0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 28 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 28(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 28(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 28.

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 28 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 56;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 56 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 609 12;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 609 12 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 218 24;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 218 24 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 436 48;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 436 48 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 872 96;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 872 96 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 745 92;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 745 92 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 491 84;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 491 84 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 983 68;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 983 68 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 967 36;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 967 36 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 934 72;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 934 72 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 869 44;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 869 44 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 738 88;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 738 88 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 567 477 76;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 567 477 76 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 134 955 52;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 134 955 52 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 269 911 04;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 269 911 04 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 539 822 08;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 539 822 08 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 079 644 16;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 079 644 16 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 159 288 32;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 159 288 32 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 318 576 64;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 318 576 64 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 637 153 28;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 637 153 28 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 274 306 56;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 274 306 56 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 548 613 12;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 548 613 12 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 685 097 226 24;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 685 097 226 24 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 370 194 452 48;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 370 194 452 48 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 740 388 904 96;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 740 388 904 96 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 480 777 809 92;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 480 777 809 92 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 961 555 619 84;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 961 555 619 84 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 923 111 239 68;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 923 111 239 68 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 846 222 479 36;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 846 222 479 36 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 692 444 958 72;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 692 444 958 72 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 384 889 917 44;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 384 889 917 44 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 769 779 834 88;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 769 779 834 88 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 189 539 559 669 76;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 189 539 559 669 76 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 379 079 119 339 52;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 379 079 119 339 52 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 758 158 238 679 04;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 758 158 238 679 04 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 516 316 477 358 08;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 516 316 477 358 08 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 032 632 954 716 16;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 032 632 954 716 16 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 065 265 909 432 32;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 065 265 909 432 32 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 130 531 818 864 64;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 130 531 818 864 64 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 261 063 637 729 28;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 261 063 637 729 28 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 104 522 127 275 458 56;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 104 522 127 275 458 56 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 209 044 254 550 917 12;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 209 044 254 550 917 12 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 418 088 509 101 834 24;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 418 088 509 101 834 24 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 836 177 018 203 668 48;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 836 177 018 203 668 48 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 672 354 036 407 336 96;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 672 354 036 407 336 96 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 344 708 072 814 673 92;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 344 708 072 814 673 92 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 689 416 145 629 347 84;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 689 416 145 629 347 84 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 378 832 291 258 695 68;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 378 832 291 258 695 68 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 757 664 582 517 391 36;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 757 664 582 517 391 36 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 515 329 165 034 782 72;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 