0.000 000 000 000 000 013 248 735 989 328 012 498 123 88 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 013 248 735 989 328 012 498 123 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
0.000 000 000 000 000 013 248 735 989 328 012 498 123 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: 0.000 000 000 000 000 013 248 735 989 328 012 498 123 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) 0.000 000 000 000 000 013 248 735 989 328 012 498 123 88 × 2 = 0 + 0.000 000 000 000 000 026 497 471 978 656 024 996 247 76;
  • 2) 0.000 000 000 000 000 026 497 471 978 656 024 996 247 76 × 2 = 0 + 0.000 000 000 000 000 052 994 943 957 312 049 992 495 52;
  • 3) 0.000 000 000 000 000 052 994 943 957 312 049 992 495 52 × 2 = 0 + 0.000 000 000 000 000 105 989 887 914 624 099 984 991 04;
  • 4) 0.000 000 000 000 000 105 989 887 914 624 099 984 991 04 × 2 = 0 + 0.000 000 000 000 000 211 979 775 829 248 199 969 982 08;
  • 5) 0.000 000 000 000 000 211 979 775 829 248 199 969 982 08 × 2 = 0 + 0.000 000 000 000 000 423 959 551 658 496 399 939 964 16;
  • 6) 0.000 000 000 000 000 423 959 551 658 496 399 939 964 16 × 2 = 0 + 0.000 000 000 000 000 847 919 103 316 992 799 879 928 32;
  • 7) 0.000 000 000 000 000 847 919 103 316 992 799 879 928 32 × 2 = 0 + 0.000 000 000 000 001 695 838 206 633 985 599 759 856 64;
  • 8) 0.000 000 000 000 001 695 838 206 633 985 599 759 856 64 × 2 = 0 + 0.000 000 000 000 003 391 676 413 267 971 199 519 713 28;
  • 9) 0.000 000 000 000 003 391 676 413 267 971 199 519 713 28 × 2 = 0 + 0.000 000 000 000 006 783 352 826 535 942 399 039 426 56;
  • 10) 0.000 000 000 000 006 783 352 826 535 942 399 039 426 56 × 2 = 0 + 0.000 000 000 000 013 566 705 653 071 884 798 078 853 12;
  • 11) 0.000 000 000 000 013 566 705 653 071 884 798 078 853 12 × 2 = 0 + 0.000 000 000 000 027 133 411 306 143 769 596 157 706 24;
  • 12) 0.000 000 000 000 027 133 411 306 143 769 596 157 706 24 × 2 = 0 + 0.000 000 000 000 054 266 822 612 287 539 192 315 412 48;
  • 13) 0.000 000 000 000 054 266 822 612 287 539 192 315 412 48 × 2 = 0 + 0.000 000 000 000 108 533 645 224 575 078 384 630 824 96;
  • 14) 0.000 000 000 000 108 533 645 224 575 078 384 630 824 96 × 2 = 0 + 0.000 000 000 000 217 067 290 449 150 156 769 261 649 92;
  • 15) 0.000 000 000 000 217 067 290 449 150 156 769 261 649 92 × 2 = 0 + 0.000 000 000 000 434 134 580 898 300 313 538 523 299 84;
  • 16) 0.000 000 000 000 434 134 580 898 300 313 538 523 299 84 × 2 = 0 + 0.000 000 000 000 868 269 161 796 600 627 077 046 599 68;
  • 17) 0.000 000 000 000 868 269 161 796 600 627 077 046 599 68 × 2 = 0 + 0.000 000 000 001 736 538 323 593 201 254 154 093 199 36;
  • 18) 0.000 000 000 001 736 538 323 593 201 254 154 093 199 36 × 2 = 0 + 0.000 000 000 003 473 076 647 186 402 508 308 186 398 72;
  • 19) 0.000 000 000 003 473 076 647 186 402 508 308 186 398 72 × 2 = 0 + 0.000 000 000 006 946 153 294 372 805 016 616 372 797 44;
  • 20) 0.000 000 000 006 946 153 294 372 805 016 616 372 797 44 × 2 = 0 + 0.000 000 000 013 892 306 588 745 610 033 232 745 594 88;
  • 21) 0.000 000 000 013 892 306 588 745 610 033 232 745 594 88 × 2 = 0 + 0.000 000 000 027 784 613 177 491 220 066 465 491 189 76;
