0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 134;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 134 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 268;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 268 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 536;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 536 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 361 072;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 361 072 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 722 144;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 722 144 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 444 288;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 444 288 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 888 576;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 888 576 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 777 152;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 777 152 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 554 304;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 554 304 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 108 608;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 108 608 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 217 216;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 217 216 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 434 432;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 434 432 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 868 864;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 868 864 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 737 728;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 737 728 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 683 475 456;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 683 475 456 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 366 950 912;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 366 950 912 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 733 901 824;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 733 901 824 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 467 803 648;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 467 803 648 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 935 607 296;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 935 607 296 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 871 214 592;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 871 214 592 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 742 429 184;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 742 429 184 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 484 858 368;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 484 858 368 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 969 716 736;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 969 716 736 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 939 433 472;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 939 433 472 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 747 878 866 944;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 747 878 866 944 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 495 757 733 888;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 495 757 733 888 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 991 515 467 776;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 991 515 467 776 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 983 030 935 552;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 983 030 935 552 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 966 061 871 104;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 966 061 871 104 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 932 123 742 208;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 932 123 742 208 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 864 247 484 416;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 864 247 484 416 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 728 494 968 832;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 728 494 968 832 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 175 456 989 937 664;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 175 456 989 937 664 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 350 913 979 875 328;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 350 913 979 875 328 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 701 827 959 750 656;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 701 827 959 750 656 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 403 655 919 501 312;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 403 655 919 501 312 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 807 311 839 002 624;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 807 311 839 002 624 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 614 623 678 005 248;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 614 623 678 005 248 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 229 247 356 010 496;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 229 247 356 010 496 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 458 494 712 020 992;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 458 494 712 020 992 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 916 989 424 041 984;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 916 989 424 041 984 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 833 978 848 083 968;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 833 978 848 083 968 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 667 957 696 167 936;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 667 957 696 167 936 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 343 335 915 392 335 872;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 343 335 915 392 335 872 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 686 671 830 784 671 744;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 686 671 830 784 671 744 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 373 343 661 569 343 488;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 373 343 661 569 343 488 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 746 687 323 138 686 976;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 746 687 323 138 686 976 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 493 374 646 277 373 952;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 493 374 646 277 373 952 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 986 749 292 554 747 904;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 986 749 292 554 747 904 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 