0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 368 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 368 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 736 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 736 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 472 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 472 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 945 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 945 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 891 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 891 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 451 782 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 451 782 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 903 564 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 903 564 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 807 129 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 807 129 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 614 259 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 614 259 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 228 518 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 228 518 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 457 036 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 457 036 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 914 073 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 914 073 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 828 147 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 828 147 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 656 294 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 656 294 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 687 312 588 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 687 312 588 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 374 625 177 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 374 625 177 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 749 250 355 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 749 250 355 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 498 500 710 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 498 500 710 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 997 001 420 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 997 001 420 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 994 002 841 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 994 002 841 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 988 005 683 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 988 005 683 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 976 011 366 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 976 011 366 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 952 022 732 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 952 022 732 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 904 045 465 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 904 045 465 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 808 090 931 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 808 090 931 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 503 616 181 862 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 503 616 181 862 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 007 232 363 724 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 007 232 363 724 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 014 464 727 449 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 014 464 727 449 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 028 929 454 899 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 028 929 454 899 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 057 858 909 798 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 057 858 909 798 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 115 717 819 596 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 115 717 819 596 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 231 435 639 193 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 231 435 639 193 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 462 871 278 387 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 462 871 278 387 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 925 742 556 774 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 925 742 556 774 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 851 485 113 548 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 851 485 113 548 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 411 702 970 227 097 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 411 702 970 227 097 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 823 405 940 454 195 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 823 405 940 454 195 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 646 811 880 908 390 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 646 811 880 908 390 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 293 623 761 816 780 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 293 623 761 816 780 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 587 247 523 633 561 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 587 247 523 633 561 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 174 495 047 267 123 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 174 495 047 267 123 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 348 990 094 534 246 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 348 990 094 534 246 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 697 980 189 068 492 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 697 980 189 068 492 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 395 960 378 136 985 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 395 960 378 136 985 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 690 791 920 756 273 971 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 690 791 920 756 273 971 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 381 583 841 512 547 942 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 381 583 841 512 547 942 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 763 167 683 025 095 884 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 763 167 683 025 095 884 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 526 335 366 050 191 769 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 526 335 366 050 191 769 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 052 670 732 100 383 539 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 052 670 732 100 383 539 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 105 341 464 200 767 078 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 