515 329 165 034 782 72 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 139 030 658 330 069 565 44;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 139 030 658 330 069 565 44 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 278 061 316 660 139 130 88;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 278 061 316 660 139 130 88 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 556 122 633 320 278 261 76;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 556 122 633 320 278 261 76 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 112 245 266 640 556 523 52;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 112 245 266 640 556 523 52 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 224 490 533 281 113 047 04;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 224 490 533 281 113 047 04 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 448 981 066 562 226 094 08;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 448 981 066 562 226 094 08 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 897 962 133 124 452 188 16;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 897 962 133 124 452 188 16 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 795 924 266 248 904 376 32;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 795 924 266 248 904 376 32 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 591 848 532 497 808 752 64;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 591 848 532 497 808 752 64 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 183 697 064 995 617 505 28;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 183 697 064 995 617 505 28 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 998 367 394 129 991 235 010 56;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 998 367 394 129 991 235 010 56 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 996 734 788 259 982 470 021 12;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 996 734 788 259 982 470 021 12 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 993 469 576 519 964 940 042 24;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 993 469 576 519 964 940 042 24 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 986 939 153 039 929 880 084 48;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 986 939 153 039 929 880 084 48 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 973 878 306 079 859 760 168 96;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 973 878 306 079 859 760 168 96 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 947 756 612 159 719 520 337 92;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 947 756 612 159 719 520 337 92 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 895 513 224 319 439 040 675 84;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 895 513 224 319 439 040 675 84 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 791 026 448 638 878 081 351 68;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 791 026 448 638 878 081 351 68 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 582 052 897 277 756 162 703 36;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 582 052 897 277 756 162 703 36 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 164 105 794 555 512 325 406 72;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 164 105 794 555 512 325 406 72 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 998 328 211 589 111 024 650 813 44;
  • 72) 0.615 579 605 102 539 062 499 999 999 998 328 211 589 111 024 650 813 44 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 996 656 423 178 222 049 301 626 88;
  • 73) 0.231 159 210 205 078 124 999 999 999 996 656 423 178 222 049 301 626 88 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 993 312 846 356 444 098 603 253 76;
  • 74) 0.462 318 420 410 156 249 999 999 999 993 312 846 356 444 098 603 253 76 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 986 625 692 712 888 197 206 507 52;
  • 75) 0.924 636 840 820 312 499 999 999 999 986 625 692 712 888 197 206 507 52 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 973 251 385 425 776 394 413 015 04;
  • 76) 0.849 273 681 640 624 999 999 999 999 973 251 385 425 776 394 413 015 04 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 946 502 770 851 552 788 826 030 08;
  • 77) 0.698 547 363 281 249 999 999 999 999 946 502 770 851 552 788 826 030 08 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 893 005 541 703 105 577 652 060 16;
  • 78) 0.397 094 726 562 499 999 999 999 999 893 005 541 703 105 577 652 060 16 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 786 011 083 406 211 155 304 120 32;
  • 79) 0.794 189 453 124 999 999 999 999 999 786 011 083 406 211 155 304 120 32 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 572 022 166 812 422 310 608 240 64;
  • 80) 0.588 378 906 249 999 999 999 999 999 572 022 166 812 422 310 608 240 64 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 144 044 333 624 844 621 216 481 28;
  • 81) 0.176 757 812 499 999 999 999 999 999 144 044 333 624 844 621 216 481 28 × 2 = 0 + 0.353 515 624 999 999 999 999 999 998 288 088 667 249 689 242 432 962 56;
  • 82) 0.353 515 624 999 999 999 999 999 998 288 088 667 249 689 242 432 962 56 × 2 = 0 + 0.707 031 249 999 999 999 999 999 996 576 177 334 499 378 484 865 925 12;
  • 83) 0.707 031 249 999 999 999 999 999 996 576 177 334 499 378 484 865 925 12 × 2 = 1 + 0.414 062 499 999 999 999 999 999 993 152 354 668 998 756 969 731 850 24;
  • 84) 0.414 062 499 999 999 999 999 999 993 152 354 668 998 756 969 731 850 24 × 2 = 0 + 0.828 124 999 999 999 999 999 999 986 304 709 337 997 513 939 463 700 48;
  • 85) 0.828 124 999 999 999 999 999 999 986 304 709 337 997 513 939 463 700 48 × 2 = 1 + 0.656 249 999 999 999 999 999 999 972 609 418 675 995 027 878 927 400 96;
  • 86) 0.656 249 999 999 999 999 999 999 972 609 418 675 995 027 878 927 400 96 × 2 = 1 + 0.312 499 999 999 999 999 999 999 945 218 837 351 990 055 757 854 801 92;
  • 87) 0.312 499 999 999 999 999 999 999 945 218 837 351 990 055 757 854 801 92 × 2 = 0 + 0.624 999 999 999 999 999 999 999 890 437 674 703 980 111 515 709 603 84;
  • 88) 0.624 999 999 999 999 999 999 999 890 437 674 703 980 111 515 709 603 84 × 2 = 1 + 0.249 999 999 999 999 999 999 999 780 875 349 407 960 223 031 419 207 68;
  • 89) 0.249 999 999 999 999 999 999 999 780 875 349 407 960 223 031 419 207 68 × 2 = 0 + 0.499 999 999 999 999 999 999 999 561 750 698 815 920 446 062 838 415 36;
  • 90) 0.499 999 999 999 999 999 999 999 561 750 698 815 920 446 062 838 415 36 × 2 = 0 + 0.999 999 999 999 999 999 999 999 123 501 397 631 840 892 125 676 830 72;
  • 91) 0.999 999 999 999 999 999 999 999 123 501 397 631 840 892 125 676 830 72 × 2 = 1 + 0.999 999 999 999 999 999 999 998 247 002 795 263 681 784 251 353 661 44;

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 28(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 28(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 28(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 28 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