0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 278 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 278 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 556 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 556 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 112 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 112 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 225 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 225 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 451 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 451 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 902 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 902 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 804 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 804 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 795 609 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 795 609 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 591 219 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 591 219 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 182 438 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 182 438 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 364 876 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 364 876 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 729 753 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 729 753 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 459 507 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 459 507 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 919 014 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 919 014 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 838 028 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 838 028 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 676 057 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 676 057 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 352 115 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 352 115 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 486 704 230 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 486 704 230 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 973 408 460 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 973 408 460 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 946 816 921 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 946 816 921 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 893 633 843 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 893 633 843 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 787 267 686 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 787 267 686 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 574 535 372 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 574 535 372 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 149 070 745 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 149 070 745 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 298 141 491 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 298 141 491 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 596 282 982 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 596 282 982 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 192 565 964 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 192 565 964 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 385 131 929 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 385 131 929 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 004 770 263 859 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 004 770 263 859 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 009 540 527 718 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 009 540 527 718 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 019 081 055 436 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 019 081 055 436 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 038 162 110 873 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 038 162 110 873 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 076 324 221 747 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 076 324 221 747 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 152 648 443 494 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 152 648 443 494 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 305 296 886 988 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 305 296 886 988 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 610 593 773 977 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 610 593 773 977 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 817 221 187 547 955 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 817 221 187 547 955 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 634 442 375 095 910 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 634 442 375 095 910 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 268 884 750 191 820 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 268 884 750 191 820 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 537 769 500 383 641 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 537 769 500 383 641 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 075 539 000 767 283 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 075 539 000 767 283 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 151 078 001 534 566 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 151 078 001 534 566 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 302 156 003 069 132 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 302 156 003 069 132 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 604 312 006 138 265 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 604 312 006 138 265 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 208 624 012 276 531 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 208 624 012 276 531 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 417 248 024 553 062 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 417 248 024 553 062 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 756 834 496 049 106 124 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 756 834 496 049 106 124 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 513 668 992 098 212 249 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 513 668 992 098 212 249 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 027 337 984 196 424 499 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 027 337 984 196 424 499 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 054 675 968 392 848 998 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 054 675 968 392 848 998 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 109 351 936 785 697 996 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 109 351 936 785 697 996 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 218 703 873 571 395 993 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 218 703 873 571 395 993 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 437 407 747 142 791 987 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 437 407 747 142 791 987 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 874 815 494 285 583 974 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 874 815 494 285 583 974 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 749 630 988 571 167 948 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 749 630 988 571 167 948 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 499 261 977 142 335 897 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 499 261 977 142 335 897 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 622 998 523 954 284 671 795 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 622 998 523 954 284 671 795 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 245 997 047 908 569 343 590 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 245 997 047 908 569 343 590 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 491 994 095 817 138 687 180 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 491 994 095 817 138 687 180 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 983 988 191 634 277 374 361 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 983 988 191 634 277 374 361 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 967 976 383 268 554 748 723 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 967 976 383 268 554 748 723 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 935 952 766 537 109 497 446 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 935 952 766 537 109 497 446 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 871 905 533 074 218 994 892 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 871 905 533 074 218 994 892 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 743 811 066 148 437 989 785 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 743 811 066 148 437 989 785 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 487 622 132 296 875 979 571 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 487 622 132 296 875 979 571 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 998 975 244 264 593 751 959 142 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 998 975 244 264 593 751 959 142 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 997 950 488 529 187 503 918 284 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 997 950 488 529 187 503 918 284 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 995 900 977 058 375 007 836 569 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 995 900 977 058 375 007 836 569 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 991 801 954 116 750 015 673 139 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 991 801 954 116 750 015 673 139 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 983 603 908 233 500 031 346 278 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 983 603 908 233 500 031 346 278 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 967 207 816 467 000 062 692 556 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 967 207 816 467 000 062 692 556 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 934 415 632 934 000 125 385 113 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 934 415 632 934 000 125 385 113 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 868 831 265 868 000 250 770 227 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 868 831 265 868 000 250 770 227 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 737 662 531 736 000 501 540 454 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 737 662 531 736 000 501 540 454 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 475 325 063 472 001 003 080 908 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 475 325 063 472 001 003 080 908 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 998 950 650 126 944 002 006 161 817 6;
  • 77) 0.698 547 363 281 249 999 999 999 998 950 650 126 944 002 006 161 817 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 997 901 300 253 888 004 012 323 635 2;
  • 78) 0.397 094 726 562 499 999 999 999 997 901 300 253 888 004 012 323 635 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 995 802 600 507 776 008 024 647 270 4;
  • 79) 0.794 189 453 124 999 999 999 999 995 802 600 507 776 008 024 647 270 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 991 605 201 015 552 016 049 294 540 8;
  • 80) 0.588 378 906 249 999 999 999 999 991 605 201 015 552 016 049 294 540 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 983 210 402 031 104 032 098 589 081 6;
  • 81) 0.176 757 812 499 999 999 999 999 983 210 402 031 104 032 098 589 081 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 966 420 804 062 208 064 197 178 163 2;
  • 82) 0.353 515 624 999 999 999 999 999 966 420 804 062 208 064 197 178 163 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 932 841 608 124 416 128 394 356 326 4;
  • 83) 0.707 031 249 999 999 999 999 999 932 841 608 124 416 128 394 356 326 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 865 683 216 248 832 256 788 712 652 8;
  • 84) 0.414 062 499 999 999 999 999 999 865 683 216 248 832 256 788 712 652 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 731 366 432 497 664 513 577 425 305 6;
  • 85) 0.828 124 999 999 999 999 999 999 731 366 432 497 664 513 577 425 305 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 462 732 864 995 329 027 154 850 611 2;
  • 86) 0.656 249 999 999 999 999 999 999 462 732 864 995 329 027 154 850 611 2 × 2 = 1 + 0.312 499 999 999 999 999 999 998 925 465 729 990 658 054 309 701 222 4;
  • 87) 0.312 499 999 999 999 999 999 998 925 465 729 990 658 054 309 701 222 4 × 2 = 0 + 0.624 999 999 999 999 999 999 997 850 931 459 981 316 108 619 402 444 8;
  • 88) 0.624 999 999 999 999 999 999 997 850 931 459 981 316 108 619 402 444 8 × 2 = 1 + 0.249 999 999 999 999 999 999 995 701 862 919 962 632 217 238 804 889 6;
  • 89) 0.249 999 999 999 999 999 999 995 701 862 919 962 632 217 238 804 889 6 × 2 = 0 + 0.499 999 999 999 999 999 999 991 403 725 839 925 264 434 477 609 779 2;
  • 90) 0.499 999 999 999 999 999 999 991 403 725 839 925 264 434 477 609 779 2 × 2 = 0 + 0.999 999 999 999 999 999 999 982 807 451 679 850 528 868 955 219 558 4;
  • 91) 0.999 999 999 999 999 999 999 982 807 451 679 850 528 868 955 219 558 4 × 2 = 1 + 0.999 999 999 999 999 999 999 965 614 903 359 701 057 737 910 439 116 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 1 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