0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 942;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 942 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 884;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 884 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 768;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 768 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 536;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 536 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 072;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 072 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 144;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 144 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 580 288;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 580 288 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 160 576;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 160 576 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 321 152;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 321 152 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 642 304;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 642 304 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 284 608;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 284 608 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 569 216;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 569 216 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 138 432;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 138 432 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 276 864;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 276 864 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 553 728;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 553 728 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 107 456;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 107 456 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 170 214 912;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 170 214 912 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 340 429 824;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 340 429 824 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 680 859 648;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 680 859 648 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 361 719 296;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 361 719 296 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 723 438 592;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 723 438 592 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 446 877 184;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 446 877 184 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 893 754 368;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 893 754 368 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 787 508 736;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 787 508 736 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 575 017 472;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 575 017 472 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 150 034 944;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 150 034 944 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 054 300 069 888;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 054 300 069 888 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 108 600 139 776;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 108 600 139 776 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 217 200 279 552;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 217 200 279 552 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 434 400 559 104;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 434 400 559 104 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 868 801 118 208;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 868 801 118 208 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 737 602 236 416;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 737 602 236 416 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 475 204 472 832;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 475 204 472 832 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 950 408 945 664;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 950 408 945 664 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 900 817 891 328;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 900 817 891 328 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 801 635 782 656;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 801 635 782 656 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 127 603 271 565 312;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 127 603 271 565 312 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 255 206 543 130 624;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 255 206 543 130 624 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 510 413 086 261 248;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 510 413 086 261 248 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 020 826 172 522 496;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 020 826 172 522 496 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 041 652 345 044 992;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 041 652 345 044 992 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 083 304 690 089 984;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 083 304 690 089 984 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 166 609 380 179 968;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 166 609 380 179 968 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 333 218 760 359 936;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 333 218 760 359 936 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 696 666 437 520 719 872;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 696 666 437 520 719 872 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 393 332 875 041 439 744;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 393 332 875 041 439 744 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 786 665 750 082 879 488;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 786 665 750 082 879 488 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 573 331 500 165 758 976;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 573 331 500 165 758 976 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 146 663 000 331 517 952;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 146 663 000 331 517 952 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 293 326 000 663 035 904;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 293 326 000 663 035 904 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 586 652 001 326 071 808;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 586 652 001 326 071 808 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 173 304 002 652 143 616;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 173 304 002 652 143 616 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 346 608 005 304 287 232;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 346 608 005 304 287 232 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 693 216 010 608 574 464;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 693 216 010 608 574 464 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 386 432 021 217 148 928;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 386 432 021 217 148 928 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 498 772 864 042 434 297 856;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 498 772 864 042 434 297 856 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 997 545 728 084 868 595 712;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 997 545 728 084 868 595 712 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 995 091 456 169 737 191 424;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 995 091 456 169 737 191 424 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 990 182 912 339 474 382 848;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 990 182 912 339 474 382 848 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 980 365 824 678 948 765 696;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 980 365 824 678 948 765 696 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 960 731 649 357 897 531 392;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 960 731 649 357 897 531 392 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 921 463 298 715 795 062 784;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 921 463 298 715 795 062 784 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 842 926 597 431 590 125 568;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 842 926 597 431 590 125 568 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 685 853 194 863 180 251 136;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 685 853 194 863 180 251 136 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 371 706 389 726 360 502 272;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 371 706 389 726 360 502 272 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 998 743 412 779 452 721 004 544;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 998 743 412 779 452 721 004 544 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 997 486 825 558 905 442 009 088;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 997 486 825 558 905 442 009 088 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 994 973 651 117 810 884 018 176;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 994 973 651 117 810 884 018 176 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 989 947 302 235 621 768 036 352;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 989 947 302 235 621 768 036 352 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 979 894 604 471 243 536 072 704;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 979 894 604 471 243 536 072 704 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 959 789 208 942 487 072 145 408;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 959 789 208 942 487 072 145 408 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 919 578 417 884 974 144 290 816;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 919 578 417 884 974 144 290 816 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 839 156 835 769 948 288 581 632;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 839 156 835 769 948 288 581 632 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 678 313 671 539 896 577 163 264;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 678 313 671 539 896 577 163 264 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 356 627 343 079 793 154 326 528;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 356 627 343 079 793 154 326 528 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 998 713 254 686 159 586 308 653 056;
  • 77) 0.698 547 363 281 249 999 999 999 999 998 713 254 686 159 586 308 653 056 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 997 426 509 372 319 172 617 306 112;
  • 78) 0.397 094 726 562 499 999 999 999 999 997 426 509 372 319 172 617 306 112 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 994 853 018 744 638 345 234 612 224;
  • 79) 0.794 189 453 124 999 999 999 999 999 994 853 018 744 638 345 234 612 224 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 989 706 037 489 276 690 469 224 448;
  • 80) 0.588 378 906 249 999 999 999 999 999 989 706 037 489 276 690 469 224 448 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 979 412 074 978 553 380 938 448 896;
  • 81) 0.176 757 812 499 999 999 999 999 999 979 412 074 978 553 380 938 448 896 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 958 824 149 957 106 761 876 897 792;
  • 82) 0.353 515 624 999 999 999 999 999 999 958 824 149 957 106 761 876 897 792 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 917 648 299 914 213 523 753 795 584;
  • 83) 0.707 031 249 999 999 999 999 999 999 917 648 299 914 213 523 753 795 584 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 835 296 599 828 427 047 507 591 168;
  • 84) 0.414 062 499 999 999 999 999 999 999 835 296 599 828 427 047 507 591 168 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 670 593 199 656 854 095 015 182 336;
  • 85) 0.828 124 999 999 999 999 999 999 999 670 593 199 656 854 095 015 182 336 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 341 186 399 313 708 190 030 364 672;
  • 86) 0.656 249 999 999 999 999 999 999 999 341 186 399 313 708 190 030 364 672 × 2 = 1 + 0.312 499 999 999 999 999 999 999 998 682 372 798 627 416 380 060 729 344;
  • 87) 0.312 499 999 999 999 999 999 999 998 682 372 798 627 416 380 060 729 344 × 2 = 0 + 0.624 999 999 999 999 999 999 999 997 364 745 597 254 832 760 121 458 688;
  • 88) 0.624 999 999 999 999 999 999 999 997 364 745 597 254 832 760 121 458 688 × 2 = 1 + 0.249 999 999 999 999 999 999 999 994 729 491 194 509 665 520 242 917 376;
  • 89) 0.249 999 999 999 999 999 999 999 994 729 491 194 509 665 520 242 917 376 × 2 = 0 + 0.499 999 999 999 999 999 999 999 989 458 982 389 019 331 040 485 834 752;
  • 90) 0.499 999 999 999 999 999 999 999 989 458 982 389 019 331 040 485 834 752 × 2 = 0 + 0.999 999 999 999 999 999 999 999 978 917 964 778 038 662 080 971 669 504;
  • 91) 0.999 999 999 999 999 999 999 999 978 917 964 778 038 662 080 971 669 504 × 2 = 1 + 0.999 999 999 999 999 999 999 999 957 835 929 556 077 324 161 943 339 008;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 971 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