  • 22) 0.000 000 000 027 784 613 177 491 220 066 465 491 189 76 × 2 = 0 + 0.000 000 000 055 569 226 354 982 440 132 930 982 379 52;
  • 23) 0.000 000 000 055 569 226 354 982 440 132 930 982 379 52 × 2 = 0 + 0.000 000 000 111 138 452 709 964 880 265 861 964 759 04;
  • 24) 0.000 000 000 111 138 452 709 964 880 265 861 964 759 04 × 2 = 0 + 0.000 000 000 222 276 905 419 929 760 531 723 929 518 08;
  • 25) 0.000 000 000 222 276 905 419 929 760 531 723 929 518 08 × 2 = 0 + 0.000 000 000 444 553 810 839 859 521 063 447 859 036 16;
  • 26) 0.000 000 000 444 553 810 839 859 521 063 447 859 036 16 × 2 = 0 + 0.000 000 000 889 107 621 679 719 042 126 895 718 072 32;
  • 27) 0.000 000 000 889 107 621 679 719 042 126 895 718 072 32 × 2 = 0 + 0.000 000 001 778 215 243 359 438 084 253 791 436 144 64;
  • 28) 0.000 000 001 778 215 243 359 438 084 253 791 436 144 64 × 2 = 0 + 0.000 000 003 556 430 486 718 876 168 507 582 872 289 28;
  • 29) 0.000 000 003 556 430 486 718 876 168 507 582 872 289 28 × 2 = 0 + 0.000 000 007 112 860 973 437 752 337 015 165 744 578 56;
  • 30) 0.000 000 007 112 860 973 437 752 337 015 165 744 578 56 × 2 = 0 + 0.000 000 014 225 721 946 875 504 674 030 331 489 157 12;
  • 31) 0.000 000 014 225 721 946 875 504 674 030 331 489 157 12 × 2 = 0 + 0.000 000 028 451 443 893 751 009 348 060 662 978 314 24;
  • 32) 0.000 000 028 451 443 893 751 009 348 060 662 978 314 24 × 2 = 0 + 0.000 000 056 902 887 787 502 018 696 121 325 956 628 48;
  • 33) 0.000 000 056 902 887 787 502 018 696 121 325 956 628 48 × 2 = 0 + 0.000 000 113 805 775 575 004 037 392 242 651 913 256 96;
  • 34) 0.000 000 113 805 775 575 004 037 392 242 651 913 256 96 × 2 = 0 + 0.000 000 227 611 551 150 008 074 784 485 303 826 513 92;
  • 35) 0.000 000 227 611 551 150 008 074 784 485 303 826 513 92 × 2 = 0 + 0.000 000 455 223 102 300 016 149 568 970 607 653 027 84;
  • 36) 0.000 000 455 223 102 300 016 149 568 970 607 653 027 84 × 2 = 0 + 0.000 000 910 446 204 600 032 299 137 941 215 306 055 68;
  • 37) 0.000 000 910 446 204 600 032 299 137 941 215 306 055 68 × 2 = 0 + 0.000 001 820 892 409 200 064 598 275 882 430 612 111 36;
  • 38) 0.000 001 820 892 409 200 064 598 275 882 430 612 111 36 × 2 = 0 + 0.000 003 641 784 818 400 129 196 551 764 861 224 222 72;
  • 39) 0.000 003 641 784 818 400 129 196 551 764 861 224 222 72 × 2 = 0 + 0.000 007 283 569 636 800 258 393 103 529 722 448 445 44;
  • 40) 0.000 007 283 569 636 800 258 393 103 529 722 448 445 44 × 2 = 0 + 0.000 014 567 139 273 600 516 786 207 059 444 896 890 88;
  • 41) 0.000 014 567 139 273 600 516 786 207 059 444 896 890 88 × 2 = 0 + 0.000 029 134 278 547 201 033 572 414 118 889 793 781 76;
  • 42) 0.000 029 134 278 547 201 033 572 414 118 889 793 781 76 × 2 = 0 + 0.000 058 268 557 094 402 067 144 828 237 779 587 563 52;
  • 43) 0.000 058 268 557 094 402 067 144 828 237 779 587 563 52 × 2 = 0 + 0.000 116 537 114 188 804 134 289 656 475 559 175 127 04;
  • 44) 0.000 116 537 114 188 804 134 289 656 475 559 175 127 04 × 2 = 0 + 0.000 233 074 228 377 608 268 579 312 951 118 350 254 08;
  • 45) 0.000 233 074 228 377 608 268 579 312 951 118 350 254 08 × 2 = 0 + 0.000 466 148 456 755 216 537 158 625 902 236 700 508 16;
