0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 386;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 386 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 772;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 772 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 544;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 544 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 088;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 088 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 902 176;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 902 176 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 804 352;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 804 352 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 608 704;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 608 704 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 217 408;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 217 408 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 434 816;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 434 816 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 869 632;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 869 632 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 739 264;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 739 264 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 478 528;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 478 528 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 957 056;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 957 056 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 149 914 112;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 149 914 112 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 299 828 224;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 299 828 224 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 599 656 448;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 599 656 448 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 199 312 896;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 199 312 896 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 398 625 792;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 398 625 792 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 797 251 584;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 797 251 584 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 594 503 168;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 594 503 168 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 189 006 336;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 189 006 336 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 378 012 672;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 378 012 672 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 756 025 344;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 756 025 344 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 385 512 050 688;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 385 512 050 688 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 771 024 101 376;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 771 024 101 376 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 542 048 202 752;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 542 048 202 752 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 084 096 405 504;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 084 096 405 504 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 168 192 811 008;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 168 192 811 008 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 336 385 622 016;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 336 385 622 016 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 672 771 244 032;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 672 771 244 032 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 345 542 488 064;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 345 542 488 064 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 691 084 976 128;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 691 084 976 128 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 382 169 952 256;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 382 169 952 256 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 394 764 339 904 512;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 394 764 339 904 512 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 789 528 679 809 024;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 789 528 679 809 024 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 579 057 359 618 048;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 579 057 359 618 048 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 158 114 719 236 096;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 158 114 719 236 096 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 316 229 438 472 192;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 316 229 438 472 192 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 632 458 876 944 384;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 632 458 876 944 384 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 264 917 753 888 768;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 264 917 753 888 768 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 529 835 507 777 536;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 529 835 507 777 536 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 213 059 671 015 555 072;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 213 059 671 015 555 072 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 426 119 342 031 110 144;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 426 119 342 031 110 144 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 852 238 684 062 220 288;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 852 238 684 062 220 288 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 704 477 368 124 440 576;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 704 477 368 124 440 576 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 408 954 736 248 881 152;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 408 954 736 248 881 152 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 817 909 472 497 762 304;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 817 909 472 497 762 304 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 635 818 944 995 524 608;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 635 818 944 995 524 608 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 271 637 889 991 049 216;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 271 637 889 991 049 216 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 543 275 779 982 098 432;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 543 275 779 982 098 432 