0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 74;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 74 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 48;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 48 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 222 96;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 222 96 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 445 92;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 445 92 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 891 84;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 891 84 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 783 68;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 783 68 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 567 36;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 567 36 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 134 72;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 134 72 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 269 44;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 269 44 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 538 88;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 538 88 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 077 76;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 077 76 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 155 52;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 155 52 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 311 04;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 311 04 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 144 622 08;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 144 622 08 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 289 244 16;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 289 244 16 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 578 488 32;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 578 488 32 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 156 976 64;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 156 976 64 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 313 953 28;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 313 953 28 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 627 906 56;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 627 906 56 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 255 813 12;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 255 813 12 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 511 626 24;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 511 626 24 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 023 252 48;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 023 252 48 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 046 504 96;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 046 504 96 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 093 009 92;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 093 009 92 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 760 186 019 84;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 760 186 019 84 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 520 372 039 68;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 520 372 039 68 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 040 744 079 36;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 040 744 079 36 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 081 488 158 72;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 081 488 158 72 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 162 976 317 44;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 162 976 317 44 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 325 952 634 88;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 325 952 634 88 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 651 905 269 76;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 651 905 269 76 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 303 810 539 52;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 303 810 539 52 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 607 621 079 04;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 607 621 079 04 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 215 242 158 08;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 215 242 158 08 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 778 430 484 316 16;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 778 430 484 316 16 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 556 860 968 632 32;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 556 860 968 632 32 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 113 721 937 264 64;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 113 721 937 264 64 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 227 443 874 529 28;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 227 443 874 529 28 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 454 887 749 058 56;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 454 887 749 058 56 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 909 775 498 117 12;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 909 775 498 117 12 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 105 819 550 996 234 24;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 105 819 550 996 234 24 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 211 639 101 992 468 48;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 211 639 101 992 468 48 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 423 278 203 984 936 96;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 423 278 203 984 936 96 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 846 556 407 969 873 92;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 846 556 407 969 873 92 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 693 112 815 939 747 84;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 693 112 815 939 747 84 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 386 225 631 879 495 68;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 386 225 631 879 495 68 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 772 451 263 758 991 36;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 772 451 263 758 991 36 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 544 902 527 517 982 72;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 544 902 527 517 982 72 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 089 805 055 035 965 44;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 089 805 055 035 965 44 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 179 610 110 071 930 88;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 179 610 110 071 930 88 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 