  • 46) 0.000 466 148 456 755 216 537 158 625 902 236 700 508 16 × 2 = 0 + 0.000 932 296 913 510 433 074 317 251 804 473 401 016 32;
  • 47) 0.000 932 296 913 510 433 074 317 251 804 473 401 016 32 × 2 = 0 + 0.001 864 593 827 020 866 148 634 503 608 946 802 032 64;
  • 48) 0.001 864 593 827 020 866 148 634 503 608 946 802 032 64 × 2 = 0 + 0.003 729 187 654 041 732 297 269 007 217 893 604 065 28;
  • 49) 0.003 729 187 654 041 732 297 269 007 217 893 604 065 28 × 2 = 0 + 0.007 458 375 308 083 464 594 538 014 435 787 208 130 56;
  • 50) 0.007 458 375 308 083 464 594 538 014 435 787 208 130 56 × 2 = 0 + 0.014 916 750 616 166 929 189 076 028 871 574 416 261 12;
  • 51) 0.014 916 750 616 166 929 189 076 028 871 574 416 261 12 × 2 = 0 + 0.029 833 501 232 333 858 378 152 057 743 148 832 522 24;
  • 52) 0.029 833 501 232 333 858 378 152 057 743 148 832 522 24 × 2 = 0 + 0.059 667 002 464 667 716 756 304 115 486 297 665 044 48;
  • 53) 0.059 667 002 464 667 716 756 304 115 486 297 665 044 48 × 2 = 0 + 0.119 334 004 929 335 433 512 608 230 972 595 330 088 96;
  • 54) 0.119 334 004 929 335 433 512 608 230 972 595 330 088 96 × 2 = 0 + 0.238 668 009 858 670 867 025 216 461 945 190 660 177 92;
  • 55) 0.238 668 009 858 670 867 025 216 461 945 190 660 177 92 × 2 = 0 + 0.477 336 019 717 341 734 050 432 923 890 381 320 355 84;
  • 56) 0.477 336 019 717 341 734 050 432 923 890 381 320 355 84 × 2 = 0 + 0.954 672 039 434 683 468 100 865 847 780 762 640 711 68;
  • 57) 0.954 672 039 434 683 468 100 865 847 780 762 640 711 68 × 2 = 1 + 0.909 344 078 869 366 936 201 731 695 561 525 281 423 36;
  • 58) 0.909 344 078 869 366 936 201 731 695 561 525 281 423 36 × 2 = 1 + 0.818 688 157 738 733 872 403 463 391 123 050 562 846 72;
  • 59) 0.818 688 157 738 733 872 403 463 391 123 050 562 846 72 × 2 = 1 + 0.637 376 315 477 467 744 806 926 782 246 101 125 693 44;
  • 60) 0.637 376 315 477 467 744 806 926 782 246 101 125 693 44 × 2 = 1 + 0.274 752 630 954 935 489 613 853 564 492 202 251 386 88;
  • 61) 0.274 752 630 954 935 489 613 853 564 492 202 251 386 88 × 2 = 0 + 0.549 505 261 909 870 979 227 707 128 984 404 502 773 76;
  • 62) 0.549 505 261 909 870 979 227 707 128 984 404 502 773 76 × 2 = 1 + 0.099 010 523 819 741 958 455 414 257 968 809 005 547 52;
  • 63) 0.099 010 523 819 741 958 455 414 257 968 809 005 547 52 × 2 = 0 + 0.198 021 047 639 483 916 910 828 515 937 618 011 095 04;
  • 64) 0.198 021 047 639 483 916 910 828 515 937 618 011 095 04 × 2 = 0 + 0.396 042 095 278 967 833 821 657 031 875 236 022 190 08;
  • 65) 0.396 042 095 278 967 833 821 657 031 875 236 022 190 08 × 2 = 0 + 0.792 084 190 557 935 667 643 314 063 750 472 044 380 16;
  • 66) 0.792 084 190 557 935 667 643 314 063 750 472 044 380 16 × 2 = 1 + 0.584 168 381 115 871 335 286 628 127 500 944 088 760 32;
  • 67) 0.584 168 381 115 871 335 286 628 127 500 944 088 760 32 × 2 = 1 + 0.168 336 762 231 742 670 573 256 255 001 888 177 520 64;
  • 68) 0.168 336 762 231 742 670 573 256 255 001 888 177 520 64 × 2 = 0 + 0.336 673 524 463 485 341 146 512 510 003 776 355 041 28;
  • 69) 0.336 673 524 463 485 341 146 512 510 003 776 355 041 28 × 2 = 0 + 0.673 347 048 926 970 682 293 025 020 007 552 710 082 56;
