0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 9 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 9(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 9(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 9.

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 9 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 615 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 615 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 231 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 231 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 462 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 462 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 924 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 924 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 849 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 849 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 699 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 699 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 398 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 398 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 796 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 796 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 593 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 593 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 395 187 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 395 187 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 790 374 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 790 374 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 580 748 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 580 748 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 161 497 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 161 497 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 322 995 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 322 995 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 645 990 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 645 990 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 291 980 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 291 980 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 583 961 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 583 961 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 167 923 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 167 923 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 335 846 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 335 846 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 671 692 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 671 692 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 849 343 385 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 849 343 385 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 698 686 771 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 698 686 771 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 397 373 542 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 397 373 542 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 794 747 084 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 794 747 084 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 589 494 169 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 589 494 169 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 178 988 339 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 178 988 339 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 357 976 678 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 357 976 678 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 715 953 356 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 715 953 356 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 431 906 713 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 431 906 713 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 863 813 427 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 863 813 427 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 101 727 626 854 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 101 727 626 854 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 203 455 253 708 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 203 455 253 708 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 406 910 507 417 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 406 910 507 417 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 813 821 014 835 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 813 821 014 835 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 627 642 029 670 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 627 642 029 670 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 255 284 059 340 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 255 284 059 340 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 510 568 118 681 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 510 568 118 681 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 277 021 136 237 363 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 277 021 136 237 363 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 554 042 272 474 726 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 554 042 272 474 726 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 108 084 544 949 452 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 108 084 544 949 452 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 216 169 089 898 905 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 216 169 089 898 905 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 432 338 179 797 811 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 432 338 179 797 811 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 864 676 359 595 622 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 864 676 359 595 622 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 729 352 719 191 244 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 729 352 719 191 244 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 458 705 438 382 489 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 458 705 438 382 489 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 917 410 876 764 979 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 917 410 876 764 979 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 834 821 753 529 958 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 834 821 753 529 958 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 669 643 507 059 916 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 669 643 507 059 916 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 071 339 287 014 119 833 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 071 339 287 014 119 833 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 142 678 574 028 239 667 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 142 678 574 028 239 667 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 285 357 148 056 479 334 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 285 357 148 056 479 334 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 570 714 296 112 958 668 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 570 714 296 112 958 668 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 141 428 592 225 917 337 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 141 428 592 225 917 337 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 282 857 184 451 834 675 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 282 857 184 451 834 675 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 565 714 368 903 669 350 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 565 714 368 903 669 350 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 131 428 737 807 338 700 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 131 428 737 807 338 700 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 262 857 475 614 677 401 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 262 857 475 614 677 401 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 525 714 951 229 354 803 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 525 714 951 229 354 803 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 001 051 429 902 458 709 606 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 001 051 429 902 458 709 606 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 002 102 859 804 917 419 212 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 002 102 859 804 917 419 212 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 004 205 719 609 834 838 425 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 004 205 719 609 834 838 425 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 008 411 439 219 669 676 851 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 008 411 439 219 669 676 851 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 016 822 878 439 339 353 702 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 016 822 878 439 339 353 702 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 033 645 756 878 678 707 404 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 033 645 756 878 678 707 404 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 067 291 513 757 357 414 809 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 067 291 513 757 357 414 809 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 134 583 027 514 714 829 619 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 134 583 027 514 714 829 619 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 269 166 055 029 429 659 238 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 269 166 055 029 429 659 238 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 538 332 110 058 859 318 476 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 538 332 110 058 859 318 476 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 001 076 664 220 117 718 636 953 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 001 076 664 220 117 718 636 953 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 002 153 328 440 235 437 273 907 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 002 153 328 440 235 437 273 907 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 004 306 656 880 470 874 547 814 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 004 306 656 880 470 874 547 814 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 008 613 313 760 941 749 095 628 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 008 613 313 760 941 749 095 628 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 017 226 627 521 883 498 191 257 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 017 226 627 521 883 498 191 257 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 034 453 255 043 766 996 382 515 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 034 453 255 043 766 996 382 515 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 068 906 510 087 533 992 765 030 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 068 906 510 087 533 992 765 030 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 137 813 020 175 067 985 530 060 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 137 813 020 175 067 985 530 060 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 275 626 040 350 135 971 060 121 6;
  • 79) 0.794 189 453 125 000 000 000 000 000 275 626 040 350 135 971 060 121 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 551 252 080 700 271 942 120 243 2;
  • 80) 0.588 378 906 250 000 000 000 000 000 551 252 080 700 271 942 120 243 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 001 102 504 161 400 543 884 240 486 4;
  • 81) 0.176 757 812 500 000 000 000 000 001 102 504 161 400 543 884 240 486 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 002 205 008 322 801 087 768 480 972 8;
  • 82) 0.353 515 625 000 000 000 000 000 002 205 008 322 801 087 768 480 972 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 004 410 016 645 602 175 536 961 945 6;
  • 83) 0.707 031 250 000 000 000 000 000 004 410 016 645 602 175 536 961 945 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 008 820 033 291 204 351 073 923 891 2;
  • 84) 0.414 062 500 000 000 000 000 000 008 820 033 291 204 351 073 923 891 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 017 640 066 582 408 702 147 847 782 4;
  • 85) 0.828 125 000 000 000 000 000 000 017 640 066 582 408 702 147 847 782 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 035 280 133 164 817 404 295 695 564 8;
  • 86) 0.656 250 000 000 000 000 000 000 035 280 133 164 817 404 295 695 564 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 070 560 266 329 634 808 591 391 129 6;
  • 87) 0.312 500 000 000 000 000 000 000 070 560 266 329 634 808 591 391 129 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 141 120 532 659 269 617 182 782 259 2;
  • 88) 0.625 000 000 000 000 000 000 000 141 120 532 659 269 617 182 782 259 2 × 2 = 1 + 0.250 000 000 000 000 000 000 000 282 241 065 318 539 234 365 564 518 4;
  • 89) 0.250 000 000 000 000 000 000 000 282 241 065 318 539 234 365 564 518 4 × 2 = 0 + 0.500 000 000 000 000 000 000 000 564 482 130 637 078 468 731 129 036 8;
  • 90) 0.500 000 000 000 000 000 000 000 564 482 130 637 078 468 731 129 036 8 × 2 = 1 + 0.000 000 000 000 000 000 000 001 128 964 261 274 156 937 462 258 073 6;
  • 91) 0.000 000 000 000 000 000 000 001 128 964 261 274 156 937 462 258 073 6 × 2 = 0 + 0.000 000 000 000 000 000 000 002 257 928 522 548 313 874 924 516 147 2;

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 9(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 9(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 9(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 9 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