0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 95 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 95(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 95(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 95.

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 95 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 9;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 9 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 718 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 718 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 436 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 436 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 873 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 873 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 747 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 747 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 494 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 494 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 988 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 988 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 977 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 977 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 955 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 955 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 910 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 910 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 820 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 820 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 679 641 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 679 641 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 359 283 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 359 283 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 718 566 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 718 566 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 437 132 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 437 132 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 874 265 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 874 265 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 748 531 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 748 531 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 497 062 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 497 062 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 994 124 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 994 124 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 988 249 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 988 249 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 976 499 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 976 499 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 743 952 998 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 743 952 998 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 487 905 996 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 487 905 996 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 975 811 993 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 975 811 993 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 951 623 987 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 951 623 987 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 903 247 974 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 903 247 974 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 806 495 948 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 806 495 948 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 612 991 897 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 612 991 897 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 225 983 795 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 225 983 795 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 451 967 590 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 451 967 590 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 348 903 935 180 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 348 903 935 180 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 697 807 870 361 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 697 807 870 361 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 395 615 740 723 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 395 615 740 723 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 791 231 481 446 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 791 231 481 446 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 582 462 962 892 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 582 462 962 892 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 164 925 925 785 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 164 925 925 785 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 329 851 851 571 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 329 851 851 571 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 659 703 703 142 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 659 703 703 142 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 319 407 406 284 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 319 407 406 284 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 638 814 812 569 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 638 814 812 569 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 341 277 629 625 139 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 341 277 629 625 139 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 682 555 259 250 278 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 682 555 259 250 278 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 365 110 518 500 556 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 365 110 518 500 556 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 730 221 037 001 113 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 730 221 037 001 113 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 460 442 074 002 227 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 460 442 074 002 227 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 920 884 148 004 454 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 920 884 148 004 454 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 841 768 296 008 908 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 841 768 296 008 908 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 683 536 592 017 817 