0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 108;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 108 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 216;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 216 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 432;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 432 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 864;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 864 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 728;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 728 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 795 456;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 795 456 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 590 912;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 590 912 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 181 824;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 181 824 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 363 648;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 363 648 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 727 296;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 727 296 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 454 592;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 454 592 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 909 184;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 909 184 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 818 368;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 818 368 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 636 736;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 636 736 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 295 273 472;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 295 273 472 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 590 546 944;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 590 546 944 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 181 093 888;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 181 093 888 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 362 187 776;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 362 187 776 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 724 375 552;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 724 375 552 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 448 751 104;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 448 751 104 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 897 502 208;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 897 502 208 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 795 004 416;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 795 004 416 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 590 008 832;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 590 008 832 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 180 017 664;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 180 017 664 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 766 360 035 328;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 766 360 035 328 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 532 720 070 656;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 532 720 070 656 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 065 440 141 312;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 065 440 141 312 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 130 880 282 624;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 130 880 282 624 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 261 760 565 248;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 261 760 565 248 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 523 521 130 496;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 523 521 130 496 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 047 042 260 992;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 047 042 260 992 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 094 084 521 984;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 094 084 521 984 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 188 169 043 968;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 188 169 043 968 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 376 338 087 936;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 376 338 087 936 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 752 676 175 872;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 752 676 175 872 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 569 505 352 351 744;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 569 505 352 351 744 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 139 010 704 703 488;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 139 010 704 703 488 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 278 021 409 406 976;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 278 021 409 406 976 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 556 042 818 813 952;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 556 042 818 813 952 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 112 085 637 627 904;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 112 085 637 627 904 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 224 171 275 255 808;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 224 171 275 255 808 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 448 342 550 511 616;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 448 342 550 511 616 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 896 685 101 023 232;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 896 685 101 023 232 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 793 370 202 046 464;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 793 370 202 046 464 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 699 586 740 404 092 928;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 699 586 740 404 092 928 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 399 173 480 808 185 856;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 399 173 480 808 185 856 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 798 346 961 616 371 712;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 798 346 961 616 371 712 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 596 693 923 232 743 424;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 596 693 923 232 743 424 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 193 387 846 465 486 848;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 193 387 846 465 486 848 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 386 775 692 930 973 696;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 386 775 692 930 973 696 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 773 551 385 861 947 392;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 773 551 385 861 947 392 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 547 102 771 723 894 784;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 547 102 771 723 894 784 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 563 094 205 543 447 789 568;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 563 094 205 543 447 789 568 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 126 188 411 086 895 579 136;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 126 188 411 086 895 579 136 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 252 376 822 173 791 158 272;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 252 376 822 173 791 158 272 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 504 753 644 347 582 316 544;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 504 753 644 347 582 316 544 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 009 507 288 695 164 633 088;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 009 507 288 695 164 633 088 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 019 014 577 390 329 266 176;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 019 014 577 390 329 266 176 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 038 029 154 780 658 532 352;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 038 029 154 780 658 532 352 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 076 058 309 561 317 064 704;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 076 058 309 561 317 064 704 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 152 116 619 122 634 129 408;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 152 116 619 122 634 129 408 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 304 233 238 245 268 258 816;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 304 233 238 245 268 258 816 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 608 466 476 490 536 517 632;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 608 466 476 490 536 517 632 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 001 216 932 952 981 073 035 264;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 001 216 932 952 981 073 035 264 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 002 433 865 905 962 146 070 528;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 002 433 865 905 962 146 070 528 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 004 867 731 811 924 292 141 056;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 004 867 731 811 924 292 141 056 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 009 735 463 623 848 584 282 112;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 009 735 463 623 848 584 282 112 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 019 470 927 247 697 168 564 224;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 019 470 927 247 697 168 564 224 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 038 941 854 495 394 337 128 448;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 038 941 854 495 394 337 128 448 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 077 883 708 990 788 674 256 896;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 077 883 708 990 788 674 256 896 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 155 767 417 981 577 348 513 792;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 155 767 417 981 577 348 513 792 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 311 534 835 963 154 697 027 584;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 311 534 835 963 154 697 027 584 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 623 069 671 926 309 394 055 168;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 623 069 671 926 309 394 055 168 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 001 246 139 343 852 618 788 110 336;
  • 75) 0.924 636 840 820 312 500 000 000 000 001 246 139 343 852 618 788 110 336 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 002 492 278 687 705 237 576 220 672;
  • 76) 0.849 273 681 640 625 000 000 000 000 002 492 278 687 705 237 576 220 672 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 004 984 557 375 410 475 152 441 344;
  • 77) 0.698 547 363 281 250 000 000 000 000 004 984 557 375 410 475 152 441 344 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 009 969 114 750 820 950 304 882 688;
  • 78) 0.397 094 726 562 500 000 000 000 000 009 969 114 750 820 950 304 882 688 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 019 938 229 501 641 900 609 765 376;
  • 79) 0.794 189 453 125 000 000 000 000 000 019 938 229 501 641 900 609 765 376 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 039 876 459 003 283 801 219 530 752;
  • 80) 0.588 378 906 250 000 000 000 000 000 039 876 459 003 283 801 219 530 752 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 079 752 918 006 567 602 439 061 504;
  • 81) 0.176 757 812 500 000 000 000 000 000 079 752 918 006 567 602 439 061 504 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 159 505 836 013 135 204 878 123 008;
  • 82) 0.353 515 625 000 000 000 000 000 000 159 505 836 013 135 204 878 123 008 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 319 011 672 026 270 409 756 246 016;
  • 83) 0.707 031 250 000 000 000 000 000 000 319 011 672 026 270 409 756 246 016 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 638 023 344 052 540 819 512 492 032;
  • 84) 0.414 062 500 000 000 000 000 000 000 638 023 344 052 540 819 512 492 032 × 2 = 0 + 0.828 125 000 000 000 000 000 000 001 276 046 688 105 081 639 024 984 064;
  • 85) 0.828 125 000 000 000 000 000 000 001 276 046 688 105 081 639 024 984 064 × 2 = 1 + 0.656 250 000 000 000 000 000 000 002 552 093 376 210 163 278 049 968 128;
  • 86) 0.656 250 000 000 000 000 000 000 002 552 093 376 210 163 278 049 968 128 × 2 = 1 + 0.312 500 000 000 000 000 000 000 005 104 186 752 420 326 556 099 936 256;
  • 87) 0.312 500 000 000 000 000 000 000 005 104 186 752 420 326 556 099 936 256 × 2 = 0 + 0.625 000 000 000 000 000 000 000 010 208 373 504 840 653 112 199 872 512;
  • 88) 0.625 000 000 000 000 000 000 000 010 208 373 504 840 653 112 199 872 512 × 2 = 1 + 0.250 000 000 000 000 000 000 000 020 416 747 009 681 306 224 399 745 024;
  • 89) 0.250 000 000 000 000 000 000 000 020 416 747 009 681 306 224 399 745 024 × 2 = 0 + 0.500 000 000 000 000 000 000 000 040 833 494 019 362 612 448 799 490 048;
  • 90) 0.500 000 000 000 000 000 000 000 040 833 494 019 362 612 448 799 490 048 × 2 = 1 + 0.000 000 000 000 000 000 000 000 081 666 988 038 725 224 897 598 980 096;
  • 91) 0.000 000 000 000 000 000 000 000 081 666 988 038 725 224 897 598 980 096 × 2 = 0 + 0.000 000 000 000 000 000 000 000 163 333 976 077 450 449 795 197 960 192;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 054 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