0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 271;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 271 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 542;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 542 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 084;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 084 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 168;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 168 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 336;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 336 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 672;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 672 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 344;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 344 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 794 688;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 794 688 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 589 376;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 589 376 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 178 752;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 178 752 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 357 504;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 357 504 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 715 008;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 715 008 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 430 016;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 430 016 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 860 032;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 860 032 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 720 064;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 720 064 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 440 128;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 440 128 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 880 256;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 880 256 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 485 760 512;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 485 760 512 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 971 521 024;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 971 521 024 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 943 042 048;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 943 042 048 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 886 084 096;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 886 084 096 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 772 168 192;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 772 168 192 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 544 336 384;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 544 336 384 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 088 672 768;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 088 672 768 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 177 345 536;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 177 345 536 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 354 691 072;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 354 691 072 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 000 709 382 144;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 000 709 382 144 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 001 418 764 288;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 001 418 764 288 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 002 837 528 576;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 002 837 528 576 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 005 675 057 152;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 005 675 057 152 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 011 350 114 304;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 011 350 114 304 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 022 700 228 608;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 022 700 228 608 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 045 400 457 216;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 045 400 457 216 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 090 800 914 432;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 090 800 914 432 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 181 601 828 864;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 181 601 828 864 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 363 203 657 728;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 363 203 657 728 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 816 726 407 315 456;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 816 726 407 315 456 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 633 452 814 630 912;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 633 452 814 630 912 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 266 905 629 261 824;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 266 905 629 261 824 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 533 811 258 523 648;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 533 811 258 523 648 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 067 622 517 047 296;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 067 622 517 047 296 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 135 245 034 094 592;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 135 245 034 094 592 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 270 490 068 189 184;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 270 490 068 189 184 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 540 980 136 378 368;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 540 980 136 378 368 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 081 960 272 756 736;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 081 960 272 756 736 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 163 920 545 513 472;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 163 920 545 513 472 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 756 327 841 091 026 944;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 756 327 841 091 026 944 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 512 655 682 182 053 888;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 512 655 682 182 053 888 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 025 311 364 364 107 776;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 025 311 364 364 107 776 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 050 622 728 728 215 552;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 