0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 11 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 11(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 11(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 11.

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 11 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 22;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 22 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 44;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 44 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 216 88;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 216 88 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 433 76;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 433 76 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 867 52;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 867 52 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 735 04;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 735 04 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 470 08;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 470 08 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 940 16;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 940 16 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 880 32;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 880 32 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 760 64;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 760 64 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 521 28;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 521 28 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 042 56;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 042 56 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 566 085 12;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 566 085 12 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 132 170 24;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 132 170 24 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 264 340 48;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 264 340 48 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 528 680 96;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 528 680 96 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 057 361 92;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 057 361 92 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 114 723 84;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 114 723 84 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 229 447 68;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 229 447 68 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 458 895 36;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 458 895 36 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 917 790 72;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 917 790 72 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 841 835 581 44;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 841 835 581 44 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 683 671 162 88;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 683 671 162 88 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 367 342 325 76;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 367 342 325 76 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 734 684 651 52;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 734 684 651 52 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 469 369 303 04;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 469 369 303 04 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 938 738 606 08;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 938 738 606 08 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 877 477 212 16;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 877 477 212 16 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 754 954 424 32;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 754 954 424 32 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 509 908 848 64;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 509 908 848 64 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 019 817 697 28;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 019 817 697 28 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 039 635 394 56;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 039 635 394 56 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 188 079 270 789 12;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 188 079 270 789 12 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 376 158 541 578 24;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 376 158 541 578 24 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 752 317 083 156 48;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 752 317 083 156 48 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 504 634 166 312 96;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 504 634 166 312 96 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 009 268 332 625 92;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 009 268 332 625 92 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 018 536 665 251 84;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 018 536 665 251 84 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 037 073 330 503 68;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 037 073 330 503 68 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 074 146 661 007 36;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 074 146 661 007 36 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 104 148 293 322 014 72;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 104 148 293 322 014 72 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 208 296 586 644 029 44;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 208 296 586 644 029 44 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 416 593 173 288 058 88;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 416 593 173 288 058 88 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 833 186 346 576 117 76;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 833 186 346 576 117 76 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 666 372 693 152 235 52;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 666 372 693 152 235 52 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 332 745 386 304 471 04;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 332 745 386 304 471 04 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 665 490 772 608 942 08;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 665 490 772 608 942 08 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 330 981 545 217 884 16;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 330 981 545 217 884 16 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 661 963 090 435 768 32;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 661 963 090 435 768 32 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 323 926 180 871 536 64;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 