0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 662;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 662 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 324;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 324 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 648;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 678 648 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 357 296;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 357 296 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 714 592;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 714 592 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 429 184;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 429 184 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 858 368;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 858 368 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 716 736;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 716 736 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 433 472;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 433 472 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 866 944;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 866 944 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 733 888;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 733 888 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 467 776;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 467 776 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 935 552;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 918 935 552 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 871 104;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 837 871 104 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 675 742 208;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 675 742 208 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 351 484 416;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 351 484 416 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 702 968 832;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 702 968 832 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 405 937 664;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 405 937 664 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 811 875 328;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 811 875 328 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 623 750 656;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 623 750 656 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 247 501 312;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 247 501 312 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 495 002 624;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 495 002 624 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 990 005 248;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 684 990 005 248 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 980 010 496;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 369 980 010 496 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 739 960 020 992;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 739 960 020 992 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 479 920 041 984;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 479 920 041 984 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 959 840 083 968;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 959 840 083 968 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 919 680 167 936;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 919 680 167 936 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 839 360 335 872;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 839 360 335 872 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 678 720 671 744;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 678 720 671 744 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 357 441 343 488;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 357 441 343 488 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 086 714 882 686 976;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 086 714 882 686 976 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 173 429 765 373 952;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 173 429 765 373 952 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 346 859 530 747 904;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 346 859 530 747 904 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 693 719 061 495 808;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 693 719 061 495 808 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 387 438 122 991 616;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 387 438 122 991 616 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 774 876 245 983 232;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 774 876 245 983 232 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 549 752 491 966 464;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 549 752 491 966 464 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 099 504 983 932 928;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 099 504 983 932 928 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 199 009 967 865 856;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 199 009 967 865 856 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 398 019 935 731 712;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 398 019 935 731 712 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 584 796 039 871 463 424;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 584 796 039 871 463 424 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 169 592 079 742 926 848;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 169 592 079 742 926 848 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 339 184 159 485 853 696;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 339 184 159 485 853 696 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 678 368 318 971 707 392;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 678 368 318 971 707 392 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 356 736 637 943 414 784;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 356 736 637 943 414 784 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 713 473 275 886 829 568;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 713 473 275 886 829 568 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 426 946 551 773 659 136;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 426 946 551 773 659 136 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 853 893 103 547 318 272;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 853 893 103 547 318 272 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 707 786 207 094 636 544;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 707 786 207 094 636 544 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 415 572 414 189 273 088;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 415 572 414 189 273 088 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 486 831 144 828 378 546 176;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 486 