0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 661 76;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 661 76 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 323 52;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 323 52 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 647 04;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 647 04 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 294 08;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 294 08 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 588 16;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 588 16 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 176 32;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 176 32 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 352 64;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 352 64 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 652 705 28;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 652 705 28 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 305 410 56;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 305 410 56 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 610 821 12;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 610 821 12 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 221 642 24;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 221 642 24 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 443 284 48;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 443 284 48 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 886 568 96;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 886 568 96 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 773 137 92;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 773 137 92 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 546 275 84;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 546 275 84 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 092 551 68;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 092 551 68 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 185 103 36;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 185 103 36 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 370 206 72;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 370 206 72 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 040 740 413 44;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 040 740 413 44 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 081 480 826 88;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 081 480 826 88 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 162 961 653 76;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 162 961 653 76 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 325 923 307 52;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 325 923 307 52 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 651 846 615 04;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 651 846 615 04 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 303 693 230 08;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 303 693 230 08 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 607 386 460 16;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 607 386 460 16 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 214 772 920 32;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 214 772 920 32 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 429 545 840 64;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 429 545 840 64 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 316 859 091 681 28;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 316 859 091 681 28 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 633 718 183 362 56;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 633 718 183 362 56 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 267 436 366 725 12;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 267 436 366 725 12 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 534 872 733 450 24;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 534 872 733 450 24 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 069 745 466 900 48;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 069 745 466 900 48 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 139 490 933 800 96;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 139 490 933 800 96 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 278 981 867 601 92;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 278 981 867 601 92 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 557 963 735 203 84;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 557 963 735 203 84 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 115 927 470 407 68;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 115 927 470 407 68 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 231 854 940 815 36;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 231 854 940 815 36 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 660 463 709 881 630 72;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 660 463 709 881 630 72 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 320 927 419 763 261 44;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 320 927 419 763 261 44 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 641 854 839 526 522 88;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 641 854 839 526 522 88 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 283 709 679 053 045 76;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 283 709 679 053 045 76 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 567 419 358 106 091 52;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 567 419 358 106 091 52 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 134 838 716 212 183 04;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 134 838 716 212 183 04 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 269 677 432 424 366 08;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 269 677 432 424 366 08 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 539 354 864 848 732 16;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 539 354 864 848 732 16 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 078 709 729 697 464 32;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 078 709 729 697 464 32 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 157 419 459 394 928 64;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 157 419 459 394 928 64 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 404 314 838 918 789 857 28;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 404 314 838 918 789 857 28 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 808 629 677 837 579 714 