973 498 585 109 495 808;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 973 498 585 109 495 808 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 946 997 170 218 991 616;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 946 997 170 218 991 616 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 893 994 340 437 983 232;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 893 994 340 437 983 232 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 787 988 680 875 966 464;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 787 988 680 875 966 464 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 951 575 977 361 751 932 928;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 951 575 977 361 751 932 928 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 903 151 954 723 503 865 856;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 903 151 954 723 503 865 856 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 806 303 909 447 007 731 712;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 806 303 909 447 007 731 712 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 612 607 818 894 015 463 424;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 612 607 818 894 015 463 424 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 225 215 637 788 030 926 848;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 225 215 637 788 030 926 848 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 450 431 275 576 061 853 696;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 450 431 275 576 061 853 696 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 900 862 551 152 123 707 392;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 900 862 551 152 123 707 392 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 801 725 102 304 247 414 784;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 801 725 102 304 247 414 784 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 603 450 204 608 494 829 568;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 603 450 204 608 494 829 568 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 206 900 409 216 989 659 136;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 206 900 409 216 989 659 136 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 998 413 800 818 433 979 318 272;
  • 65) 0.707 934 215 664 863 586 425 781 249 998 413 800 818 433 979 318 272 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 996 827 601 636 867 958 636 544;
  • 66) 0.415 868 431 329 727 172 851 562 499 996 827 601 636 867 958 636 544 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 993 655 203 273 735 917 273 088;
  • 67) 0.831 736 862 659 454 345 703 124 999 993 655 203 273 735 917 273 088 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 987 310 406 547 471 834 546 176;
  • 68) 0.663 473 725 318 908 691 406 249 999 987 310 406 547 471 834 546 176 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 974 620 813 094 943 669 092 352;
  • 69) 0.326 947 450 637 817 382 812 499 999 974 620 813 094 943 669 092 352 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 949 241 626 189 887 338 184 704;
  • 70) 0.653 894 901 275 634 765 624 999 999 949 241 626 189 887 338 184 704 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 898 483 252 379 774 676 369 408;
  • 71) 0.307 789 802 551 269 531 249 999 999 898 483 252 379 774 676 369 408 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 796 966 504 759 549 352 738 816;
  • 72) 0.615 579 605 102 539 062 499 999 999 796 966 504 759 549 352 738 816 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 593 933 009 519 098 705 477 632;
  • 73) 0.231 159 210 205 078 124 999 999 999 593 933 009 519 098 705 477 632 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 187 866 019 038 197 410 955 264;
  • 74) 0.462 318 420 410 156 249 999 999 999 187 866 019 038 197 410 955 264 × 2 = 0 + 0.924 636 840 820 312 499 999 999 998 375 732 038 076 394 821 910 528;
  • 75) 0.924 636 840 820 312 499 999 999 998 375 732 038 076 394 821 910 528 × 2 = 1 + 0.849 273 681 640 624 999 999 999 996 751 464 076 152 789 643 821 056;
  • 76) 0.849 273 681 640 624 999 999 999 996 751 464 076 152 789 643 821 056 × 2 = 1 + 0.698 547 363 281 249 999 999 999 993 502 928 152 305 579 287 642 112;
  • 77) 0.698 547 363 281 249 999 999 999 993 502 928 152 305 579 287 642 112 × 2 = 1 + 0.397 094 726 562 499 999 999 999 987 005 856 304 611 158 575 284 224;
  • 78) 0.397 094 726 562 499 999 999 999 987 005 856 304 611 158 575 284 224 × 2 = 0 + 0.794 189 453 124 999 999 999 999 974 011 712 609 222 317 150 568 448;
  • 79) 0.794 189 453 124 999 999 999 999 974 011 712 609 222 317 150 568 448 × 2 = 1 + 0.588 378 906 249 999 999 999 999 948 023 425 218 444 634 301 136 896;
  • 80) 0.588 378 906 249 999 999 999 999 948 023 425 218 444 634 301 136 896 × 2 = 1 + 0.176 757 812 499 999 999 999 999 896 046 850 436 889 268 602 273 792;
  • 81) 0.176 757 812 499 999 999 999 999 896 046 850 436 889 268 602 273 792 × 2 = 0 + 0.353 515 624 999 999 999 999 999 792 093 700 873 778 537 204 547 584;
  • 82) 0.353 515 624 999 999 999 999 999 792 093 700 873 778 537 204 547 584 × 2 = 0 + 0.707 031 249 999 999 999 999 999 584 187 401 747 557 074 409 095 168;
  • 83) 0.707 031 249 999 999 999 999 999 584 187 401 747 557 074 409 095 168 × 2 = 1 + 0.414 062 499 999 999 999 999 999 168 374 803 495 114 148 818 190 336;
  • 84) 0.414 062 499 999 999 999 999 999 168 374 803 495 114 148 818 190 336 × 2 = 0 + 0.828 124 999 999 999 999 999 998 336 749 606 990 228 297 636 380 672;
  • 85) 0.828 124 999 999 999 999 999 998 336 749 606 990 228 297 636 380 672 × 2 = 1 + 0.656 249 999 999 999 999 999 996 673 499 213 980 456 595 272 761 344;
  • 86) 0.656 249 999 999 999 999 999 996 673 499 213 980 456 595 272 761 344 × 2 = 1 + 0.312 499 999 999 999 999 999 993 346 998 427 960 913 190 545 522 688;
  • 87) 0.312 499 999 999 999 999 999 993 346 998 427 960 913 190 545 522 688 × 2 = 0 + 0.624 999 999 999 999 999 999 986 693 996 855 921 826 381 091 045 376;
  • 88) 0.624 999 999 999 999 999 999 986 693 996 855 921 826 381 091 045 376 × 2 = 1 + 0.249 999 999 999 999 999 999 973 387 993 711 843 652 762 182 090 752;
  • 89) 0.249 999 999 999 999 999 999 973 387 993 711 843 652 762 182 090 752 × 2 = 0 + 0.499 999 999 999 999 999 999 946 775 987 423 687 305 524 364 181 504;
  • 90) 0.499 999 999 999 999 999 999 946 775 987 423 687 305 524 364 181 504 × 2 = 0 + 0.999 999 999 999 999 999 999 893 551 974 847 374 611 048 728 363 008;
  • 91) 0.999 999 999 999 999 999 999 893 551 974 847 374 611 048 728 363 008 × 2 = 1 + 0.999 999 999 999 999 999 999 787 103 949 694 749 222 097 456 726 016;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 067 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