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 683 536 592 017 817 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 367 073 184 035 635 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 367 073 184 035 635 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 734 146 368 071 270 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 734 146 368 071 270 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 949 468 292 736 142 540 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 949 468 292 736 142 540 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 898 936 585 472 285 081 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 898 936 585 472 285 081 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 797 873 170 944 570 163 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 797 873 170 944 570 163 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 595 746 341 889 140 326 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 595 746 341 889 140 326 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 191 492 683 778 280 652 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 191 492 683 778 280 652 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 382 985 367 556 561 305 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 382 985 367 556 561 305 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 765 970 735 113 122 611 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 765 970 735 113 122 611 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 531 941 470 226 245 222 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 531 941 470 226 245 222 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 063 882 940 452 490 444 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 063 882 940 452 490 444 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 127 765 880 904 980 889 6;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 127 765 880 904 980 889 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 996 255 531 761 809 961 779 2;
  • 65) 0.707 934 215 664 863 586 425 781 249 996 255 531 761 809 961 779 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 992 511 063 523 619 923 558 4;
  • 66) 0.415 868 431 329 727 172 851 562 499 992 511 063 523 619 923 558 4 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 985 022 127 047 239 847 116 8;
  • 67) 0.831 736 862 659 454 345 703 124 999 985 022 127 047 239 847 116 8 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 970 044 254 094 479 694 233 6;
  • 68) 0.663 473 725 318 908 691 406 249 999 970 044 254 094 479 694 233 6 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 940 088 508 188 959 388 467 2;
  • 69) 0.326 947 450 637 817 382 812 499 999 940 088 508 188 959 388 467 2 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 880 177 016 377 918 776 934 4;
  • 70) 0.653 894 901 275 634 765 624 999 999 880 177 016 377 918 776 934 4 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 760 354 032 755 837 553 868 8;
  • 71) 0.307 789 802 551 269 531 249 999 999 760 354 032 755 837 553 868 8 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 520 708 065 511 675 107 737 6;
  • 72) 0.615 579 605 102 539 062 499 999 999 520 708 065 511 675 107 737 6 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 041 416 131 023 350 215 475 2;
  • 73) 0.231 159 210 205 078 124 999 999 999 041 416 131 023 350 215 475 2 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 082 832 262 046 700 430 950 4;
  • 74) 0.462 318 420 410 156 249 999 999 998 082 832 262 046 700 430 950 4 × 2 = 0 + 0.924 636 840 820 312 499 999 999 996 165 664 524 093 400 861 900 8;
  • 75) 0.924 636 840 820 312 499 999 999 996 165 664 524 093 400 861 900 8 × 2 = 1 + 0.849 273 681 640 624 999 999 999 992 331 329 048 186 801 723 801 6;
  • 76) 0.849 273 681 640 624 999 999 999 992 331 329 048 186 801 723 801 6 × 2 = 1 + 0.698 547 363 281 249 999 999 999 984 662 658 096 373 603 447 603 2;
  • 77) 0.698 547 363 281 249 999 999 999 984 662 658 096 373 603 447 603 2 × 2 = 1 + 0.397 094 726 562 499 999 999 999 969 325 316 192 747 206 895 206 4;
  • 78) 0.397 094 726 562 499 999 999 999 969 325 316 192 747 206 895 206 4 × 2 = 0 + 0.794 189 453 124 999 999 999 999 938 650 632 385 494 413 790 412 8;
  • 79) 0.794 189 453 124 999 999 999 999 938 650 632 385 494 413 790 412 8 × 2 = 1 + 0.588 378 906 249 999 999 999 999 877 301 264 770 988 827 580 825 6;
  • 80) 0.588 378 906 249 999 999 999 999 877 301 264 770 988 827 580 825 6 × 2 = 1 + 0.176 757 812 499 999 999 999 999 754 602 529 541 977 655 161 651 2;
  • 81) 0.176 757 812 499 999 999 999 999 754 602 529 541 977 655 161 651 2 × 2 = 0 + 0.353 515 624 999 999 999 999 999 509 205 059 083 955 310 323 302 4;
  • 82) 0.353 515 624 999 999 999 999 999 509 205 059 083 955 310 323 302 4 × 2 = 0 + 0.707 031 249 999 999 999 999 999 018 410 118 167 910 620 646 604 8;
  • 83) 0.707 031 249 999 999 999 999 999 018 410 118 167 910 620 646 604 8 × 2 = 1 + 0.414 062 499 999 999 999 999 998 036 820 236 335 821 241 293 209 6;
  • 84) 0.414 062 499 999 999 999 999 998 036 820 236 335 821 241 293 209 6 × 2 = 0 + 0.828 124 999 999 999 999 999 996 073 640 472 671 642 482 586 419 2;
  • 85) 0.828 124 999 999 999 999 999 996 073 640 472 671 642 482 586 419 2 × 2 = 1 + 0.656 249 999 999 999 999 999 992 147 280 945 343 284 965 172 838 4;
  • 86) 0.656 249 999 999 999 999 999 992 147 280 945 343 284 965 172 838 4 × 2 = 1 + 0.312 499 999 999 999 999 999 984 294 561 890 686 569 930 345 676 8;
  • 87) 0.312 499 999 999 999 999 999 984 294 561 890 686 569 930 345 676 8 × 2 = 0 + 0.624 999 999 999 999 999 999 968 589 123 781 373 139 860 691 353 6;
  • 88) 0.624 999 999 999 999 999 999 968 589 123 781 373 139 860 691 353 6 × 2 = 1 + 0.249 999 999 999 999 999 999 937 178 247 562 746 279 721 382 707 2;
  • 89) 0.249 999 999 999 999 999 999 937 178 247 562 746 279 721 382 707 2 × 2 = 0 + 0.499 999 999 999 999 999 999 874 356 495 125 492 559 442 765 414 4;
  • 90) 0.499 999 999 999 999 999 999 874 356 495 125 492 559 442 765 414 4 × 2 = 0 + 0.999 999 999 999 999 999 999 748 712 990 250 985 118 885 530 828 8;
  • 91) 0.999 999 999 999 999 999 999 748 712 990 250 985 118 885 530 828 8 × 2 = 1 + 0.999 999 999 999 999 999 999 497 425 980 501 970 237 771 061 657 6;

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 95(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 95(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 95(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 95 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