56;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 808 629 677 837 579 714 56 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 617 259 355 675 159 429 12;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 617 259 355 675 159 429 12 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 234 518 711 350 318 858 24;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 234 518 711 350 318 858 24 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 469 037 422 700 637 716 48;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 469 037 422 700 637 716 48 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 938 074 845 401 275 432 96;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 938 074 845 401 275 432 96 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 876 149 690 802 550 865 92;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 876 149 690 802 550 865 92 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 752 299 381 605 101 731 84;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 752 299 381 605 101 731 84 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 504 598 763 210 203 463 68;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 504 598 763 210 203 463 68 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 009 197 526 420 406 927 36;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 009 197 526 420 406 927 36 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 018 395 052 840 813 854 72;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 018 395 052 840 813 854 72 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 060 036 790 105 681 627 709 44;
  • 60) 0.834 622 944 239 526 987 075 805 664 060 036 790 105 681 627 709 44 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 120 073 580 211 363 255 418 88;
  • 61) 0.669 245 888 479 053 974 151 611 328 120 073 580 211 363 255 418 88 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 240 147 160 422 726 510 837 76;
  • 62) 0.338 491 776 958 107 948 303 222 656 240 147 160 422 726 510 837 76 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 480 294 320 845 453 021 675 52;
  • 63) 0.676 983 553 916 215 896 606 445 312 480 294 320 845 453 021 675 52 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 960 588 641 690 906 043 351 04;
  • 64) 0.353 967 107 832 431 793 212 890 624 960 588 641 690 906 043 351 04 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 921 177 283 381 812 086 702 08;
  • 65) 0.707 934 215 664 863 586 425 781 249 921 177 283 381 812 086 702 08 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 842 354 566 763 624 173 404 16;
  • 66) 0.415 868 431 329 727 172 851 562 499 842 354 566 763 624 173 404 16 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 684 709 133 527 248 346 808 32;
  • 67) 0.831 736 862 659 454 345 703 124 999 684 709 133 527 248 346 808 32 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 369 418 267 054 496 693 616 64;
  • 68) 0.663 473 725 318 908 691 406 249 999 369 418 267 054 496 693 616 64 × 2 = 1 + 0.326 947 450 637 817 382 812 499 998 738 836 534 108 993 387 233 28;
  • 69) 0.326 947 450 637 817 382 812 499 998 738 836 534 108 993 387 233 28 × 2 = 0 + 0.653 894 901 275 634 765 624 999 997 477 673 068 217 986 774 466 56;
  • 70) 0.653 894 901 275 634 765 624 999 997 477 673 068 217 986 774 466 56 × 2 = 1 + 0.307 789 802 551 269 531 249 999 994 955 346 136 435 973 548 933 12;
  • 71) 0.307 789 802 551 269 531 249 999 994 955 346 136 435 973 548 933 12 × 2 = 0 + 0.615 579 605 102 539 062 499 999 989 910 692 272 871 947 097 866 24;
  • 72) 0.615 579 605 102 539 062 499 999 989 910 692 272 871 947 097 866 24 × 2 = 1 + 0.231 159 210 205 078 124 999 999 979 821 384 545 743 894 195 732 48;
  • 73) 0.231 159 210 205 078 124 999 999 979 821 384 545 743 894 195 732 48 × 2 = 0 + 0.462 318 420 410 156 249 999 999 959 642 769 091 487 788 391 464 96;
  • 74) 0.462 318 420 410 156 249 999 999 959 642 769 091 487 788 391 464 96 × 2 = 0 + 0.924 636 840 820 312 499 999 999 919 285 538 182 975 576 782 929 92;
  • 75) 0.924 636 840 820 312 499 999 999 919 285 538 182 975 576 782 929 92 × 2 = 1 + 0.849 273 681 640 624 999 999 999 838 571 076 365 951 153 565 859 84;
  • 76) 0.849 273 681 640 624 999 999 999 838 571 076 365 951 153 565 859 84 × 2 = 1 + 0.698 547 363 281 249 999 999 999 677 142 152 731 902 307 131 719 68;
  • 77) 0.698 547 363 281 249 999 999 999 677 142 152 731 902 307 131 719 68 × 2 = 1 + 0.397 094 726 562 499 999 999 999 354 284 305 463 804 614 263 439 36;
  • 78) 0.397 094 726 562 499 999 999 999 354 284 305 463 804 614 263 439 36 × 2 = 0 + 0.794 189 453 124 999 999 999 998 708 568 610 927 609 228 526 878 72;
  • 79) 0.794 189 453 124 999 999 999 998 708 568 610 927 609 228 526 878 72 × 2 = 1 + 0.588 378 906 249 999 999 999 997 417 137 221 855 218 457 053 757 44;
  • 80) 0.588 378 906 249 999 999 999 997 417 137 221 855 218 457 053 757 44 × 2 = 1 + 0.176 757 812 499 999 999 999 994 834 274 443 710 436 914 107 514 88;
  • 81) 0.176 757 812 499 999 999 999 994 834 274 443 710 436 914 107 514 88 × 2 = 0 + 0.353 515 624 999 999 999 999 989 668 548 887 420 873 828 215 029 76;
  • 82) 0.353 515 624 999 999 999 999 989 668 548 887 420 873 828 215 029 76 × 2 = 0 + 0.707 031 249 999 999 999 999 979 337 097 774 841 747 656 430 059 52;
  • 83) 0.707 031 249 999 999 999 999 979 337 097 774 841 747 656 430 059 52 × 2 = 1 + 0.414 062 499 999 999 999 999 958 674 195 549 683 495 312 860 119 04;
  • 84) 0.414 062 499 999 999 999 999 958 674 195 549 683 495 312 860 119 04 × 2 = 0 + 0.828 124 999 999 999 999 999 917 348 391 099 366 990 625 720 238 08;
  • 85) 0.828 124 999 999 999 999 999 917 348 391 099 366 990 625 720 238 08 × 2 = 1 + 0.656 249 999 999 999 999 999 834 696 782 198 733 981 251 440 476 16;
  • 86) 0.656 249 999 999 999 999 999 834 696 782 198 733 981 251 440 476 16 × 2 = 1 + 0.312 499 999 999 999 999 999 669 393 564 397 467 962 502 880 952 32;
  • 87) 0.312 499 999 999 999 999 999 669 393 564 397 467 962 502 880 952 32 × 2 = 0 + 0.624 999 999 999 999 999 999 338 787 128 794 935 925 005 761 904 64;
  • 88) 0.624 999 999 999 999 999 999 338 787 128 794 935 925 005 761 904 64 × 2 = 1 + 0.249 999 999 999 999 999 998 677 574 257 589 871 850 011 523 809 28;
  • 89) 0.249 999 999 999 999 999 998 677 574 257 589 871 850 011 523 809 28 × 2 = 0 + 0.499 999 999 999 999 999 997 355 148 515 179 743 700 023 047 618 56;
  • 90) 0.499 999 999 999 999 999 997 355 148 515 179 743 700 023 047 618 56 × 2 = 0 + 0.999 999 999 999 999 999 994 710 297 030 359 487 400 046 095 237 12;
  • 91) 0.999 999 999 999 999 999 994 710 297 030 359 487 400 046 095 237 12 × 2 = 1 + 0.999 999 999 999 999 999 989 420 594 060 718 974 800 092 190 474 24;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 330 88 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