0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 012;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 012 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 024;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 024 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 048;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 048 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 096;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 096 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 720 192;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 720 192 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 440 384;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 440 384 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 880 768;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 880 768 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 761 536;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 761 536 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 523 072;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 523 072 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 046 144;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 046 144 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 092 288;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 092 288 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 184 576;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 184 576 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 369 152;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 369 152 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 738 304;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 738 304 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 681 476 608;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 681 476 608 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 362 953 216;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 362 953 216 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 725 906 432;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 725 906 432 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 451 812 864;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 451 812 864 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 903 625 728;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 903 625 728 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 807 251 456;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 807 251 456 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 614 502 912;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 614 502 912 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 229 005 824;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 229 005 824 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 458 011 648;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 458 011 648 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 916 023 296;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 916 023 296 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 745 832 046 592;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 745 832 046 592 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 491 664 093 184;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 491 664 093 184 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 983 328 186 368;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 983 328 186 368 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 966 656 372 736;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 966 656 372 736 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 933 312 745 472;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 933 312 745 472 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 866 625 490 944;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 866 625 490 944 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 733 250 981 888;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 733 250 981 888 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 466 501 963 776;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 466 501 963 776 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 933 003 927 552;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 933 003 927 552 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 349 866 007 855 104;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 349 866 007 855 104 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 699 732 015 710 208;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 699 732 015 710 208 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 399 464 031 420 416;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 399 464 031 420 416 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 798 928 062 840 832;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 798 928 062 840 832 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 597 856 125 681 664;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 597 856 125 681 664 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 195 712 251 363 328;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 195 712 251 363 328 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 391 424 502 726 656;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 391 424 502 726 656 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 782 849 005 453 312;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 782 849 005 453 312 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 565 698 010 906 624;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 565 698 010 906 624 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 131 396 021 813 248;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 131 396 021 813 248 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 342 262 792 043 626 496;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 342 262 792 043 626 496 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 684 525 584 087 252 992;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 684 525 584 087 252 992 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 369 051 168 174 505 984;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 369 051 168 174 505 984 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 738 102 336 349 011 968;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 738 102 336 349 011 968 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 476 204 672 698 023 936;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 476 204 672 698 023 936 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 952 409 345 396 047 872;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 952 409 345 396 047 872 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 