0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 4 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 4(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 4(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 4.

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 4 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 278 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 278 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 557 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 557 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 115 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 115 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 230 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 230 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 460 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 460 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 921 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 921 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 843 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 843 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 795 686 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 795 686 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 591 372 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 591 372 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 182 745 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 182 745 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 365 491 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 365 491 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 730 982 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 730 982 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 461 964 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 461 964 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 923 929 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 923 929 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 847 859 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 847 859 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 695 718 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 695 718 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 391 436 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 391 436 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 486 782 873 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 486 782 873 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 973 565 747 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 973 565 747 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 947 131 494 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 947 131 494 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 894 262 988 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 894 262 988 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 788 525 977 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 788 525 977 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 577 051 955 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 577 051 955 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 154 103 910 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 154 103 910 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 308 207 820 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 308 207 820 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 616 415 641 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 616 415 641 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 232 831 283 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 232 831 283 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 465 662 566 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 002 465 662 566 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 004 931 325 132 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 004 931 325 132 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 009 862 650 265 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 009 862 650 265 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 019 725 300 531 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 019 725 300 531 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 039 450 601 062 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 039 450 601 062 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 078 901 202 124 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 078 901 202 124 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 157 802 404 249 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 157 802 404 249 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 315 604 808 499 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 315 604 808 499 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 631 209 616 998 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 631 209 616 998 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 817 262 419 233 996 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 817 262 419 233 996 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 634 524 838 467 993 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 634 524 838 467 993 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 269 049 676 935 987 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 269 049 676 935 987 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 538 099 353 871 974 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 538 099 353 871 974 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 076 198 707 743 948 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 076 198 707 743 948 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 152 397 415 487 897 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 152 397 415 487 897 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 304 794 830 975 795 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 304 794 830 975 795 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 609 589 661 951 590 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 609 589 661 951 590 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 219 179 323 903 180 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 219 179 323 903 180 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 438 358 647 806 361 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 438 358 647 806 361 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 756 876 717 295 612 723 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 756 876 717 295 612 723 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 513 753 434 591 225 446 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 513 753 434 591 225 446 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 027 506 869 182 450 892 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 027 506 869 182 450 892 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 055 013 738 364 901 785 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 