 642 810 394 148 864 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 499 285 620 788 297 728;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 499 285 620 788 297 728 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 998 571 241 576 595 456;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 998 571 241 576 595 456 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 997 142 483 153 190 912;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 997 142 483 153 190 912 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 994 284 966 306 381 824;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 994 284 966 306 381 824 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 988 569 932 612 763 648;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 988 569 932 612 763 648 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 977 139 865 225 527 296;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 977 139 865 225 527 296 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 951 954 279 730 451 054 592;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 951 954 279 730 451 054 592 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 903 908 559 460 902 109 184;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 903 908 559 460 902 109 184 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 807 817 118 921 804 218 368;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 807 817 118 921 804 218 368 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 615 634 237 843 608 436 736;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 615 634 237 843 608 436 736 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 231 268 475 687 216 873 472;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 231 268 475 687 216 873 472 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 462 536 951 374 433 746 944;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 462 536 951 374 433 746 944 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 925 073 902 748 867 493 888;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 925 073 902 748 867 493 888 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 850 147 805 497 734 987 776;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 850 147 805 497 734 987 776 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 700 295 610 995 469 975 552;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 700 295 610 995 469 975 552 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 400 591 221 990 939 951 104;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 400 591 221 990 939 951 104 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 998 801 182 443 981 879 902 208;
  • 65) 0.707 934 215 664 863 586 425 781 249 998 801 182 443 981 879 902 208 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 997 602 364 887 963 759 804 416;
  • 66) 0.415 868 431 329 727 172 851 562 499 997 602 364 887 963 759 804 416 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 995 204 729 775 927 519 608 832;
  • 67) 0.831 736 862 659 454 345 703 124 999 995 204 729 775 927 519 608 832 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 990 409 459 551 855 039 217 664;
  • 68) 0.663 473 725 318 908 691 406 249 999 990 409 459 551 855 039 217 664 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 980 818 919 103 710 078 435 328;
  • 69) 0.326 947 450 637 817 382 812 499 999 980 818 919 103 710 078 435 328 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 961 637 838 207 420 156 870 656;
  • 70) 0.653 894 901 275 634 765 624 999 999 961 637 838 207 420 156 870 656 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 923 275 676 414 840 313 741 312;
  • 71) 0.307 789 802 551 269 531 249 999 999 923 275 676 414 840 313 741 312 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 846 551 352 829 680 627 482 624;
  • 72) 0.615 579 605 102 539 062 499 999 999 846 551 352 829 680 627 482 624 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 693 102 705 659 361 254 965 248;
  • 73) 0.231 159 210 205 078 124 999 999 999 693 102 705 659 361 254 965 248 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 386 205 411 318 722 509 930 496;
  • 74) 0.462 318 420 410 156 249 999 999 999 386 205 411 318 722 509 930 496 × 2 = 0 + 0.924 636 840 820 312 499 999 999 998 772 410 822 637 445 019 860 992;
  • 75) 0.924 636 840 820 312 499 999 999 998 772 410 822 637 445 019 860 992 × 2 = 1 + 0.849 273 681 640 624 999 999 999 997 544 821 645 274 890 039 721 984;
  • 76) 0.849 273 681 640 624 999 999 999 997 544 821 645 274 890 039 721 984 × 2 = 1 + 0.698 547 363 281 249 999 999 999 995 089 643 290 549 780 079 443 968;
  • 77) 0.698 547 363 281 249 999 999 999 995 089 643 290 549 780 079 443 968 × 2 = 1 + 0.397 094 726 562 499 999 999 999 990 179 286 581 099 560 158 887 936;
  • 78) 0.397 094 726 562 499 999 999 999 990 179 286 581 099 560 158 887 936 × 2 = 0 + 0.794 189 453 124 999 999 999 999 980 358 573 162 199 120 317 775 872;
  • 79) 0.794 189 453 124 999 999 999 999 980 358 573 162 199 120 317 775 872 × 2 = 1 + 0.588 378 906 249 999 999 999 999 960 717 146 324 398 240 635 551 744;
  • 80) 0.588 378 906 249 999 999 999 999 960 717 146 324 398 240 635 551 744 × 2 = 1 + 0.176 757 812 499 999 999 999 999 921 434 292 648 796 481 271 103 488;
  • 81) 0.176 757 812 499 999 999 999 999 921 434 292 648 796 481 271 103 488 × 2 = 0 + 0.353 515 624 999 999 999 999 999 842 868 585 297 592 962 542 206 976;
  • 82) 0.353 515 624 999 999 999 999 999 842 868 585 297 592 962 542 206 976 × 2 = 0 + 0.707 031 249 999 999 999 999 999 685 737 170 595 185 925 084 413 952;
  • 83) 0.707 031 249 999 999 999 999 999 685 737 170 595 185 925 084 413 952 × 2 = 1 + 0.414 062 499 999 999 999 999 999 371 474 341 190 371 850 168 827 904;
  • 84) 0.414 062 499 999 999 999 999 999 371 474 341 190 371 850 168 827 904 × 2 = 0 + 0.828 124 999 999 999 999 999 998 742 948 682 380 743 700 337 655 808;
  • 85) 0.828 124 999 999 999 999 999 998 742 948 682 380 743 700 337 655 808 × 2 = 1 + 0.656 249 999 999 999 999 999 997 485 897 364 761 487 400 675 311 616;
  • 86) 0.656 249 999 999 999 999 999 997 485 897 364 761 487 400 675 311 616 × 2 = 1 + 0.312 499 999 999 999 999 999 994 971 794 729 522 974 801 350 623 232;
  • 87) 0.312 499 999 999 999 999 999 994 971 794 729 522 974 801 350 623 232 × 2 = 0 + 0.624 999 999 999 999 999 999 989 943 589 459 045 949 602 701 246 464;
  • 88) 0.624 999 999 999 999 999 999 989 943 589 459 045 949 602 701 246 464 × 2 = 1 + 0.249 999 999 999 999 999 999 979 887 178 918 091 899 205 402 492 928;
  • 89) 0.249 999 999 999 999 999 999 979 887 178 918 091 899 205 402 492 928 × 2 = 0 + 0.499 999 999 999 999 999 999 959 774 357 836 183 798 410 804 985 856;
  • 90) 0.499 999 999 999 999 999 999 959 774 357 836 183 798 410 804 985 856 × 2 = 0 + 0.999 999 999 999 999 999 999 919 548 715 672 367 596 821 609 971 712;
  • 91) 0.999 999 999 999 999 999 999 919 548 715 672 367 596 821 609 971 712 × 2 = 1 + 0.999 999 999 999 999 999 999 839 097 431 344 735 193 643 219 943 424;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 088 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