0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 88 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 88 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 303 76;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 303 76 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 607 52;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 607 52 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 215 04;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 215 04 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 430 08;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 430 08 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 860 16;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 860 16 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 720 32;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 720 32 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 440 64;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 440 64 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 881 28;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 881 28 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 762 56;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 762 56 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 525 12;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 525 12 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 050 24;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 050 24 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 782 100 48;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 782 100 48 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 564 200 96;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 564 200 96 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 128 401 92;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 128 401 92 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 256 803 84;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 256 803 84 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 513 607 68;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 513 607 68 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 027 215 36;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 027 215 36 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 054 430 72;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 054 430 72 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 108 861 44;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 108 861 44 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 217 722 88;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 217 722 88 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 435 445 76;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 435 445 76 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 840 870 891 52;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 840 870 891 52 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 681 741 783 04;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 681 741 783 04 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 363 483 566 08;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 363 483 566 08 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 726 967 132 16;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 726 967 132 16 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 453 934 264 32;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 453 934 264 32 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 907 868 528 64;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 907 868 528 64 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 815 737 057 28;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 815 737 057 28 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 631 474 114 56;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 631 474 114 56 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 262 948 229 12;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 262 948 229 12 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 046 525 896 458 24;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 046 525 896 458 24 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 093 051 792 916 48;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 093 051 792 916 48 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 186 103 585 832 96;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 186 103 585 832 96 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 372 207 171 665 92;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 372 207 171 665 92 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 744 414 343 331 84;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 744 414 343 331 84 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 488 828 686 663 68;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 488 828 686 663 68 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 977 657 373 327 36;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 977 657 373 327 36 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 637 955 314 746 654 72;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 637 955 314 746 654 72 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 275 910 629 493 309 44;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 275 910 629 493 309 44 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 551 821 258 986 618 88;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 551 821 258 986 618 88 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 103 642 517 973 237 76;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 103 642 517 973 237 76 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 207 285 035 946 475 52;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 207 285 035 946 475 52 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 414 570 071 892 951 04;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 414 570 071 892 951 04 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 829 140 143 785 902 08;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 829 140 143 785 902 08 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 658 280 287 571 804 16;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 658 280 287 571 804 16 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 316 560 575 143 608 32;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 316 560 575 143 608 32 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 633 121 150 287 216 64;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 633 121 150 287 216 64 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 266 242 300 574 433 28;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 266 242 300 574 433 28 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 532 484 601 148 866 56;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 532 484 601 148 866 56 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 064 969 202 297 733 