904 818 690 792 095 744;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 904 818 690 792 095 744 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 809 637 381 584 191 488;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 809 637 381 584 191 488 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 619 274 763 168 382 976;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 619 274 763 168 382 976 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 238 549 526 336 765 952;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 238 549 526 336 765 952 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 950 477 099 052 673 531 904;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 950 477 099 052 673 531 904 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 900 954 198 105 347 063 808;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 900 954 198 105 347 063 808 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 801 908 396 210 694 127 616;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 801 908 396 210 694 127 616 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 603 816 792 421 388 255 232;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 603 816 792 421 388 255 232 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 207 633 584 842 776 510 464;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 207 633 584 842 776 510 464 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 415 267 169 685 553 020 928;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 415 267 169 685 553 020 928 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 830 534 339 371 106 041 856;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 830 534 339 371 106 041 856 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 661 068 678 742 212 083 712;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 661 068 678 742 212 083 712 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 322 137 357 484 424 167 424;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 322 137 357 484 424 167 424 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 644 274 714 968 848 334 848;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 644 274 714 968 848 334 848 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 997 288 549 429 937 696 669 696;
  • 65) 0.707 934 215 664 863 586 425 781 249 997 288 549 429 937 696 669 696 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 994 577 098 859 875 393 339 392;
  • 66) 0.415 868 431 329 727 172 851 562 499 994 577 098 859 875 393 339 392 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 989 154 197 719 750 786 678 784;
  • 67) 0.831 736 862 659 454 345 703 124 999 989 154 197 719 750 786 678 784 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 978 308 395 439 501 573 357 568;
  • 68) 0.663 473 725 318 908 691 406 249 999 978 308 395 439 501 573 357 568 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 956 616 790 879 003 146 715 136;
  • 69) 0.326 947 450 637 817 382 812 499 999 956 616 790 879 003 146 715 136 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 913 233 581 758 006 293 430 272;
  • 70) 0.653 894 901 275 634 765 624 999 999 913 233 581 758 006 293 430 272 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 826 467 163 516 012 586 860 544;
  • 71) 0.307 789 802 551 269 531 249 999 999 826 467 163 516 012 586 860 544 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 652 934 327 032 025 173 721 088;
  • 72) 0.615 579 605 102 539 062 499 999 999 652 934 327 032 025 173 721 088 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 305 868 654 064 050 347 442 176;
  • 73) 0.231 159 210 205 078 124 999 999 999 305 868 654 064 050 347 442 176 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 611 737 308 128 100 694 884 352;
  • 74) 0.462 318 420 410 156 249 999 999 998 611 737 308 128 100 694 884 352 × 2 = 0 + 0.924 636 840 820 312 499 999 999 997 223 474 616 256 201 389 768 704;
  • 75) 0.924 636 840 820 312 499 999 999 997 223 474 616 256 201 389 768 704 × 2 = 1 + 0.849 273 681 640 624 999 999 999 994 446 949 232 512 402 779 537 408;
  • 76) 0.849 273 681 640 624 999 999 999 994 446 949 232 512 402 779 537 408 × 2 = 1 + 0.698 547 363 281 249 999 999 999 988 893 898 465 024 805 559 074 816;
  • 77) 0.698 547 363 281 249 999 999 999 988 893 898 465 024 805 559 074 816 × 2 = 1 + 0.397 094 726 562 499 999 999 999 977 787 796 930 049 611 118 149 632;
  • 78) 0.397 094 726 562 499 999 999 999 977 787 796 930 049 611 118 149 632 × 2 = 0 + 0.794 189 453 124 999 999 999 999 955 575 593 860 099 222 236 299 264;
  • 79) 0.794 189 453 124 999 999 999 999 955 575 593 860 099 222 236 299 264 × 2 = 1 + 0.588 378 906 249 999 999 999 999 911 151 187 720 198 444 472 598 528;
  • 80) 0.588 378 906 249 999 999 999 999 911 151 187 720 198 444 472 598 528 × 2 = 1 + 0.176 757 812 499 999 999 999 999 822 302 375 440 396 888 945 197 056;
  • 81) 0.176 757 812 499 999 999 999 999 822 302 375 440 396 888 945 197 056 × 2 = 0 + 0.353 515 624 999 999 999 999 999 644 604 750 880 793 777 890 394 112;
  • 82) 0.353 515 624 999 999 999 999 999 644 604 750 880 793 777 890 394 112 × 2 = 0 + 0.707 031 249 999 999 999 999 999 289 209 501 761 587 555 780 788 224;
  • 83) 0.707 031 249 999 999 999 999 999 289 209 501 761 587 555 780 788 224 × 2 = 1 + 0.414 062 499 999 999 999 999 998 578 419 003 523 175 111 561 576 448;
  • 84) 0.414 062 499 999 999 999 999 998 578 419 003 523 175 111 561 576 448 × 2 = 0 + 0.828 124 999 999 999 999 999 997 156 838 007 046 350 223 123 152 896;
  • 85) 0.828 124 999 999 999 999 999 997 156 838 007 046 350 223 123 152 896 × 2 = 1 + 0.656 249 999 999 999 999 999 994 313 676 014 092 700 446 246 305 792;
  • 86) 0.656 249 999 999 999 999 999 994 313 676 014 092 700 446 246 305 792 × 2 = 1 + 0.312 499 999 999 999 999 999 988 627 352 028 185 400 892 492 611 584;
  • 87) 0.312 499 999 999 999 999 999 988 627 352 028 185 400 892 492 611 584 × 2 = 0 + 0.624 999 999 999 999 999 999 977 254 704 056 370 801 784 985 223 168;
  • 88) 0.624 999 999 999 999 999 999 977 254 704 056 370 801 784 985 223 168 × 2 = 1 + 0.249 999 999 999 999 999 999 954 509 408 112 741 603 569 970 446 336;
  • 89) 0.249 999 999 999 999 999 999 954 509 408 112 741 603 569 970 446 336 × 2 = 0 + 0.499 999 999 999 999 999 999 909 018 816 225 483 207 139 940 892 672;
  • 90) 0.499 999 999 999 999 999 999 909 018 816 225 483 207 139 940 892 672 × 2 = 0 + 0.999 999 999 999 999 999 999 818 037 632 450 966 414 279 881 785 344;
  • 91) 0.999 999 999 999 999 999 999 818 037 632 450 966 414 279 881 785 344 × 2 = 1 + 0.999 999 999 999 999 999 999 636 075 264 901 932 828 559 763 570 688;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 006 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