050 622 728 728 215 552 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 101 245 457 456 431 104;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 101 245 457 456 431 104 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 202 490 914 912 862 208;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 202 490 914 912 862 208 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 404 981 829 825 724 416;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 404 981 829 825 724 416 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 809 963 659 651 448 832;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 809 963 659 651 448 832 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 619 927 319 302 897 664;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 619 927 319 302 897 664 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 239 854 638 605 795 328;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 239 854 638 605 795 328 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 622 479 709 277 211 590 656;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 622 479 709 277 211 590 656 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 244 959 418 554 423 181 312;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 244 959 418 554 423 181 312 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 489 918 837 108 846 362 624;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 489 918 837 108 846 362 624 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 979 837 674 217 692 725 248;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 979 837 674 217 692 725 248 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 959 675 348 435 385 450 496;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 959 675 348 435 385 450 496 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 919 350 696 870 770 900 992;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 919 350 696 870 770 900 992 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 838 701 393 741 541 801 984;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 838 701 393 741 541 801 984 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 677 402 787 483 083 603 968;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 677 402 787 483 083 603 968 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 354 805 574 966 167 207 936;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 354 805 574 966 167 207 936 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 998 709 611 149 932 334 415 872;
  • 67) 0.831 736 862 659 454 345 703 124 999 998 709 611 149 932 334 415 872 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 997 419 222 299 864 668 831 744;
  • 68) 0.663 473 725 318 908 691 406 249 999 997 419 222 299 864 668 831 744 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 994 838 444 599 729 337 663 488;
  • 69) 0.326 947 450 637 817 382 812 499 999 994 838 444 599 729 337 663 488 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 989 676 889 199 458 675 326 976;
  • 70) 0.653 894 901 275 634 765 624 999 999 989 676 889 199 458 675 326 976 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 979 353 778 398 917 350 653 952;
  • 71) 0.307 789 802 551 269 531 249 999 999 979 353 778 398 917 350 653 952 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 958 707 556 797 834 701 307 904;
  • 72) 0.615 579 605 102 539 062 499 999 999 958 707 556 797 834 701 307 904 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 917 415 113 595 669 402 615 808;
  • 73) 0.231 159 210 205 078 124 999 999 999 917 415 113 595 669 402 615 808 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 834 830 227 191 338 805 231 616;
  • 74) 0.462 318 420 410 156 249 999 999 999 834 830 227 191 338 805 231 616 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 669 660 454 382 677 610 463 232;
  • 75) 0.924 636 840 820 312 499 999 999 999 669 660 454 382 677 610 463 232 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 339 320 908 765 355 220 926 464;
  • 76) 0.849 273 681 640 624 999 999 999 999 339 320 908 765 355 220 926 464 × 2 = 1 + 0.698 547 363 281 249 999 999 999 998 678 641 817 530 710 441 852 928;
  • 77) 0.698 547 363 281 249 999 999 999 998 678 641 817 530 710 441 852 928 × 2 = 1 + 0.397 094 726 562 499 999 999 999 997 357 283 635 061 420 883 705 856;
  • 78) 0.397 094 726 562 499 999 999 999 997 357 283 635 061 420 883 705 856 × 2 = 0 + 0.794 189 453 124 999 999 999 999 994 714 567 270 122 841 767 411 712;
  • 79) 0.794 189 453 124 999 999 999 999 994 714 567 270 122 841 767 411 712 × 2 = 1 + 0.588 378 906 249 999 999 999 999 989 429 134 540 245 683 534 823 424;
  • 80) 0.588 378 906 249 999 999 999 999 989 429 134 540 245 683 534 823 424 × 2 = 1 + 0.176 757 812 499 999 999 999 999 978 858 269 080 491 367 069 646 848;
  • 81) 0.176 757 812 499 999 999 999 999 978 858 269 080 491 367 069 646 848 × 2 = 0 + 0.353 515 624 999 999 999 999 999 957 716 538 160 982 734 139 293 696;
  • 82) 0.353 515 624 999 999 999 999 999 957 716 538 160 982 734 139 293 696 × 2 = 0 + 0.707 031 249 999 999 999 999 999 915 433 076 321 965 468 278 587 392;
  • 83) 0.707 031 249 999 999 999 999 999 915 433 076 321 965 468 278 587 392 × 2 = 1 + 0.414 062 499 999 999 999 999 999 830 866 152 643 930 936 557 174 784;
  • 84) 0.414 062 499 999 999 999 999 999 830 866 152 643 930 936 557 174 784 × 2 = 0 + 0.828 124 999 999 999 999 999 999 661 732 305 287 861 873 114 349 568;
  • 85) 0.828 124 999 999 999 999 999 999 661 732 305 287 861 873 114 349 568 × 2 = 1 + 0.656 249 999 999 999 999 999 999 323 464 610 575 723 746 228 699 136;
  • 86) 0.656 249 999 999 999 999 999 999 323 464 610 575 723 746 228 699 136 × 2 = 1 + 0.312 499 999 999 999 999 999 998 646 929 221 151 447 492 457 398 272;
  • 87) 0.312 499 999 999 999 999 999 998 646 929 221 151 447 492 457 398 272 × 2 = 0 + 0.624 999 999 999 999 999 999 997 293 858 442 302 894 984 914 796 544;
  • 88) 0.624 999 999 999 999 999 999 997 293 858 442 302 894 984 914 796 544 × 2 = 1 + 0.249 999 999 999 999 999 999 994 587 716 884 605 789 969 829 593 088;
  • 89) 0.249 999 999 999 999 999 999 994 587 716 884 605 789 969 829 593 088 × 2 = 0 + 0.499 999 999 999 999 999 999 989 175 433 769 211 579 939 659 186 176;
  • 90) 0.499 999 999 999 999 999 999 989 175 433 769 211 579 939 659 186 176 × 2 = 0 + 0.999 999 999 999 999 999 999 978 350 867 538 423 159 879 318 372 352;
  • 91) 0.999 999 999 999 999 999 999 978 350 867 538 423 159 879 318 372 352 × 2 = 1 + 0.999 999 999 999 999 999 999 956 701 735 076 846 319 758 636 744 704;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 135 5 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