0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 798;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 798 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 596;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 596 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 192;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 192 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 446 384;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 446 384 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 892 768;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 892 768 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 785 536;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 785 536 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 571 072;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 571 072 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 142 144;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 142 144 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 284 288;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 284 288 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 568 576;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 568 576 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 137 152;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 137 152 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 274 304;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 274 304 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 548 608;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 548 608 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 145 097 216;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 145 097 216 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 290 194 432;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 290 194 432 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 580 388 864;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 580 388 864 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 160 777 728;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 160 777 728 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 321 555 456;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 321 555 456 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 643 110 912;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 643 110 912 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 286 221 824;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 286 221 824 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 572 443 648;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 572 443 648 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 144 887 296;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 144 887 296 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 289 774 592;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 289 774 592 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 579 549 184;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 579 549 184 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 761 159 098 368;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 761 159 098 368 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 522 318 196 736;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 522 318 196 736 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 044 636 393 472;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 044 636 393 472 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 089 272 786 944;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 089 272 786 944 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 178 545 573 888;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 178 545 573 888 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 357 091 147 776;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 357 091 147 776 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 714 182 295 552;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 714 182 295 552 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 428 364 591 104;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 428 364 591 104 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 856 729 182 208;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 856 729 182 208 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 713 458 364 416;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 713 458 364 416 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 779 426 916 728 832;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 779 426 916 728 832 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 558 853 833 457 664;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 558 853 833 457 664 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 117 707 666 915 328;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 117 707 666 915 328 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 235 415 333 830 656;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 235 415 333 830 656 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 470 830 667 661 312;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 470 830 667 661 312 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 941 661 335 322 624;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 941 661 335 322 624 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 105 883 322 670 645 248;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 105 883 322 670 645 248 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 211 766 645 341 290 496;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 211 766 645 341 290 496 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 423 533 290 682 580 992;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 423 533 290 682 580 992 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 847 066 581 365 161 984;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 847 066 581 365 161 984 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 694 133 162 730 323 968;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 694 133 162 730 323 968 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 388 266 325 460 647 936;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 388 266 325 460 647 936 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 776 532 650 921 295 872;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 776 532 650 921 295 872 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 553 065 301 842 591 744;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 553 065 301 842 591 744 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 106 130 603 685 183 488;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 106 130 603 685 183 488 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 212 261 207 370 366 976;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 212 261 207 370 366 976 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 424 522 414 740 733 952;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 424 522 414 740 733 952 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 280 849 044 829 481 467 904;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 280 849 044 829 481 467 904 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 561 698 089 658 962 935 808;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 561 698 089 658 962 935 808 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 123 396 179 317 925 871 616;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 123 396 179 317 925 871 616 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 246 792 358 635 851 743 232;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 246 792 358 635 851 743 232 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 493 584 717 271 703 486 464;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 493 584 717 271 703 486 464 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 987 169 434 543 406 972 928;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 987 169 434 543 406 972 928 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 974 338 869 086 813 945 856;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 974 338 869 086 813 945 856 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 948 677 738 173 627 891 712;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 948 677 738 173 627 891 712 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 897 355 476 347 255 783 424;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 897 355 476 347 255 783 424 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 794 710 952 694 511 566 848;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 794 710 952 694 511 566 848 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 589 421 905 389 023 133 696;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 589 421 905 389 023 133 696 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 178 843 810 778 046 267 392;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 178 843 810 778 046 267 392 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 998 357 687 621 556 092 534 784;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 998 357 687 621 556 092 534 784 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 996 715 375 243 112 185 069 568;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 996 715 375 243 112 185 069 568 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 993 430 750 486 224 370 139 136;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 993 430 750 486 224 370 139 136 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 986 861 500 972 448 740 278 272;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 986 861 500 972 448 740 278 272 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 973 723 001 944 897 480 556 544;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 973 723 001 944 897 480 556 544 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 947 446 003 889 794 961 113 088;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 947 446 003 889 794 961 113 088 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 894 892 007 779 589 922 226 176;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 894 892 007 779 589 922 226 176 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 789 784 015 559 179 844 452 352;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 789 784 015 559 179 844 452 352 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 579 568 031 118 359 688 904 704;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 579 568 031 118 359 688 904 704 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 159 136 062 236 719 377 809 408;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 159 136 062 236 719 377 809 408 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 998 318 272 124 473 438 755 618 816;
  • 75) 0.924 636 840 820 312 499 999 999 999 998 318 272 124 473 438 755 618 816 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 996 636 544 248 946 877 511 237 632;
  • 76) 0.849 273 681 640 624 999 999 999 999 996 636 544 248 946 877 511 237 632 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 993 273 088 497 893 755 022 475 264;
  • 77) 0.698 547 363 281 249 999 999 999 999 993 273 088 497 893 755 022 475 264 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 986 546 176 995 787 510 044 950 528;
  • 78) 0.397 094 726 562 499 999 999 999 999 986 546 176 995 787 510 044 950 528 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 973 092 353 991 575 020 089 901 056;
  • 79) 0.794 189 453 124 999 999 999 999 999 973 092 353 991 575 020 089 901 056 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 946 184 707 983 150 040 179 802 112;
  • 80) 0.588 378 906 249 999 999 999 999 999 946 184 707 983 150 040 179 802 112 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 892 369 415 966 300 080 359 604 224;
  • 81) 0.176 757 812 499 999 999 999 999 999 892 369 415 966 300 080 359 604 224 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 784 738 831 932 600 160 719 208 448;
  • 82) 0.353 515 624 999 999 999 999 999 999 784 738 831 932 600 160 719 208 448 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 569 477 663 865 200 321 438 416 896;
  • 83) 0.707 031 249 999 999 999 999 999 999 569 477 663 865 200 321 438 416 896 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 138 955 327 730 400 642 876 833 792;
  • 84) 0.414 062 499 999 999 999 999 999 999 138 955 327 730 400 642 876 833 792 × 2 = 0 + 0.828 124 999 999 999 999 999 999 998 277 910 655 460 801 285 753 667 584;
  • 85) 0.828 124 999 999 999 999 999 999 998 277 910 655 460 801 285 753 667 584 × 2 = 1 + 0.656 249 999 999 999 999 999 999 996 555 821 310 921 602 571 507 335 168;
  • 86) 0.656 249 999 999 999 999 999 999 996 555 821 310 921 602 571 507 335 168 × 2 = 1 + 0.312 499 999 999 999 999 999 999 993 111 642 621 843 205 143 014 670 336;
  • 87) 0.312 499 999 999 999 999 999 999 993 111 642 621 843 205 143 014 670 336 × 2 = 0 + 0.624 999 999 999 999 999 999 999 986 223 285 243 686 410 286 029 340 672;
  • 88) 0.624 999 999 999 999 999 999 999 986 223 285 243 686 410 286 029 340 672 × 2 = 1 + 0.249 999 999 999 999 999 999 999 972 446 570 487 372 820 572 058 681 344;
  • 89) 0.249 999 999 999 999 999 999 999 972 446 570 487 372 820 572 058 681 344 × 2 = 0 + 0.499 999 999 999 999 999 999 999 944 893 140 974 745 641 144 117 362 688;
  • 90) 0.499 999 999 999 999 999 999 999 944 893 140 974 745 641 144 117 362 688 × 2 = 0 + 0.999 999 999 999 999 999 999 999 889 786 281 949 491 282 288 234 725 376;
  • 91) 0.999 999 999 999 999 999 999 999 889 786 281 949 491 282 288 234 725 376 × 2 = 1 + 0.999 999 999 999 999 999 999 999 779 572 563 898 982 564 576 469 450 752;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 899 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