0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 666 02;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 666 02 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 332 04;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 332 04 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 664 08;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 664 08 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 328 16;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 328 16 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 656 32;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 656 32 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 312 64;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 312 64 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 625 28;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 625 28 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 250 56;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 250 56 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 501 12;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 501 12 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 002 24;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 002 24 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 226 004 48;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 226 004 48 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 452 008 96;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 452 008 96 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 904 017 92;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 904 017 92 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 808 035 84;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 808 035 84 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 616 071 68;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 616 071 68 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 232 143 36;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 232 143 36 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 464 286 72;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 464 286 72 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 928 573 44;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 928 573 44 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 857 146 88;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 857 146 88 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 714 293 76;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 714 293 76 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 167 428 587 52;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 167 428 587 52 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 334 857 175 04;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 334 857 175 04 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 669 714 350 08;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 669 714 350 08 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 339 428 700 16;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 339 428 700 16 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 678 857 400 32;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 678 857 400 32 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 357 714 800 64;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 357 714 800 64 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 715 429 601 28;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 715 429 601 28 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 430 859 202 56;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 430 859 202 56 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 634 861 718 405 12;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 634 861 718 405 12 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 269 723 436 810 24;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 269 723 436 810 24 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 539 446 873 620 48;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 539 446 873 620 48 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 078 893 747 240 96;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 078 893 747 240 96 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 157 787 494 481 92;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 157 787 494 481 92 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 315 574 988 963 84;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 315 574 988 963 84 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 631 149 977 927 68;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 631 149 977 927 68 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 262 299 955 855 36;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 262 299 955 855 36 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 524 599 911 710 72;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 524 599 911 710 72 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 049 199 823 421 44;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 049 199 823 421 44 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 322 098 399 646 842 88;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 322 098 399 646 842 88 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 644 196 799 293 685 76;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 644 196 799 293 685 76 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 288 393 598 587 371 52;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 288 393 598 587 371 52 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 576 787 197 174 743 04;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 576 787 197 174 743 04 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 153 574 394 349 486 08;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 153 574 394 349 486 08 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 307 148 788 698 972 16;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 307 148 788 698 972 16 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 614 297 577 397 944 32;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 614 297 577 397 944 32 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 228 595 154 795 888 64;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 228 595 154 795 888 64 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 457 190 309 591 777 28;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 457 190 309 591 777 28 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 404 914 380 619 183 554 56;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 404 914 380 619 183 554 56 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 809 828 761 238 367 109 12;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 