0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 83;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 83 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 66;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 66 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 32;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 32 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 64;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 64 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 717 28;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 717 28 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 434 56;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 434 56 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 869 12;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 869 12 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 738 24;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 738 24 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 476 48;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 476 48 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 952 96;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 952 96 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 905 92;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 905 92 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 811 84;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 811 84 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 623 68;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 623 68 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 247 36;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 839 247 36 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 678 494 72;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 678 494 72 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 356 989 44;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 356 989 44 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 713 978 88;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 713 978 88 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 427 957 76;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 427 957 76 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 855 915 52;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 855 915 52 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 711 831 04;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 711 831 04 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 423 662 08;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 423 662 08 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 847 324 16;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 847 324 16 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 694 648 32;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 694 648 32 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 389 296 64;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 389 296 64 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 778 593 28;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 778 593 28 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 485 557 186 56;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 485 557 186 56 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 971 114 373 12;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 971 114 373 12 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 942 228 746 24;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 942 228 746 24 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 884 457 492 48;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 884 457 492 48 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 768 914 984 96;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 768 914 984 96 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 537 829 969 92;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 537 829 969 92 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 075 659 939 84;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 075 659 939 84 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 151 319 879 68;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 151 319 879 68 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 348 302 639 759 36;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 348 302 639 759 36 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 696 605 279 518 72;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 696 605 279 518 72 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 393 210 559 037 44;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 393 210 559 037 44 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 786 421 118 074 88;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 786 421 118 074 88 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 572 842 236 149 76;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 572 842 236 149 76 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 145 684 472 299 52;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 145 684 472 299 52 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 291 368 944 599 04;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 291 368 944 599 04 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 582 737 889 198 08;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 582 737 889 198 08 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 165 475 778 396 16;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 165 475 778 396 16 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 330 951 556 792 32;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 330 951 556 792 32 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 340 661 903 113 584 64;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 340 661 903 113 584 64 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 681 323 806 227 169 28;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 681 323 806 227 169 28 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 362 647 612 454 338 56;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 362 647 612 454 338 56 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 725 295 224 908 677 12;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 725 295 224 908 677 12 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 450 590 449 817 354 24;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 450 590 449 817 354 24 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 901 180 899 634 708 48;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 901 180 899 634 708 48 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 802 361 