  • 70) 0.673 347 048 926 970 682 293 025 020 007 552 710 082 56 × 2 = 1 + 0.346 694 097 853 941 364 586 050 040 015 105 420 165 12;
  • 71) 0.346 694 097 853 941 364 586 050 040 015 105 420 165 12 × 2 = 0 + 0.693 388 195 707 882 729 172 100 080 030 210 840 330 24;
  • 72) 0.693 388 195 707 882 729 172 100 080 030 210 840 330 24 × 2 = 1 + 0.386 776 391 415 765 458 344 200 160 060 421 680 660 48;
  • 73) 0.386 776 391 415 765 458 344 200 160 060 421 680 660 48 × 2 = 0 + 0.773 552 782 831 530 916 688 400 320 120 843 361 320 96;
  • 74) 0.773 552 782 831 530 916 688 400 320 120 843 361 320 96 × 2 = 1 + 0.547 105 565 663 061 833 376 800 640 241 686 722 641 92;
  • 75) 0.547 105 565 663 061 833 376 800 640 241 686 722 641 92 × 2 = 1 + 0.094 211 131 326 123 666 753 601 280 483 373 445 283 84;
  • 76) 0.094 211 131 326 123 666 753 601 280 483 373 445 283 84 × 2 = 0 + 0.188 422 262 652 247 333 507 202 560 966 746 890 567 68;
  • 77) 0.188 422 262 652 247 333 507 202 560 966 746 890 567 68 × 2 = 0 + 0.376 844 525 304 494 667 014 405 121 933 493 781 135 36;
  • 78) 0.376 844 525 304 494 667 014 405 121 933 493 781 135 36 × 2 = 0 + 0.753 689 050 608 989 334 028 810 243 866 987 562 270 72;
  • 79) 0.753 689 050 608 989 334 028 810 243 866 987 562 270 72 × 2 = 1 + 0.507 378 101 217 978 668 057 620 487 733 975 124 541 44;
  • 80) 0.507 378 101 217 978 668 057 620 487 733 975 124 541 44 × 2 = 1 + 0.014 756 202 435 957 336 115 240 975 467 950 249 082 88;
  • 81) 0.014 756 202 435 957 336 115 240 975 467 950 249 082 88 × 2 = 0 + 0.029 512 404 871 914 672 230 481 950 935 900 498 165 76;
  • 82) 0.029 512 404 871 914 672 230 481 950 935 900 498 165 76 × 2 = 0 + 0.059 024 809 743 829 344 460 963 901 871 800 996 331 52;
  • 83) 0.059 024 809 743 829 344 460 963 901 871 800 996 331 52 × 2 = 0 + 0.118 049 619 487 658 688 921 927 803 743 601 992 663 04;
  • 84) 0.118 049 619 487 658 688 921 927 803 743 601 992 663 04 × 2 = 0 + 0.236 099 238 975 317 377 843 855 607 487 203 985 326 08;
  • 85) 0.236 099 238 975 317 377 843 855 607 487 203 985 326 08 × 2 = 0 + 0.472 198 477 950 634 755 687 711 214 974 407 970 652 16;
  • 86) 0.472 198 477 950 634 755 687 711 214 974 407 970 652 16 × 2 = 0 + 0.944 396 955 901 269 511 375 422 429 948 815 941 304 32;
  • 87) 0.944 396 955 901 269 511 375 422 429 948 815 941 304 32 × 2 = 1 + 0.888 793 911 802 539 022 750 844 859 897 631 882 608 64;
  • 88) 0.888 793 911 802 539 022 750 844 859 897 631 882 608 64 × 2 = 1 + 0.777 587 823 605 078 045 501 689 719 795 263 765 217 28;
  • 89) 0.777 587 823 605 078 045 501 689 719 795 263 765 217 28 × 2 = 1 + 0.555 175 647 210 156 091 003 379 439 590 527 530 434 56;
  • 90) 0.555 175 647 210 156 091 003 379 439 590 527 530 434 56 × 2 = 1 + 0.110 351 294 420 312 182 006 758 879 181 055 060 869 12;
  • 91) 0.110 351 294 420 312 182 006 758 879 181 055 060 869 12 × 2 = 0 + 0.220 702 588 840 624 364 013 517 758 362 110 121 738 24;
  • 92) 0.220 702 588 840 624 364 013 517 758 362 110 121 738 24 × 2 = 0 + 0.441 405 177 681 248 728 027 035 516 724 220 243 476 48;
  • 93) 0.441 405 177 681 248 728 027 035 516 724 220 243 476 48 × 2 = 0 + 0.882 810 355 362 497 456 054 071 033 448 440 486 952 96;