105 341 464 200 767 078 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 210 682 928 401 534 156 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 210 682 928 401 534 156 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 421 365 856 803 068 313 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 421 365 856 803 068 313 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 842 731 713 606 136 627 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 842 731 713 606 136 627 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 685 463 427 212 273 254 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 685 463 427 212 273 254 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 907 370 926 854 424 546 508 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 907 370 926 854 424 546 508 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 814 741 853 708 849 093 017 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 814 741 853 708 849 093 017 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 629 483 707 417 698 186 035 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 629 483 707 417 698 186 035 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 258 967 414 835 396 372 070 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 258 967 414 835 396 372 070 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 517 934 829 670 792 744 140 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 517 934 829 670 792 744 140 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 035 869 659 341 585 488 281 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 035 869 659 341 585 488 281 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 071 739 318 683 170 976 563 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 071 739 318 683 170 976 563 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 143 478 637 366 341 953 126 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 143 478 637 366 341 953 126 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 286 957 274 732 683 906 252 8;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 286 957 274 732 683 906 252 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 573 914 549 465 367 812 505 6;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 573 914 549 465 367 812 505 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 001 147 829 098 930 735 625 011 2;
  • 66) 0.415 868 431 329 727 172 851 562 500 001 147 829 098 930 735 625 011 2 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 002 295 658 197 861 471 250 022 4;
  • 67) 0.831 736 862 659 454 345 703 125 000 002 295 658 197 861 471 250 022 4 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 004 591 316 395 722 942 500 044 8;
  • 68) 0.663 473 725 318 908 691 406 250 000 004 591 316 395 722 942 500 044 8 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 009 182 632 791 445 885 000 089 6;
  • 69) 0.326 947 450 637 817 382 812 500 000 009 182 632 791 445 885 000 089 6 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 018 365 265 582 891 770 000 179 2;
  • 70) 0.653 894 901 275 634 765 625 000 000 018 365 265 582 891 770 000 179 2 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 036 730 531 165 783 540 000 358 4;
  • 71) 0.307 789 802 551 269 531 250 000 000 036 730 531 165 783 540 000 358 4 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 073 461 062 331 567 080 000 716 8;
  • 72) 0.615 579 605 102 539 062 500 000 000 073 461 062 331 567 080 000 716 8 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 146 922 124 663 134 160 001 433 6;
  • 73) 0.231 159 210 205 078 125 000 000 000 146 922 124 663 134 160 001 433 6 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 293 844 249 326 268 320 002 867 2;
  • 74) 0.462 318 420 410 156 250 000 000 000 293 844 249 326 268 320 002 867 2 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 587 688 498 652 536 640 005 734 4;
  • 75) 0.924 636 840 820 312 500 000 000 000 587 688 498 652 536 640 005 734 4 × 2 = 1 + 0.849 273 681 640 625 000 000 000 001 175 376 997 305 073 280 011 468 8;
  • 76) 0.849 273 681 640 625 000 000 000 001 175 376 997 305 073 280 011 468 8 × 2 = 1 + 0.698 547 363 281 250 000 000 000 002 350 753 994 610 146 560 022 937 6;
  • 77) 0.698 547 363 281 250 000 000 000 002 350 753 994 610 146 560 022 937 6 × 2 = 1 + 0.397 094 726 562 500 000 000 000 004 701 507 989 220 293 120 045 875 2;
  • 78) 0.397 094 726 562 500 000 000 000 004 701 507 989 220 293 120 045 875 2 × 2 = 0 + 0.794 189 453 125 000 000 000 000 009 403 015 978 440 586 240 091 750 4;
  • 79) 0.794 189 453 125 000 000 000 000 009 403 015 978 440 586 240 091 750 4 × 2 = 1 + 0.588 378 906 250 000 000 000 000 018 806 031 956 881 172 480 183 500 8;
  • 80) 0.588 378 906 250 000 000 000 000 018 806 031 956 881 172 480 183 500 8 × 2 = 1 + 0.176 757 812 500 000 000 000 000 037 612 063 913 762 344 960 367 001 6;
  • 81) 0.176 757 812 500 000 000 000 000 037 612 063 913 762 344 960 367 001 6 × 2 = 0 + 0.353 515 625 000 000 000 000 000 075 224 127 827 524 689 920 734 003 2;
  • 82) 0.353 515 625 000 000 000 000 000 075 224 127 827 524 689 920 734 003 2 × 2 = 0 + 0.707 031 250 000 000 000 000 000 150 448 255 655 049 379 841 468 006 4;
  • 83) 0.707 031 250 000 000 000 000 000 150 448 255 655 049 379 841 468 006 4 × 2 = 1 + 0.414 062 500 000 000 000 000 000 300 896 511 310 098 759 682 936 012 8;
  • 84) 0.414 062 500 000 000 000 000 000 300 896 511 310 098 759 682 936 012 8 × 2 = 0 + 0.828 125 000 000 000 000 000 000 601 793 022 620 197 519 365 872 025 6;
  • 85) 0.828 125 000 000 000 000 000 000 601 793 022 620 197 519 365 872 025 6 × 2 = 1 + 0.656 250 000 000 000 000 000 001 203 586 045 240 395 038 731 744 051 2;
  • 86) 0.656 250 000 000 000 000 000 001 203 586 045 240 395 038 731 744 051 2 × 2 = 1 + 0.312 500 000 000 000 000 000 002 407 172 090 480 790 077 463 488 102 4;
  • 87) 0.312 500 000 000 000 000 000 002 407 172 090 480 790 077 463 488 102 4 × 2 = 0 + 0.625 000 000 000 000 000 000 004 814 344 180 961 580 154 926 976 204 8;
  • 88) 0.625 000 000 000 000 000 000 004 814 344 180 961 580 154 926 976 204 8 × 2 = 1 + 0.250 000 000 000 000 000 000 009 628 688 361 923 160 309 853 952 409 6;
  • 89) 0.250 000 000 000 000 000 000 009 628 688 361 923 160 309 853 952 409 6 × 2 = 0 + 0.500 000 000 000 000 000 000 019 257 376 723 846 320 619 707 904 819 2;
  • 90) 0.500 000 000 000 000 000 000 019 257 376 723 846 320 619 707 904 819 2 × 2 = 1 + 0.000 000 000 000 000 000 000 038 514 753 447 692 641 239 415 809 638 4;
  • 91) 0.000 000 000 000 000 000 000 038 514 753 447 692 641 239 415 809 638 4 × 2 = 0 + 0.000 000 000 000 000 000 000 077 029 506 895 385 282 478 831 619 276 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 184 1 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