799 269 416 96;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 802 361 799 269 416 96 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 604 723 598 538 833 92;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 604 723 598 538 833 92 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 209 447 197 077 667 84;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 209 447 197 077 667 84 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 418 894 394 155 335 68;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 418 894 394 155 335 68 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 948 837 788 788 310 671 36;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 948 837 788 788 310 671 36 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 897 675 577 576 621 342 72;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 897 675 577 576 621 342 72 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 795 351 155 153 242 685 44;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 795 351 155 153 242 685 44 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 590 702 310 306 485 370 88;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 590 702 310 306 485 370 88 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 181 404 620 612 970 741 76;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 181 404 620 612 970 741 76 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 362 809 241 225 941 483 52;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 362 809 241 225 941 483 52 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 725 618 482 451 882 967 04;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 725 618 482 451 882 967 04 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 451 236 964 903 765 934 08;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 451 236 964 903 765 934 08 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 902 473 929 807 531 868 16;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 902 473 929 807 531 868 16 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 997 804 947 859 615 063 736 32;
  • 64) 0.353 967 107 832 431 793 212 890 624 997 804 947 859 615 063 736 32 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 995 609 895 719 230 127 472 64;
  • 65) 0.707 934 215 664 863 586 425 781 249 995 609 895 719 230 127 472 64 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 991 219 791 438 460 254 945 28;
  • 66) 0.415 868 431 329 727 172 851 562 499 991 219 791 438 460 254 945 28 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 982 439 582 876 920 509 890 56;
  • 67) 0.831 736 862 659 454 345 703 124 999 982 439 582 876 920 509 890 56 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 964 879 165 753 841 019 781 12;
  • 68) 0.663 473 725 318 908 691 406 249 999 964 879 165 753 841 019 781 12 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 929 758 331 507 682 039 562 24;
  • 69) 0.326 947 450 637 817 382 812 499 999 929 758 331 507 682 039 562 24 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 859 516 663 015 364 079 124 48;
  • 70) 0.653 894 901 275 634 765 624 999 999 859 516 663 015 364 079 124 48 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 719 033 326 030 728 158 248 96;
  • 71) 0.307 789 802 551 269 531 249 999 999 719 033 326 030 728 158 248 96 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 438 066 652 061 456 316 497 92;
  • 72) 0.615 579 605 102 539 062 499 999 999 438 066 652 061 456 316 497 92 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 876 133 304 122 912 632 995 84;
  • 73) 0.231 159 210 205 078 124 999 999 998 876 133 304 122 912 632 995 84 × 2 = 0 + 0.462 318 420 410 156 249 999 999 997 752 266 608 245 825 265 991 68;
  • 74) 0.462 318 420 410 156 249 999 999 997 752 266 608 245 825 265 991 68 × 2 = 0 + 0.924 636 840 820 312 499 999 999 995 504 533 216 491 650 531 983 36;
  • 75) 0.924 636 840 820 312 499 999 999 995 504 533 216 491 650 531 983 36 × 2 = 1 + 0.849 273 681 640 624 999 999 999 991 009 066 432 983 301 063 966 72;
  • 76) 0.849 273 681 640 624 999 999 999 991 009 066 432 983 301 063 966 72 × 2 = 1 + 0.698 547 363 281 249 999 999 999 982 018 132 865 966 602 127 933 44;
  • 77) 0.698 547 363 281 249 999 999 999 982 018 132 865 966 602 127 933 44 × 2 = 1 + 0.397 094 726 562 499 999 999 999 964 036 265 731 933 204 255 866 88;
  • 78) 0.397 094 726 562 499 999 999 999 964 036 265 731 933 204 255 866 88 × 2 = 0 + 0.794 189 453 124 999 999 999 999 928 072 531 463 866 408 511 733 76;
  • 79) 0.794 189 453 124 999 999 999 999 928 072 531 463 866 408 511 733 76 × 2 = 1 + 0.588 378 906 249 999 999 999 999 856 145 062 927 732 817 023 467 52;
  • 80) 0.588 378 906 249 999 999 999 999 856 145 062 927 732 817 023 467 52 × 2 = 1 + 0.176 757 812 499 999 999 999 999 712 290 125 855 465 634 046 935 04;
  • 81) 0.176 757 812 499 999 999 999 999 712 290 125 855 465 634 046 935 04 × 2 = 0 + 0.353 515 624 999 999 999 999 999 424 580 251 710 931 268 093 870 08;
  • 82) 0.353 515 624 999 999 999 999 999 424 580 251 710 931 268 093 870 08 × 2 = 0 + 0.707 031 249 999 999 999 999 998 849 160 503 421 862 536 187 740 16;
  • 83) 0.707 031 249 999 999 999 999 998 849 160 503 421 862 536 187 740 16 × 2 = 1 + 0.414 062 499 999 999 999 999 997 698 321 006 843 725 072 375 480 32;
  • 84) 0.414 062 499 999 999 999 999 997 698 321 006 843 725 072 375 480 32 × 2 = 0 + 0.828 124 999 999 999 999 999 995 396 642 013 687 450 144 750 960 64;
  • 85) 0.828 124 999 999 999 999 999 995 396 642 013 687 450 144 750 960 64 × 2 = 1 + 0.656 249 999 999 999 999 999 990 793 284 027 374 900 289 501 921 28;
  • 86) 0.656 249 999 999 999 999 999 990 793 284 027 374 900 289 501 921 28 × 2 = 1 + 0.312 499 999 999 999 999 999 981 586 568 054 749 800 579 003 842 56;
  • 87) 0.312 499 999 999 999 999 999 981 586 568 054 749 800 579 003 842 56 × 2 = 0 + 0.624 999 999 999 999 999 999 963 173 136 109 499 601 158 007 685 12;
  • 88) 0.624 999 999 999 999 999 999 963 173 136 109 499 601 158 007 685 12 × 2 = 1 + 0.249 999 999 999 999 999 999 926 346 272 218 999 202 316 015 370 24;
  • 89) 0.249 999 999 999 999 999 999 926 346 272 218 999 202 316 015 370 24 × 2 = 0 + 0.499 999 999 999 999 999 999 852 692 544 437 998 404 632 030 740 48;
  • 90) 0.499 999 999 999 999 999 999 852 692 544 437 998 404 632 030 740 48 × 2 = 0 + 0.999 999 999 999 999 999 999 705 385 088 875 996 809 264 061 480 96;
  • 91) 0.999 999 999 999 999 999 999 705 385 088 875 996 809 264 061 480 96 × 2 = 1 + 0.999 999 999 999 999 999 999 410 770 177 751 993 618 528 122 961 92;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 915 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