359 220 220 143 861 76;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 359 220 220 143 861 76 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 280 718 440 440 287 723 52;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 280 718 440 440 287 723 52 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 561 436 880 880 575 447 04;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 561 436 880 880 575 447 04 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 122 873 761 761 150 894 08;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 122 873 761 761 150 894 08 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 245 747 523 522 301 788 16;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 245 747 523 522 301 788 16 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 491 495 047 044 603 576 32;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 491 495 047 044 603 576 32 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 982 990 094 089 207 152 64;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 982 990 094 089 207 152 64 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 965 980 188 178 414 305 28;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 965 980 188 178 414 305 28 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 931 960 376 356 828 610 56;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 931 960 376 356 828 610 56 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 863 920 752 713 657 221 12;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 863 920 752 713 657 221 12 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 727 841 505 427 314 442 24;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 727 841 505 427 314 442 24 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 455 683 010 854 628 884 48;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 455 683 010 854 628 884 48 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 998 911 366 021 709 257 768 96;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 998 911 366 021 709 257 768 96 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 997 822 732 043 418 515 537 92;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 997 822 732 043 418 515 537 92 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 995 645 464 086 837 031 075 84;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 995 645 464 086 837 031 075 84 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 991 290 928 173 674 062 151 68;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 991 290 928 173 674 062 151 68 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 982 581 856 347 348 124 303 36;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 982 581 856 347 348 124 303 36 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 965 163 712 694 696 248 606 72;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 965 163 712 694 696 248 606 72 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 930 327 425 389 392 497 213 44;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 930 327 425 389 392 497 213 44 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 860 654 850 778 784 994 426 88;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 860 654 850 778 784 994 426 88 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 721 309 701 557 569 988 853 76;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 721 309 701 557 569 988 853 76 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 442 619 403 115 139 977 707 52;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 442 619 403 115 139 977 707 52 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 998 885 238 806 230 279 955 415 04;
  • 74) 0.462 318 420 410 156 249 999 999 999 998 885 238 806 230 279 955 415 04 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 997 770 477 612 460 559 910 830 08;
  • 75) 0.924 636 840 820 312 499 999 999 999 997 770 477 612 460 559 910 830 08 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 995 540 955 224 921 119 821 660 16;
  • 76) 0.849 273 681 640 624 999 999 999 999 995 540 955 224 921 119 821 660 16 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 991 081 910 449 842 239 643 320 32;
  • 77) 0.698 547 363 281 249 999 999 999 999 991 081 910 449 842 239 643 320 32 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 982 163 820 899 684 479 286 640 64;
  • 78) 0.397 094 726 562 499 999 999 999 999 982 163 820 899 684 479 286 640 64 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 964 327 641 799 368 958 573 281 28;
  • 79) 0.794 189 453 124 999 999 999 999 999 964 327 641 799 368 958 573 281 28 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 928 655 283 598 737 917 146 562 56;
  • 80) 0.588 378 906 249 999 999 999 999 999 928 655 283 598 737 917 146 562 56 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 857 310 567 197 475 834 293 125 12;
  • 81) 0.176 757 812 499 999 999 999 999 999 857 310 567 197 475 834 293 125 12 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 714 621 134 394 951 668 586 250 24;
  • 82) 0.353 515 624 999 999 999 999 999 999 714 621 134 394 951 668 586 250 24 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 429 242 268 789 903 337 172 500 48;
  • 83) 0.707 031 249 999 999 999 999 999 999 429 242 268 789 903 337 172 500 48 × 2 = 1 + 0.414 062 499 999 999 999 999 999 998 858 484 537 579 806 674 345 000 96;
  • 84) 0.414 062 499 999 999 999 999 999 998 858 484 537 579 806 674 345 000 96 × 2 = 0 + 0.828 124 999 999 999 999 999 999 997 716 969 075 159 613 348 690 001 92;
  • 85) 0.828 124 999 999 999 999 999 999 997 716 969 075 159 613 348 690 001 92 × 2 = 1 + 0.656 249 999 999 999 999 999 999 995 433 938 150 319 226 697 380 003 84;
  • 86) 0.656 249 999 999 999 999 999 999 995 433 938 150 319 226 697 380 003 84 × 2 = 1 + 0.312 499 999 999 999 999 999 999 990 867 876 300 638 453 394 760 007 68;
  • 87) 0.312 499 999 999 999 999 999 999 990 867 876 300 638 453 394 760 007 68 × 2 = 0 + 0.624 999 999 999 999 999 999 999 981 735 752 601 276 906 789 520 015 36;
  • 88) 0.624 999 999 999 999 999 999 999 981 735 752 601 276 906 789 520 015 36 × 2 = 1 + 0.249 999 999 999 999 999 999 999 963 471 505 202 553 813 579 040 030 72;
  • 89) 0.249 999 999 999 999 999 999 999 963 471 505 202 553 813 579 040 030 72 × 2 = 0 + 0.499 999 999 999 999 999 999 999 926 943 010 405 107 627 158 080 061 44;
  • 90) 0.499 999 999 999 999 999 999 999 926 943 010 405 107 627 158 080 061 44 × 2 = 0 + 0.999 999 999 999 999 999 999 999 853 886 020 810 215 254 316 160 122 88;
  • 91) 0.999 999 999 999 999 999 999 999 853 886 020 810 215 254 316 160 122 88 × 2 = 1 + 0.999 999 999 999 999 999 999 999 707 772 041 620 430 508 632 320 245 76;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 87 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