0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 95;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 95 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 9;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 9 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 719 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 719 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 438 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 438 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 876 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 876 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 753 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 753 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 507 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 507 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 014 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 014 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 028 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 028 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 057 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 057 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 115 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 115 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 230 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 230 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 680 460 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 680 460 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 360 921 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 360 921 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 721 843 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 721 843 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 443 686 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 443 686 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 887 372 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 887 372 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 774 745 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 774 745 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 549 491 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 549 491 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 098 982 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 098 982 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 197 964 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 197 964 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 395 929 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 395 929 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 744 791 859 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 744 791 859 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 489 583 718 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 489 583 718 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 979 167 436 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 979 167 436 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 958 334 873 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 958 334 873 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 916 669 747 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 916 669 747 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 833 339 494 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 833 339 494 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 666 678 988 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 666 678 988 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 333 357 977 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 333 357 977 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 666 715 955 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 666 715 955 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 349 333 431 910 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 349 333 431 910 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 698 666 863 820 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 698 666 863 820 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 397 333 727 641 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 397 333 727 641 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 794 667 455 283 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 794 667 455 283 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 589 334 910 566 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 589 334 910 566 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 178 669 821 132 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 178 669 821 132 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 357 339 642 265 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 357 339 642 265 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 714 679 284 531 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 714 679 284 531 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 429 358 569 062 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 429 358 569 062 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 858 717 138 124 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 858 717 138 124 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 341 717 434 276 249 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 341 717 434 276 249 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 683 434 868 552 499 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 683 434 868 552 499 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 366 869 737 104 998 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 366 869 737 104 998 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 733 739 474 209 996 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 733 739 474 209 996 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 467 478 948 419 993 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 467 478 948 419 993 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 934 957 896 839 987 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 934 957 896 839 987 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 869 915 793 679 974 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 869 915 793 679 974 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 739 831 587 359 948 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 739 831 587 359 948 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 479 663 174 719 897 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 479 663 174 719 897 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 959 326 349 439 795 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 959 326 349 439 795 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 949 918 652 698 879 590 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 949 918 652 698 879 590 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 899 837 305 397 759 180 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 899 837 305 397 759 180 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 799 674 610 795 518 361 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 799 674 610 795 518 361 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 599 349 221 591 036 723 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 599 349 221 591 036 723 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 198 698 443 182 073 446 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 198 698 443 182 073 446 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 397 396 886 364 146 892 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 397 396 886 364 146 892 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 794 793 772 728 293 785 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 794 793 772 728 293 785 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 589 587 545 456 587 571 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 589 587 545 456 587 571 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 179 175 090 913 175 142 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 179 175 090 913 175 142 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 358 350 181 826 350 284 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 358 350 181 826 350 284 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 996 716 700 363 652 700 569 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 996 716 700 363 652 700 569 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 993 433 400 727 305 401 139 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 993 433 400 727 305 401 139 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 986 866 801 454 610 802 278 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 986 866 801 454 610 802 278 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 973 733 602 909 221 604 556 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 973 733 602 909 221 604 556 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 947 467 205 818 443 209 113 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 947 467 205 818 443 209 113 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 894 934 411 636 886 418 227 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 894 934 411 636 886 418 227 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 789 868 823 273 772 836 454 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 789 868 823 273 772 836 454 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 579 737 646 547 545 672 908 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 579 737 646 547 545 672 908 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 159 475 293 095 091 345 817 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 159 475 293 095 091 345 817 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 318 950 586 190 182 691 635 2;
  • 74) 0.462 318 420 410 156 249 999 999 998 318 950 586 190 182 691 635 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 996 637 901 172 380 365 383 270 4;
  • 75) 0.924 636 840 820 312 499 999 999 996 637 901 172 380 365 383 270 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 993 275 802 344 760 730 766 540 8;
  • 76) 0.849 273 681 640 624 999 999 999 993 275 802 344 760 730 766 540 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 986 551 604 689 521 461 533 081 6;
  • 77) 0.698 547 363 281 249 999 999 999 986 551 604 689 521 461 533 081 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 973 103 209 379 042 923 066 163 2;
  • 78) 0.397 094 726 562 499 999 999 999 973 103 209 379 042 923 066 163 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 946 206 418 758 085 846 132 326 4;
  • 79) 0.794 189 453 124 999 999 999 999 946 206 418 758 085 846 132 326 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 892 412 837 516 171 692 264 652 8;
  • 80) 0.588 378 906 249 999 999 999 999 892 412 837 516 171 692 264 652 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 784 825 675 032 343 384 529 305 6;
  • 81) 0.176 757 812 499 999 999 999 999 784 825 675 032 343 384 529 305 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 569 651 350 064 686 769 058 611 2;
  • 82) 0.353 515 624 999 999 999 999 999 569 651 350 064 686 769 058 611 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 139 302 700 129 373 538 117 222 4;
  • 83) 0.707 031 249 999 999 999 999 999 139 302 700 129 373 538 117 222 4 × 2 = 1 + 0.414 062 499 999 999 999 999 998 278 605 400 258 747 076 234 444 8;
  • 84) 0.414 062 499 999 999 999 999 998 278 605 400 258 747 076 234 444 8 × 2 = 0 + 0.828 124 999 999 999 999 999 996 557 210 800 517 494 152 468 889 6;
  • 85) 0.828 124 999 999 999 999 999 996 557 210 800 517 494 152 468 889 6 × 2 = 1 + 0.656 249 999 999 999 999 999 993 114 421 601 034 988 304 937 779 2;
  • 86) 0.656 249 999 999 999 999 999 993 114 421 601 034 988 304 937 779 2 × 2 = 1 + 0.312 499 999 999 999 999 999 986 228 843 202 069 976 609 875 558 4;
  • 87) 0.312 499 999 999 999 999 999 986 228 843 202 069 976 609 875 558 4 × 2 = 0 + 0.624 999 999 999 999 999 999 972 457 686 404 139 953 219 751 116 8;
  • 88) 0.624 999 999 999 999 999 999 972 457 686 404 139 953 219 751 116 8 × 2 = 1 + 0.249 999 999 999 999 999 999 944 915 372 808 279 906 439 502 233 6;
  • 89) 0.249 999 999 999 999 999 999 944 915 372 808 279 906 439 502 233 6 × 2 = 0 + 0.499 999 999 999 999 999 999 889 830 745 616 559 812 879 004 467 2;
  • 90) 0.499 999 999 999 999 999 999 889 830 745 616 559 812 879 004 467 2 × 2 = 0 + 0.999 999 999 999 999 999 999 779 661 491 233 119 625 758 008 934 4;
  • 91) 0.999 999 999 999 999 999 999 779 661 491 233 119 625 758 008 934 4 × 2 = 1 + 0.999 999 999 999 999 999 999 559 322 982 466 239 251 516 017 868 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 975 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