809 828 761 238 367 109 12 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 619 657 522 476 734 218 24;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 619 657 522 476 734 218 24 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 239 315 044 953 468 436 48;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 239 315 044 953 468 436 48 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 478 630 089 906 936 872 96;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 478 630 089 906 936 872 96 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 957 260 179 813 873 745 92;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 957 260 179 813 873 745 92 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 914 520 359 627 747 491 84;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 914 520 359 627 747 491 84 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 829 040 719 255 494 983 68;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 829 040 719 255 494 983 68 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 658 081 438 510 989 967 36;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 658 081 438 510 989 967 36 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 316 162 877 021 979 934 72;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 316 162 877 021 979 934 72 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 632 325 754 043 959 869 44;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 632 325 754 043 959 869 44 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 061 264 651 508 087 919 738 88;
  • 60) 0.834 622 944 239 526 987 075 805 664 061 264 651 508 087 919 738 88 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 122 529 303 016 175 839 477 76;
  • 61) 0.669 245 888 479 053 974 151 611 328 122 529 303 016 175 839 477 76 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 245 058 606 032 351 678 955 52;
  • 62) 0.338 491 776 958 107 948 303 222 656 245 058 606 032 351 678 955 52 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 490 117 212 064 703 357 911 04;
  • 63) 0.676 983 553 916 215 896 606 445 312 490 117 212 064 703 357 911 04 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 980 234 424 129 406 715 822 08;
  • 64) 0.353 967 107 832 431 793 212 890 624 980 234 424 129 406 715 822 08 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 960 468 848 258 813 431 644 16;
  • 65) 0.707 934 215 664 863 586 425 781 249 960 468 848 258 813 431 644 16 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 920 937 696 517 626 863 288 32;
  • 66) 0.415 868 431 329 727 172 851 562 499 920 937 696 517 626 863 288 32 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 841 875 393 035 253 726 576 64;
  • 67) 0.831 736 862 659 454 345 703 124 999 841 875 393 035 253 726 576 64 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 683 750 786 070 507 453 153 28;
  • 68) 0.663 473 725 318 908 691 406 249 999 683 750 786 070 507 453 153 28 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 367 501 572 141 014 906 306 56;
  • 69) 0.326 947 450 637 817 382 812 499 999 367 501 572 141 014 906 306 56 × 2 = 0 + 0.653 894 901 275 634 765 624 999 998 735 003 144 282 029 812 613 12;
  • 70) 0.653 894 901 275 634 765 624 999 998 735 003 144 282 029 812 613 12 × 2 = 1 + 0.307 789 802 551 269 531 249 999 997 470 006 288 564 059 625 226 24;
  • 71) 0.307 789 802 551 269 531 249 999 997 470 006 288 564 059 625 226 24 × 2 = 0 + 0.615 579 605 102 539 062 499 999 994 940 012 577 128 119 250 452 48;
  • 72) 0.615 579 605 102 539 062 499 999 994 940 012 577 128 119 250 452 48 × 2 = 1 + 0.231 159 210 205 078 124 999 999 989 880 025 154 256 238 500 904 96;
  • 73) 0.231 159 210 205 078 124 999 999 989 880 025 154 256 238 500 904 96 × 2 = 0 + 0.462 318 420 410 156 249 999 999 979 760 050 308 512 477 001 809 92;
  • 74) 0.462 318 420 410 156 249 999 999 979 760 050 308 512 477 001 809 92 × 2 = 0 + 0.924 636 840 820 312 499 999 999 959 520 100 617 024 954 003 619 84;
  • 75) 0.924 636 840 820 312 499 999 999 959 520 100 617 024 954 003 619 84 × 2 = 1 + 0.849 273 681 640 624 999 999 999 919 040 201 234 049 908 007 239 68;
  • 76) 0.849 273 681 640 624 999 999 999 919 040 201 234 049 908 007 239 68 × 2 = 1 + 0.698 547 363 281 249 999 999 999 838 080 402 468 099 816 014 479 36;
  • 77) 0.698 547 363 281 249 999 999 999 838 080 402 468 099 816 014 479 36 × 2 = 1 + 0.397 094 726 562 499 999 999 999 676 160 804 936 199 632 028 958 72;
  • 78) 0.397 094 726 562 499 999 999 999 676 160 804 936 199 632 028 958 72 × 2 = 0 + 0.794 189 453 124 999 999 999 999 352 321 609 872 399 264 057 917 44;
  • 79) 0.794 189 453 124 999 999 999 999 352 321 609 872 399 264 057 917 44 × 2 = 1 + 0.588 378 906 249 999 999 999 998 704 643 219 744 798 528 115 834 88;
  • 80) 0.588 378 906 249 999 999 999 998 704 643 219 744 798 528 115 834 88 × 2 = 1 + 0.176 757 812 499 999 999 999 997 409 286 439 489 597 056 231 669 76;
  • 81) 0.176 757 812 499 999 999 999 997 409 286 439 489 597 056 231 669 76 × 2 = 0 + 0.353 515 624 999 999 999 999 994 818 572 878 979 194 112 463 339 52;
  • 82) 0.353 515 624 999 999 999 999 994 818 572 878 979 194 112 463 339 52 × 2 = 0 + 0.707 031 249 999 999 999 999 989 637 145 757 958 388 224 926 679 04;
  • 83) 0.707 031 249 999 999 999 999 989 637 145 757 958 388 224 926 679 04 × 2 = 1 + 0.414 062 499 999 999 999 999 979 274 291 515 916 776 449 853 358 08;
  • 84) 0.414 062 499 999 999 999 999 979 274 291 515 916 776 449 853 358 08 × 2 = 0 + 0.828 124 999 999 999 999 999 958 548 583 031 833 552 899 706 716 16;
  • 85) 0.828 124 999 999 999 999 999 958 548 583 031 833 552 899 706 716 16 × 2 = 1 + 0.656 249 999 999 999 999 999 917 097 166 063 667 105 799 413 432 32;
  • 86) 0.656 249 999 999 999 999 999 917 097 166 063 667 105 799 413 432 32 × 2 = 1 + 0.312 499 999 999 999 999 999 834 194 332 127 334 211 598 826 864 64;
  • 87) 0.312 499 999 999 999 999 999 834 194 332 127 334 211 598 826 864 64 × 2 = 0 + 0.624 999 999 999 999 999 999 668 388 664 254 668 423 197 653 729 28;
  • 88) 0.624 999 999 999 999 999 999 668 388 664 254 668 423 197 653 729 28 × 2 = 1 + 0.249 999 999 999 999 999 999 336 777 328 509 336 846 395 307 458 56;
  • 89) 0.249 999 999 999 999 999 999 336 777 328 509 336 846 395 307 458 56 × 2 = 0 + 0.499 999 999 999 999 999 998 673 554 657 018 673 692 790 614 917 12;
  • 90) 0.499 999 999 999 999 999 998 673 554 657 018 673 692 790 614 917 12 × 2 = 0 + 0.999 999 999 999 999 999 997 347 109 314 037 347 385 581 229 834 24;
  • 91) 0.999 999 999 999 999 999 997 347 109 314 037 347 385 581 229 834 24 × 2 = 1 + 0.999 999 999 999 999 999 994 694 218 628 074 694 771 162 459 668 48;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 01 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