055 013 738 364 901 785 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 110 027 476 729 803 571 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 110 027 476 729 803 571 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 220 054 953 459 607 142 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 220 054 953 459 607 142 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 440 109 906 919 214 284 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 440 109 906 919 214 284 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 880 219 813 838 428 569 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 880 219 813 838 428 569 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 760 439 627 676 857 139 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 760 439 627 676 857 139 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 520 879 255 353 714 278 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 520 879 255 353 714 278 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 623 041 758 510 707 428 556 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 623 041 758 510 707 428 556 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 246 083 517 021 414 857 113 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 246 083 517 021 414 857 113 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 492 167 034 042 829 714 227 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 492 167 034 042 829 714 227 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 984 334 068 085 659 428 454 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 984 334 068 085 659 428 454 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 968 668 136 171 318 856 908 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 968 668 136 171 318 856 908 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 937 336 272 342 637 713 817 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 937 336 272 342 637 713 817 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 874 672 544 685 275 427 635 2;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 874 672 544 685 275 427 635 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 749 345 089 370 550 855 270 4;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 749 345 089 370 550 855 270 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 498 690 178 741 101 710 540 8;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 498 690 178 741 101 710 540 8 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 998 997 380 357 482 203 421 081 6;
  • 67) 0.831 736 862 659 454 345 703 124 999 998 997 380 357 482 203 421 081 6 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 997 994 760 714 964 406 842 163 2;
  • 68) 0.663 473 725 318 908 691 406 249 999 997 994 760 714 964 406 842 163 2 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 995 989 521 429 928 813 684 326 4;
  • 69) 0.326 947 450 637 817 382 812 499 999 995 989 521 429 928 813 684 326 4 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 991 979 042 859 857 627 368 652 8;
  • 70) 0.653 894 901 275 634 765 624 999 999 991 979 042 859 857 627 368 652 8 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 983 958 085 719 715 254 737 305 6;
  • 71) 0.307 789 802 551 269 531 249 999 999 983 958 085 719 715 254 737 305 6 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 967 916 171 439 430 509 474 611 2;
  • 72) 0.615 579 605 102 539 062 499 999 999 967 916 171 439 430 509 474 611 2 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 935 832 342 878 861 018 949 222 4;
  • 73) 0.231 159 210 205 078 124 999 999 999 935 832 342 878 861 018 949 222 4 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 871 664 685 757 722 037 898 444 8;
  • 74) 0.462 318 420 410 156 249 999 999 999 871 664 685 757 722 037 898 444 8 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 743 329 371 515 444 075 796 889 6;
  • 75) 0.924 636 840 820 312 499 999 999 999 743 329 371 515 444 075 796 889 6 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 486 658 743 030 888 151 593 779 2;
  • 76) 0.849 273 681 640 624 999 999 999 999 486 658 743 030 888 151 593 779 2 × 2 = 1 + 0.698 547 363 281 249 999 999 999 998 973 317 486 061 776 303 187 558 4;
  • 77) 0.698 547 363 281 249 999 999 999 998 973 317 486 061 776 303 187 558 4 × 2 = 1 + 0.397 094 726 562 499 999 999 999 997 946 634 972 123 552 606 375 116 8;
  • 78) 0.397 094 726 562 499 999 999 999 997 946 634 972 123 552 606 375 116 8 × 2 = 0 + 0.794 189 453 124 999 999 999 999 995 893 269 944 247 105 212 750 233 6;
  • 79) 0.794 189 453 124 999 999 999 999 995 893 269 944 247 105 212 750 233 6 × 2 = 1 + 0.588 378 906 249 999 999 999 999 991 786 539 888 494 210 425 500 467 2;
  • 80) 0.588 378 906 249 999 999 999 999 991 786 539 888 494 210 425 500 467 2 × 2 = 1 + 0.176 757 812 499 999 999 999 999 983 573 079 776 988 420 851 000 934 4;
  • 81) 0.176 757 812 499 999 999 999 999 983 573 079 776 988 420 851 000 934 4 × 2 = 0 + 0.353 515 624 999 999 999 999 999 967 146 159 553 976 841 702 001 868 8;
  • 82) 0.353 515 624 999 999 999 999 999 967 146 159 553 976 841 702 001 868 8 × 2 = 0 + 0.707 031 249 999 999 999 999 999 934 292 319 107 953 683 404 003 737 6;
  • 83) 0.707 031 249 999 999 999 999 999 934 292 319 107 953 683 404 003 737 6 × 2 = 1 + 0.414 062 499 999 999 999 999 999 868 584 638 215 907 366 808 007 475 2;
  • 84) 0.414 062 499 999 999 999 999 999 868 584 638 215 907 366 808 007 475 2 × 2 = 0 + 0.828 124 999 999 999 999 999 999 737 169 276 431 814 733 616 014 950 4;
  • 85) 0.828 124 999 999 999 999 999 999 737 169 276 431 814 733 616 014 950 4 × 2 = 1 + 0.656 249 999 999 999 999 999 999 474 338 552 863 629 467 232 029 900 8;
  • 86) 0.656 249 999 999 999 999 999 999 474 338 552 863 629 467 232 029 900 8 × 2 = 1 + 0.312 499 999 999 999 999 999 998 948 677 105 727 258 934 464 059 801 6;
  • 87) 0.312 499 999 999 999 999 999 998 948 677 105 727 258 934 464 059 801 6 × 2 = 0 + 0.624 999 999 999 999 999 999 997 897 354 211 454 517 868 928 119 603 2;
  • 88) 0.624 999 999 999 999 999 999 997 897 354 211 454 517 868 928 119 603 2 × 2 = 1 + 0.249 999 999 999 999 999 999 995 794 708 422 909 035 737 856 239 206 4;
  • 89) 0.249 999 999 999 999 999 999 995 794 708 422 909 035 737 856 239 206 4 × 2 = 0 + 0.499 999 999 999 999 999 999 991 589 416 845 818 071 475 712 478 412 8;
  • 90) 0.499 999 999 999 999 999 999 991 589 416 845 818 071 475 712 478 412 8 × 2 = 0 + 0.999 999 999 999 999 999 999 983 178 833 691 636 142 951 424 956 825 6;
  • 91) 0.999 999 999 999 999 999 999 983 178 833 691 636 142 951 424 956 825 6 × 2 = 1 + 0.999 999 999 999 999 999 999 966 357 667 383 272 285 902 849 913 651 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 139 4(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 4(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 4(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 4 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