  • 94) 0.882 810 355 362 497 456 054 071 033 448 440 486 952 96 × 2 = 1 + 0.765 620 710 724 994 912 108 142 066 896 880 973 905 92;
  • 95) 0.765 620 710 724 994 912 108 142 066 896 880 973 905 92 × 2 = 1 + 0.531 241 421 449 989 824 216 284 133 793 761 947 811 84;
  • 96) 0.531 241 421 449 989 824 216 284 133 793 761 947 811 84 × 2 = 1 + 0.062 482 842 899 979 648 432 568 267 587 523 895 623 68;
  • 97) 0.062 482 842 899 979 648 432 568 267 587 523 895 623 68 × 2 = 0 + 0.124 965 685 799 959 296 865 136 535 175 047 791 247 36;
  • 98) 0.124 965 685 799 959 296 865 136 535 175 047 791 247 36 × 2 = 0 + 0.249 931 371 599 918 593 730 273 070 350 095 582 494 72;
  • 99) 0.249 931 371 599 918 593 730 273 070 350 095 582 494 72 × 2 = 0 + 0.499 862 743 199 837 187 460 546 140 700 191 164 989 44;
  • 100) 0.499 862 743 199 837 187 460 546 140 700 191 164 989 44 × 2 = 0 + 0.999 725 486 399 674 374 921 092 281 400 382 329 978 88;
  • 101) 0.999 725 486 399 674 374 921 092 281 400 382 329 978 88 × 2 = 1 + 0.999 450 972 799 348 749 842 184 562 800 764 659 957 76;
  • 102) 0.999 450 972 799 348 749 842 184 562 800 764 659 957 76 × 2 = 1 + 0.998 901 945 598 697 499 684 369 125 601 529 319 915 52;
  • 103) 0.998 901 945 598 697 499 684 369 125 601 529 319 915 52 × 2 = 1 + 0.997 803 891 197 394 999 368 738 251 203 058 639 831 04;
  • 104) 0.997 803 891 197 394 999 368 738 251 203 058 639 831 04 × 2 = 1 + 0.995 607 782 394 789 998 737 476 502 406 117 279 662 08;
  • 105) 0.995 607 782 394 789 998 737 476 502 406 117 279 662 08 × 2 = 1 + 0.991 215 564 789 579 997 474 953 004 812 234 559 324 16;
  • 106) 0.991 215 564 789 579 997 474 953 004 812 234 559 324 16 × 2 = 1 + 0.982 431 129 579 159 994 949 906 009 624 469 118 648 32;
  • 107) 0.982 431 129 579 159 994 949 906 009 624 469 118 648 32 × 2 = 1 + 0.964 862 259 158 319 989 899 812 019 248 938 237 296 64;
  • 108) 0.964 862 259 158 319 989 899 812 019 248 938 237 296 64 × 2 = 1 + 0.929 724 518 316 639 979 799 624 038 497 876 474 593 28;
  • 109) 0.929 724 518 316 639 979 799 624 038 497 876 474 593 28 × 2 = 1 + 0.859 449 036 633 279 959 599 248 076 995 752 949 186 56;

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:


0.000 000 000 000 000 013 248 735 989 328 012 498 123 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 013 248 735 989 328 012 498 123 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 013 248 735 989 328 012 498 123 88(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 0100 0110 0101 0110 0011 0000 0011 1100 0111 0000 1111 1111 1(2) × 20 =

1.1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111 =

1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


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 1100 0110

Mantissa (52 bits) =
1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111


Decimal number 0.000 000 000 000 000 013 248 735 989 328 012 498 123 88 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1110 1000 1100 1010 1100 0110 0000 0111 1000 1110 0001 1111 1111

» Convert decimal 0.000 000 000 000 000 013 248 735 989 328 012 498 123 32 to 64 bit double precision IEEE 754 binary floating point representation standard

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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