323 926 180 871 536 64 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 138 647 852 361 743 073 28;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 138 647 852 361 743 073 28 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 277 295 704 723 486 146 56;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 277 295 704 723 486 146 56 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 554 591 409 446 972 293 12;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 554 591 409 446 972 293 12 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 109 182 818 893 944 586 24;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 109 182 818 893 944 586 24 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 218 365 637 787 889 172 48;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 218 365 637 787 889 172 48 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 436 731 275 575 778 344 96;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 436 731 275 575 778 344 96 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 873 462 551 151 556 689 92;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 873 462 551 151 556 689 92 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 746 925 102 303 113 379 84;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 746 925 102 303 113 379 84 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 493 850 204 606 226 759 68;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 493 850 204 606 226 759 68 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 998 987 700 409 212 453 519 36;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 998 987 700 409 212 453 519 36 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 997 975 400 818 424 907 038 72;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 997 975 400 818 424 907 038 72 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 995 950 801 636 849 814 077 44;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 995 950 801 636 849 814 077 44 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 991 901 603 273 699 628 154 88;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 991 901 603 273 699 628 154 88 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 983 803 206 547 399 256 309 76;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 983 803 206 547 399 256 309 76 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 967 606 413 094 798 512 619 52;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 967 606 413 094 798 512 619 52 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 935 212 826 189 597 025 239 04;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 935 212 826 189 597 025 239 04 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 870 425 652 379 194 050 478 08;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 870 425 652 379 194 050 478 08 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 740 851 304 758 388 100 956 16;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 740 851 304 758 388 100 956 16 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 481 702 609 516 776 201 912 32;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 481 702 609 516 776 201 912 32 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 998 963 405 219 033 552 403 824 64;
  • 71) 0.307 789 802 551 269 531 249 999 999 998 963 405 219 033 552 403 824 64 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 997 926 810 438 067 104 807 649 28;
  • 72) 0.615 579 605 102 539 062 499 999 999 997 926 810 438 067 104 807 649 28 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 995 853 620 876 134 209 615 298 56;
  • 73) 0.231 159 210 205 078 124 999 999 999 995 853 620 876 134 209 615 298 56 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 991 707 241 752 268 419 230 597 12;
  • 74) 0.462 318 420 410 156 249 999 999 999 991 707 241 752 268 419 230 597 12 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 983 414 483 504 536 838 461 194 24;
  • 75) 0.924 636 840 820 312 499 999 999 999 983 414 483 504 536 838 461 194 24 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 966 828 967 009 073 676 922 388 48;
  • 76) 0.849 273 681 640 624 999 999 999 999 966 828 967 009 073 676 922 388 48 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 933 657 934 018 147 353 844 776 96;
  • 77) 0.698 547 363 281 249 999 999 999 999 933 657 934 018 147 353 844 776 96 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 867 315 868 036 294 707 689 553 92;
  • 78) 0.397 094 726 562 499 999 999 999 999 867 315 868 036 294 707 689 553 92 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 734 631 736 072 589 415 379 107 84;
  • 79) 0.794 189 453 124 999 999 999 999 999 734 631 736 072 589 415 379 107 84 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 469 263 472 145 178 830 758 215 68;
  • 80) 0.588 378 906 249 999 999 999 999 999 469 263 472 145 178 830 758 215 68 × 2 = 1 + 0.176 757 812 499 999 999 999 999 998 938 526 944 290 357 661 516 431 36;
  • 81) 0.176 757 812 499 999 999 999 999 998 938 526 944 290 357 661 516 431 36 × 2 = 0 + 0.353 515 624 999 999 999 999 999 997 877 053 888 580 715 323 032 862 72;
  • 82) 0.353 515 624 999 999 999 999 999 997 877 053 888 580 715 323 032 862 72 × 2 = 0 + 0.707 031 249 999 999 999 999 999 995 754 107 777 161 430 646 065 725 44;
  • 83) 0.707 031 249 999 999 999 999 999 995 754 107 777 161 430 646 065 725 44 × 2 = 1 + 0.414 062 499 999 999 999 999 999 991 508 215 554 322 861 292 131 450 88;
  • 84) 0.414 062 499 999 999 999 999 999 991 508 215 554 322 861 292 131 450 88 × 2 = 0 + 0.828 124 999 999 999 999 999 999 983 016 431 108 645 722 584 262 901 76;
  • 85) 0.828 124 999 999 999 999 999 999 983 016 431 108 645 722 584 262 901 76 × 2 = 1 + 0.656 249 999 999 999 999 999 999 966 032 862 217 291 445 168 525 803 52;
  • 86) 0.656 249 999 999 999 999 999 999 966 032 862 217 291 445 168 525 803 52 × 2 = 1 + 0.312 499 999 999 999 999 999 999 932 065 724 434 582 890 337 051 607 04;
  • 87) 0.312 499 999 999 999 999 999 999 932 065 724 434 582 890 337 051 607 04 × 2 = 0 + 0.624 999 999 999 999 999 999 999 864 131 448 869 165 780 674 103 214 08;
  • 88) 0.624 999 999 999 999 999 999 999 864 131 448 869 165 780 674 103 214 08 × 2 = 1 + 0.249 999 999 999 999 999 999 999 728 262 897 738 331 561 348 206 428 16;
  • 89) 0.249 999 999 999 999 999 999 999 728 262 897 738 331 561 348 206 428 16 × 2 = 0 + 0.499 999 999 999 999 999 999 999 456 525 795 476 663 122 696 412 856 32;
  • 90) 0.499 999 999 999 999 999 999 999 456 525 795 476 663 122 696 412 856 32 × 2 = 0 + 0.999 999 999 999 999 999 999 998 913 051 590 953 326 245 392 825 712 64;
  • 91) 0.999 999 999 999 999 999 999 998 913 051 590 953 326 245 392 825 712 64 × 2 = 1 + 0.999 999 999 999 999 999 999 997 826 103 181 906 652 490 785 651 425 28;

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 11(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 11(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 11(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 11 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