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 086 551 559 964 196 864;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 086 551 559 964 196 864 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 282 173 103 119 928 393 728;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 282 173 103 119 928 393 728 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 564 346 206 239 856 787 456;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 564 346 206 239 856 787 456 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 128 692 412 479 713 574 912;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 128 692 412 479 713 574 912 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 257 384 824 959 427 149 824;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 257 384 824 959 427 149 824 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 514 769 649 918 854 299 648;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 514 769 649 918 854 299 648 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 029 539 299 837 708 599 296;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 029 539 299 837 708 599 296 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 059 078 599 675 417 198 592;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 059 078 599 675 417 198 592 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 118 157 199 350 834 397 184;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 118 157 199 350 834 397 184 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 236 314 398 701 668 794 368;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 236 314 398 701 668 794 368 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 472 628 797 403 337 588 736;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 472 628 797 403 337 588 736 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 945 257 594 806 675 177 472;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 945 257 594 806 675 177 472 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 001 890 515 189 613 350 354 944;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 001 890 515 189 613 350 354 944 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 003 781 030 379 226 700 709 888;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 003 781 030 379 226 700 709 888 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 007 562 060 758 453 401 419 776;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 007 562 060 758 453 401 419 776 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 015 124 121 516 906 802 839 552;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 015 124 121 516 906 802 839 552 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 030 248 243 033 813 605 679 104;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 030 248 243 033 813 605 679 104 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 060 496 486 067 627 211 358 208;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 060 496 486 067 627 211 358 208 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 120 992 972 135 254 422 716 416;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 120 992 972 135 254 422 716 416 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 241 985 944 270 508 845 432 832;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 241 985 944 270 508 845 432 832 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 483 971 888 541 017 690 865 664;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 483 971 888 541 017 690 865 664 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 967 943 777 082 035 381 731 328;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 967 943 777 082 035 381 731 328 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 001 935 887 554 164 070 763 462 656;
  • 74) 0.462 318 420 410 156 250 000 000 000 001 935 887 554 164 070 763 462 656 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 003 871 775 108 328 141 526 925 312;
  • 75) 0.924 636 840 820 312 500 000 000 000 003 871 775 108 328 141 526 925 312 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 007 743 550 216 656 283 053 850 624;
  • 76) 0.849 273 681 640 625 000 000 000 000 007 743 550 216 656 283 053 850 624 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 015 487 100 433 312 566 107 701 248;
  • 77) 0.698 547 363 281 250 000 000 000 000 015 487 100 433 312 566 107 701 248 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 030 974 200 866 625 132 215 402 496;
  • 78) 0.397 094 726 562 500 000 000 000 000 030 974 200 866 625 132 215 402 496 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 061 948 401 733 250 264 430 804 992;
  • 79) 0.794 189 453 125 000 000 000 000 000 061 948 401 733 250 264 430 804 992 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 123 896 803 466 500 528 861 609 984;
  • 80) 0.588 378 906 250 000 000 000 000 000 123 896 803 466 500 528 861 609 984 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 247 793 606 933 001 057 723 219 968;
  • 81) 0.176 757 812 500 000 000 000 000 000 247 793 606 933 001 057 723 219 968 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 495 587 213 866 002 115 446 439 936;
  • 82) 0.353 515 625 000 000 000 000 000 000 495 587 213 866 002 115 446 439 936 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 991 174 427 732 004 230 892 879 872;
  • 83) 0.707 031 250 000 000 000 000 000 000 991 174 427 732 004 230 892 879 872 × 2 = 1 + 0.414 062 500 000 000 000 000 000 001 982 348 855 464 008 461 785 759 744;
  • 84) 0.414 062 500 000 000 000 000 000 001 982 348 855 464 008 461 785 759 744 × 2 = 0 + 0.828 125 000 000 000 000 000 000 003 964 697 710 928 016 923 571 519 488;
  • 85) 0.828 125 000 000 000 000 000 000 003 964 697 710 928 016 923 571 519 488 × 2 = 1 + 0.656 250 000 000 000 000 000 000 007 929 395 421 856 033 847 143 038 976;
  • 86) 0.656 250 000 000 000 000 000 000 007 929 395 421 856 033 847 143 038 976 × 2 = 1 + 0.312 500 000 000 000 000 000 000 015 858 790 843 712 067 694 286 077 952;
  • 87) 0.312 500 000 000 000 000 000 000 015 858 790 843 712 067 694 286 077 952 × 2 = 0 + 0.625 000 000 000 000 000 000 000 031 717 581 687 424 135 388 572 155 904;
  • 88) 0.625 000 000 000 000 000 000 000 031 717 581 687 424 135 388 572 155 904 × 2 = 1 + 0.250 000 000 000 000 000 000 000 063 435 163 374 848 270 777 144 311 808;
  • 89) 0.250 000 000 000 000 000 000 000 063 435 163 374 848 270 777 144 311 808 × 2 = 0 + 0.500 000 000 000 000 000 000 000 126 870 326 749 696 541 554 288 623 616;
  • 90) 0.500 000 000 000 000 000 000 000 126 870 326 749 696 541 554 288 623 616 × 2 = 1 + 0.000 000 000 000 000 000 000 000 253 740 653 499 393 083 108 577 247 232;
  • 91) 0.000 000 000 000 000 000 000 000 253 740 653 499 393 083 108 577 247 232 × 2 = 0 + 0.000 000 000 000 000 000 000 000 507 481 306 998 786 166 217 154 494 464;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 193 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