0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 963 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 963 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 926 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 926 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 852 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 852 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 705 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 705 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 411 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 411 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 822 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 822 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 644 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 644 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 163 289 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 163 289 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 326 579 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 326 579 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 653 158 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 653 158 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 306 316 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 306 316 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 612 633 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 612 633 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 225 267 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 225 267 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 450 534 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 450 534 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 901 068 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 901 068 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 802 137 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 802 137 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 604 275 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 604 275 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 343 208 550 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 343 208 550 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 686 417 100 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 686 417 100 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 372 834 201 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 372 834 201 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 745 668 403 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 745 668 403 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 491 336 806 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 491 336 806 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 982 673 612 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 982 673 612 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 965 347 225 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 965 347 225 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 930 694 451 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 930 694 451 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 861 388 902 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 861 388 902 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 722 777 804 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 722 777 804 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 111 445 555 609 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 111 445 555 609 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 222 891 111 219 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 222 891 111 219 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 445 782 222 438 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 445 782 222 438 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 891 564 444 876 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 891 564 444 876 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 783 128 889 753 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 783 128 889 753 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 566 257 779 507 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 566 257 779 507 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 132 515 559 014 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 132 515 559 014 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 265 031 118 028 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 265 031 118 028 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 530 062 236 057 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 530 062 236 057 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 060 124 472 115 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 060 124 472 115 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 258 120 248 944 230 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 258 120 248 944 230 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 516 240 497 888 460 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 516 240 497 888 460 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 032 480 995 776 921 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 032 480 995 776 921 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 064 961 991 553 843 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 064 961 991 553 843 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 129 923 983 107 686 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 129 923 983 107 686 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 259 847 966 215 372 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 259 847 966 215 372 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 519 695 932 430 745 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 519 695 932 430 745 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 039 391 864 861 491 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 039 391 864 861 491 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 078 783 729 722 982 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 078 783 729 722 982 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 788 157 567 459 445 964 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 788 157 567 459 445 964 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 576 315 134 918 891 929 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 576 315 134 918 891 929 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 152 630 269 837 783 859 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 152 630 269 837 783 859 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 305 260 539 675 567 718 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 305 260 539 675 567 718 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 610 521 079 351 135 436 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 610 521 079 351 135 436 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 221 042 158 702 270 873 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 221 042 158 702 270 873 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 442 084 317 404 541 747 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 442 084 317 404 541 747 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 884 168 634 809 083 494 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 884 168 634 809 083 494 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 768 337 269 618 166 988 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 768 337 269 618 166 988 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 536 674 539 236 333 977 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 536 674 539 236 333 977 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 073 349 078 472 667 955 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 073 349 078 472 667 955 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 998 146 698 156 945 335 910 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 998 146 698 156 945 335 910 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 996 293 396 313 890 671 820 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 996 293 396 313 890 671 820 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 992 586 792 627 781 343 641 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 992 586 792 627 781 343 641 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 985 173 585 255 562 687 283 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 985 173 585 255 562 687 283 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 970 347 170 511 125 374 566 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 970 347 170 511 125 374 566 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 940 694 341 022 250 749 132 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 940 694 341 022 250 749 132 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 881 388 682 044 501 498 265 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 881 388 682 044 501 498 265 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 762 777 364 089 002 996 531 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 762 777 364 089 002 996 531 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 525 554 728 178 005 993 062 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 525 554 728 178 005 993 062 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 051 109 456 356 011 986 124 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 051 109 456 356 011 986 124 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 998 102 218 912 712 023 972 249 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 998 102 218 912 712 023 972 249 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 996 204 437 825 424 047 944 499 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 996 204 437 825 424 047 944 499 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 992 408 875 650 848 095 888 998 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 992 408 875 650 848 095 888 998 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 984 817 751 301 696 191 777 996 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 984 817 751 301 696 191 777 996 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 969 635 502 603 392 383 555 993 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 969 635 502 603 392 383 555 993 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 939 271 005 206 784 767 111 987 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 939 271 005 206 784 767 111 987 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 878 542 010 413 569 534 223 974 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 878 542 010 413 569 534 223 974 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 757 084 020 827 139 068 447 948 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 757 084 020 827 139 068 447 948 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 514 168 041 654 278 136 895 897 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 514 168 041 654 278 136 895 897 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 028 336 083 308 556 273 791 795 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 028 336 083 308 556 273 791 795 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 998 056 672 166 617 112 547 583 590 4;
  • 79) 0.794 189 453 124 999 999 999 999 999 998 056 672 166 617 112 547 583 590 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 996 113 344 333 234 225 095 167 180 8;
  • 80) 0.588 378 906 249 999 999 999 999 999 996 113 344 333 234 225 095 167 180 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 992 226 688 666 468 450 190 334 361 6;
  • 81) 0.176 757 812 499 999 999 999 999 999 992 226 688 666 468 450 190 334 361 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 984 453 377 332 936 900 380 668 723 2;
  • 82) 0.353 515 624 999 999 999 999 999 999 984 453 377 332 936 900 380 668 723 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 968 906 754 665 873 800 761 337 446 4;
  • 83) 0.707 031 249 999 999 999 999 999 999 968 906 754 665 873 800 761 337 446 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 937 813 509 331 747 601 522 674 892 8;
  • 84) 0.414 062 499 999 999 999 999 999 999 937 813 509 331 747 601 522 674 892 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 875 627 018 663 495 203 045 349 785 6;
  • 85) 0.828 124 999 999 999 999 999 999 999 875 627 018 663 495 203 045 349 785 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 751 254 037 326 990 406 090 699 571 2;
  • 86) 0.656 249 999 999 999 999 999 999 999 751 254 037 326 990 406 090 699 571 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 502 508 074 653 980 812 181 399 142 4;
  • 87) 0.312 499 999 999 999 999 999 999 999 502 508 074 653 980 812 181 399 142 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 005 016 149 307 961 624 362 798 284 8;
  • 88) 0.624 999 999 999 999 999 999 999 999 005 016 149 307 961 624 362 798 284 8 × 2 = 1 + 0.249 999 999 999 999 999 999 999 998 010 032 298 615 923 248 725 596 569 6;
  • 89) 0.249 999 999 999 999 999 999 999 998 010 032 298 615 923 248 725 596 569 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 996 020 064 597 231 846 497 451 193 139 2;
  • 90) 0.499 999 999 999 999 999 999 999 996 020 064 597 231 846 497 451 193 139 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 992 040 129 194 463 692 994 902 386 278 4;
  • 91) 0.999 999 999 999 999 999 999 999 992 040 129 194 463 692 994 902 386 278 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 984 080 258 388 927 385 989 804 772 556 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 981 6 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