831 144 828 378 546 176 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 973 662 289 656 757 092 352;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 973 662 289 656 757 092 352 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 947 324 579 313 514 184 704;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 947 324 579 313 514 184 704 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 894 649 158 627 028 369 408;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 894 649 158 627 028 369 408 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 789 298 317 254 056 738 816;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 789 298 317 254 056 738 816 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 578 596 634 508 113 477 632;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 578 596 634 508 113 477 632 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 157 193 269 016 226 955 264;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 157 193 269 016 226 955 264 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 314 386 538 032 453 910 528;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 314 386 538 032 453 910 528 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 628 773 076 064 907 821 056;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 628 773 076 064 907 821 056 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 257 546 152 129 815 642 112;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 257 546 152 129 815 642 112 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 515 092 304 259 631 284 224;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 515 092 304 259 631 284 224 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 997 030 184 608 519 262 568 448;
  • 64) 0.353 967 107 832 431 793 212 890 624 997 030 184 608 519 262 568 448 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 994 060 369 217 038 525 136 896;
  • 65) 0.707 934 215 664 863 586 425 781 249 994 060 369 217 038 525 136 896 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 988 120 738 434 077 050 273 792;
  • 66) 0.415 868 431 329 727 172 851 562 499 988 120 738 434 077 050 273 792 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 976 241 476 868 154 100 547 584;
  • 67) 0.831 736 862 659 454 345 703 124 999 976 241 476 868 154 100 547 584 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 952 482 953 736 308 201 095 168;
  • 68) 0.663 473 725 318 908 691 406 249 999 952 482 953 736 308 201 095 168 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 904 965 907 472 616 402 190 336;
  • 69) 0.326 947 450 637 817 382 812 499 999 904 965 907 472 616 402 190 336 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 809 931 814 945 232 804 380 672;
  • 70) 0.653 894 901 275 634 765 624 999 999 809 931 814 945 232 804 380 672 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 619 863 629 890 465 608 761 344;
  • 71) 0.307 789 802 551 269 531 249 999 999 619 863 629 890 465 608 761 344 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 239 727 259 780 931 217 522 688;
  • 72) 0.615 579 605 102 539 062 499 999 999 239 727 259 780 931 217 522 688 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 479 454 519 561 862 435 045 376;
  • 73) 0.231 159 210 205 078 124 999 999 998 479 454 519 561 862 435 045 376 × 2 = 0 + 0.462 318 420 410 156 249 999 999 996 958 909 039 123 724 870 090 752;
  • 74) 0.462 318 420 410 156 249 999 999 996 958 909 039 123 724 870 090 752 × 2 = 0 + 0.924 636 840 820 312 499 999 999 993 917 818 078 247 449 740 181 504;
  • 75) 0.924 636 840 820 312 499 999 999 993 917 818 078 247 449 740 181 504 × 2 = 1 + 0.849 273 681 640 624 999 999 999 987 835 636 156 494 899 480 363 008;
  • 76) 0.849 273 681 640 624 999 999 999 987 835 636 156 494 899 480 363 008 × 2 = 1 + 0.698 547 363 281 249 999 999 999 975 671 272 312 989 798 960 726 016;
  • 77) 0.698 547 363 281 249 999 999 999 975 671 272 312 989 798 960 726 016 × 2 = 1 + 0.397 094 726 562 499 999 999 999 951 342 544 625 979 597 921 452 032;
  • 78) 0.397 094 726 562 499 999 999 999 951 342 544 625 979 597 921 452 032 × 2 = 0 + 0.794 189 453 124 999 999 999 999 902 685 089 251 959 195 842 904 064;
  • 79) 0.794 189 453 124 999 999 999 999 902 685 089 251 959 195 842 904 064 × 2 = 1 + 0.588 378 906 249 999 999 999 999 805 370 178 503 918 391 685 808 128;
  • 80) 0.588 378 906 249 999 999 999 999 805 370 178 503 918 391 685 808 128 × 2 = 1 + 0.176 757 812 499 999 999 999 999 610 740 357 007 836 783 371 616 256;
  • 81) 0.176 757 812 499 999 999 999 999 610 740 357 007 836 783 371 616 256 × 2 = 0 + 0.353 515 624 999 999 999 999 999 221 480 714 015 673 566 743 232 512;
  • 82) 0.353 515 624 999 999 999 999 999 221 480 714 015 673 566 743 232 512 × 2 = 0 + 0.707 031 249 999 999 999 999 998 442 961 428 031 347 133 486 465 024;
  • 83) 0.707 031 249 999 999 999 999 998 442 961 428 031 347 133 486 465 024 × 2 = 1 + 0.414 062 499 999 999 999 999 996 885 922 856 062 694 266 972 930 048;
  • 84) 0.414 062 499 999 999 999 999 996 885 922 856 062 694 266 972 930 048 × 2 = 0 + 0.828 124 999 999 999 999 999 993 771 845 712 125 388 533 945 860 096;
  • 85) 0.828 124 999 999 999 999 999 993 771 845 712 125 388 533 945 860 096 × 2 = 1 + 0.656 249 999 999 999 999 999 987 543 691 424 250 777 067 891 720 192;
  • 86) 0.656 249 999 999 999 999 999 987 543 691 424 250 777 067 891 720 192 × 2 = 1 + 0.312 499 999 999 999 999 999 975 087 382 848 501 554 135 783 440 384;
  • 87) 0.312 499 999 999 999 999 999 975 087 382 848 501 554 135 783 440 384 × 2 = 0 + 0.624 999 999 999 999 999 999 950 174 765 697 003 108 271 566 880 768;
  • 88) 0.624 999 999 999 999 999 999 950 174 765 697 003 108 271 566 880 768 × 2 = 1 + 0.249 999 999 999 999 999 999 900 349 531 394 006 216 543 133 761 536;
  • 89) 0.249 999 999 999 999 999 999 900 349 531 394 006 216 543 133 761 536 × 2 = 0 + 0.499 999 999 999 999 999 999 800 699 062 788 012 433 086 267 523 072;
  • 90) 0.499 999 999 999 999 999 999 800 699 062 788 012 433 086 267 523 072 × 2 = 0 + 0.999 999 999 999 999 999 999 601 398 125 576 024 866 172 535 046 144;
  • 91) 0.999 999 999 999 999 999 999 601 398 125 576 024 866 172 535 046 144 × 2 = 1 + 0.999 999 999 999 999 999 999 202 796 251 152 049 732 345 070 092 288;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831(10) =

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

6. Normalize the binary representation of the number.

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831(10) =

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

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

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


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-39 + 1 023)(10) =

984(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

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

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


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

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

Exponent (11 bits) =
011 1101 1000

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


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 831 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