12;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 064 969 202 297 733 12 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 138 129 938 404 595 466 24;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 138 129 938 404 595 466 24 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 276 259 876 809 190 932 48;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 276 259 876 809 190 932 48 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 552 519 753 618 381 864 96;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 552 519 753 618 381 864 96 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 105 039 507 236 763 729 92;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 105 039 507 236 763 729 92 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 210 079 014 473 527 459 84;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 210 079 014 473 527 459 84 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 420 158 028 947 054 919 68;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 420 158 028 947 054 919 68 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 840 316 057 894 109 839 36;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 840 316 057 894 109 839 36 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 680 632 115 788 219 678 72;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 680 632 115 788 219 678 72 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 361 264 231 576 439 357 44;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 361 264 231 576 439 357 44 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 998 722 528 463 152 878 714 88;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 998 722 528 463 152 878 714 88 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 997 445 056 926 305 757 429 76;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 997 445 056 926 305 757 429 76 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 994 890 113 852 611 514 859 52;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 994 890 113 852 611 514 859 52 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 989 780 227 705 223 029 719 04;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 989 780 227 705 223 029 719 04 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 979 560 455 410 446 059 438 08;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 979 560 455 410 446 059 438 08 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 959 120 910 820 892 118 876 16;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 959 120 910 820 892 118 876 16 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 918 241 821 641 784 237 752 32;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 918 241 821 641 784 237 752 32 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 836 483 643 283 568 475 504 64;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 836 483 643 283 568 475 504 64 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 672 967 286 567 136 951 009 28;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 672 967 286 567 136 951 009 28 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 345 934 573 134 273 902 018 56;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 345 934 573 134 273 902 018 56 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 998 691 869 146 268 547 804 037 12;
  • 71) 0.307 789 802 551 269 531 249 999 999 998 691 869 146 268 547 804 037 12 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 997 383 738 292 537 095 608 074 24;
  • 72) 0.615 579 605 102 539 062 499 999 999 997 383 738 292 537 095 608 074 24 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 994 767 476 585 074 191 216 148 48;
  • 73) 0.231 159 210 205 078 124 999 999 999 994 767 476 585 074 191 216 148 48 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 989 534 953 170 148 382 432 296 96;
  • 74) 0.462 318 420 410 156 249 999 999 999 989 534 953 170 148 382 432 296 96 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 979 069 906 340 296 764 864 593 92;
  • 75) 0.924 636 840 820 312 499 999 999 999 979 069 906 340 296 764 864 593 92 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 958 139 812 680 593 529 729 187 84;
  • 76) 0.849 273 681 640 624 999 999 999 999 958 139 812 680 593 529 729 187 84 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 916 279 625 361 187 059 458 375 68;
  • 77) 0.698 547 363 281 249 999 999 999 999 916 279 625 361 187 059 458 375 68 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 832 559 250 722 374 118 916 751 36;
  • 78) 0.397 094 726 562 499 999 999 999 999 832 559 250 722 374 118 916 751 36 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 665 118 501 444 748 237 833 502 72;
  • 79) 0.794 189 453 124 999 999 999 999 999 665 118 501 444 748 237 833 502 72 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 330 237 002 889 496 475 667 005 44;
  • 80) 0.588 378 906 249 999 999 999 999 999 330 237 002 889 496 475 667 005 44 × 2 = 1 + 0.176 757 812 499 999 999 999 999 998 660 474 005 778 992 951 334 010 88;
  • 81) 0.176 757 812 499 999 999 999 999 998 660 474 005 778 992 951 334 010 88 × 2 = 0 + 0.353 515 624 999 999 999 999 999 997 320 948 011 557 985 902 668 021 76;
  • 82) 0.353 515 624 999 999 999 999 999 997 320 948 011 557 985 902 668 021 76 × 2 = 0 + 0.707 031 249 999 999 999 999 999 994 641 896 023 115 971 805 336 043 52;
  • 83) 0.707 031 249 999 999 999 999 999 994 641 896 023 115 971 805 336 043 52 × 2 = 1 + 0.414 062 499 999 999 999 999 999 989 283 792 046 231 943 610 672 087 04;
  • 84) 0.414 062 499 999 999 999 999 999 989 283 792 046 231 943 610 672 087 04 × 2 = 0 + 0.828 124 999 999 999 999 999 999 978 567 584 092 463 887 221 344 174 08;
  • 85) 0.828 124 999 999 999 999 999 999 978 567 584 092 463 887 221 344 174 08 × 2 = 1 + 0.656 249 999 999 999 999 999 999 957 135 168 184 927 774 442 688 348 16;
  • 86) 0.656 249 999 999 999 999 999 999 957 135 168 184 927 774 442 688 348 16 × 2 = 1 + 0.312 499 999 999 999 999 999 999 914 270 336 369 855 548 885 376 696 32;
  • 87) 0.312 499 999 999 999 999 999 999 914 270 336 369 855 548 885 376 696 32 × 2 = 0 + 0.624 999 999 999 999 999 999 999 828 540 672 739 711 097 770 753 392 64;
  • 88) 0.624 999 999 999 999 999 999 999 828 540 672 739 711 097 770 753 392 64 × 2 = 1 + 0.249 999 999 999 999 999 999 999 657 081 345 479 422 195 541 506 785 28;
  • 89) 0.249 999 999 999 999 999 999 999 657 081 345 479 422 195 541 506 785 28 × 2 = 0 + 0.499 999 999 999 999 999 999 999 314 162 690 958 844 391 083 013 570 56;
  • 90) 0.499 999 999 999 999 999 999 999 314 162 690 958 844 391 083 013 570 56 × 2 = 0 + 0.999 999 999 999 999 999 999 998 628 325 381 917 688 782 166 027 141 12;
  • 91) 0.999 999 999 999 999 999 999 998 628 325 381 917 688 782 166 027 141 12 × 2 = 1 + 0.999 999 999 999 999 999 999 997 256 650 763 835 377 564 332 054 282 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 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