0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 82;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 82 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 64;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 64 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 227 28;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 227 28 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 454 56;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 454 56 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 909 12;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 909 12 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 818 24;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 818 24 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 636 48;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 636 48 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 272 96;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 272 96 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 545 92;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 545 92 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 091 84;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 091 84 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 183 68;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 183 68 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 367 36;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 367 36 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 734 72;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 734 72 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 153 469 44;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 153 469 44 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 306 938 88;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 306 938 88 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 613 877 76;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 613 877 76 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 227 755 52;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 227 755 52 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 455 511 04;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 455 511 04 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 911 022 08;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 911 022 08 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 822 044 16;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 822 044 16 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 644 088 32;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 644 088 32 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 847 288 176 64;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 847 288 176 64 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 694 576 353 28;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 694 576 353 28 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 389 152 706 56;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 389 152 706 56 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 778 305 413 12;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 778 305 413 12 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 556 610 826 24;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 556 610 826 24 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 113 221 652 48;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 113 221 652 48 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 226 443 304 96;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 226 443 304 96 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 452 886 609 92;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 452 886 609 92 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 905 773 219 84;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 905 773 219 84 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 811 546 439 68;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 811 546 439 68 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 623 092 879 36;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 623 092 879 36 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 199 246 185 758 72;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 199 246 185 758 72 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 398 492 371 517 44;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 398 492 371 517 44 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 796 984 743 034 88;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 796 984 743 034 88 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 593 969 486 069 76;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 593 969 486 069 76 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 187 938 972 139 52;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 187 938 972 139 52 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 375 877 944 279 04;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 375 877 944 279 04 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 751 755 888 558 08;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 751 755 888 558 08 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 503 511 777 116 16;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 503 511 777 116 16 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 107 007 023 554 232 32;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 107 007 023 554 232 32 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 214 014 047 108 464 64;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 214 014 047 108 464 64 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 428 028 094 216 929 28;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 428 028 094 216 929 28 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 856 056 188 433 858 56;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 856 056 188 433 858 56 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 712 112 376 867 717 12;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 712 112 376 867 717 12 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 424 224 753 735 434 24;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 424 224 753 735 434 24 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 848 449 507 470 868 48;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 848 449 507 470 868 48 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 696 899 014 941 736 96;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 696 899 014 941 736 96 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 393 798 029 883 473 92;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 393 798 029 883 473 92 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 787 596 059 766 947 84;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 787 596 059 766 947 84 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 575 192 119 533 895 68;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 575 192 119 533 895 68 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 283 150 384 239 067 791 36;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 283 150 384 239 067 791 36 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 566 300 768 478 135 582 72;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 566 300 768 478 135 582 72 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 132 601 536 956 271 165 44;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 132 601 536 956 271 165 44 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 265 203 073 912 542 330 88;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 265 203 073 912 542 330 88 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 530 406 147 825 084 661 76;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 530 406 147 825 084 661 76 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 060 812 295 650 169 323 52;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 060 812 295 650 169 323 52 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 121 624 591 300 338 647 04;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 121 624 591 300 338 647 04 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 243 249 182 600 677 294 08;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 243 249 182 600 677 294 08 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 486 498 365 201 354 588 16;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 486 498 365 201 354 588 16 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 972 996 730 402 709 176 32;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 972 996 730 402 709 176 32 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 001 945 993 460 805 418 352 64;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 001 945 993 460 805 418 352 64 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 003 891 986 921 610 836 705 28;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 003 891 986 921 610 836 705 28 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 007 783 973 843 221 673 410 56;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 007 783 973 843 221 673 410 56 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 015 567 947 686 443 346 821 12;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 015 567 947 686 443 346 821 12 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 031 135 895 372 886 693 642 24;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 031 135 895 372 886 693 642 24 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 062 271 790 745 773 387 284 48;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 062 271 790 745 773 387 284 48 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 124 543 581 491 546 774 568 96;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 124 543 581 491 546 774 568 96 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 249 087 162 983 093 549 137 92;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 249 087 162 983 093 549 137 92 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 498 174 325 966 187 098 275 84;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 498 174 325 966 187 098 275 84 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 996 348 651 932 374 196 551 68;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 996 348 651 932 374 196 551 68 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 001 992 697 303 864 748 393 103 36;
  • 73) 0.231 159 210 205 078 125 000 000 000 001 992 697 303 864 748 393 103 36 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 003 985 394 607 729 496 786 206 72;
  • 74) 0.462 318 420 410 156 250 000 000 000 003 985 394 607 729 496 786 206 72 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 007 970 789 215 458 993 572 413 44;
  • 75) 0.924 636 840 820 312 500 000 000 000 007 970 789 215 458 993 572 413 44 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 015 941 578 430 917 987 144 826 88;
  • 76) 0.849 273 681 640 625 000 000 000 000 015 941 578 430 917 987 144 826 88 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 031 883 156 861 835 974 289 653 76;
  • 77) 0.698 547 363 281 250 000 000 000 000 031 883 156 861 835 974 289 653 76 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 063 766 313 723 671 948 579 307 52;
  • 78) 0.397 094 726 562 500 000 000 000 000 063 766 313 723 671 948 579 307 52 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 127 532 627 447 343 897 158 615 04;
  • 79) 0.794 189 453 125 000 000 000 000 000 127 532 627 447 343 897 158 615 04 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 255 065 254 894 687 794 317 230 08;
  • 80) 0.588 378 906 250 000 000 000 000 000 255 065 254 894 687 794 317 230 08 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 510 130 509 789 375 588 634 460 16;
  • 81) 0.176 757 812 500 000 000 000 000 000 510 130 509 789 375 588 634 460 16 × 2 = 0 + 0.353 515 625 000 000 000 000 000 001 020 261 019 578 751 177 268 920 32;
  • 82) 0.353 515 625 000 000 000 000 000 001 020 261 019 578 751 177 268 920 32 × 2 = 0 + 0.707 031 250 000 000 000 000 000 002 040 522 039 157 502 354 537 840 64;
  • 83) 0.707 031 250 000 000 000 000 000 002 040 522 039 157 502 354 537 840 64 × 2 = 1 + 0.414 062 500 000 000 000 000 000 004 081 044 078 315 004 709 075 681 28;
  • 84) 0.414 062 500 000 000 000 000 000 004 081 044 078 315 004 709 075 681 28 × 2 = 0 + 0.828 125 000 000 000 000 000 000 008 162 088 156 630 009 418 151 362 56;
  • 85) 0.828 125 000 000 000 000 000 000 008 162 088 156 630 009 418 151 362 56 × 2 = 1 + 0.656 250 000 000 000 000 000 000 016 324 176 313 260 018 836 302 725 12;
  • 86) 0.656 250 000 000 000 000 000 000 016 324 176 313 260 018 836 302 725 12 × 2 = 1 + 0.312 500 000 000 000 000 000 000 032 648 352 626 520 037 672 605 450 24;
  • 87) 0.312 500 000 000 000 000 000 000 032 648 352 626 520 037 672 605 450 24 × 2 = 0 + 0.625 000 000 000 000 000 000 000 065 296 705 253 040 075 345 210 900 48;
  • 88) 0.625 000 000 000 000 000 000 000 065 296 705 253 040 075 345 210 900 48 × 2 = 1 + 0.250 000 000 000 000 000 000 000 130 593 410 506 080 150 690 421 800 96;
  • 89) 0.250 000 000 000 000 000 000 000 130 593 410 506 080 150 690 421 800 96 × 2 = 0 + 0.500 000 000 000 000 000 000 000 261 186 821 012 160 301 380 843 601 92;
  • 90) 0.500 000 000 000 000 000 000 000 261 186 821 012 160 301 380 843 601 92 × 2 = 1 + 0.000 000 000 000 000 000 000 000 522 373 642 024 320 602 761 687 203 84;
  • 91) 0.000 000 000 000 000 000 000 000 522 373 642 024 320 602 761 687 203 84 × 2 = 0 + 0.000 000 000 000 000 000 000 001 044 747 284 048 641 205 523 374 407 68;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(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 1010


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 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


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 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 41 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 1010

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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