0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 52;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 52 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 615 04;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 615 04 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 230 08;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 230 08 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 460 16;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 460 16 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 920 32;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 920 32 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 840 64;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 840 64 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 681 28;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 681 28 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 362 56;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 362 56 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 725 12;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 725 12 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 450 24;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 450 24 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 900 48;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 900 48 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 789 800 96;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 789 800 96 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 579 601 92;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 579 601 92 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 159 203 84;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 159 203 84 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 318 407 68;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 318 407 68 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 636 815 36;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 636 815 36 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 273 630 72;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 273 630 72 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 547 261 44;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 547 261 44 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 094 522 88;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 094 522 88 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 189 045 76;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 189 045 76 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 378 091 52;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 378 091 52 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 848 756 183 04;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 848 756 183 04 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 697 512 366 08;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 697 512 366 08 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 395 024 732 16;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 395 024 732 16 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 790 049 464 32;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 790 049 464 32 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 580 098 928 64;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 580 098 928 64 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 160 197 857 28;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 160 197 857 28 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 320 395 714 56;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 320 395 714 56 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 640 791 429 12;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 640 791 429 12 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 281 582 858 24;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 281 582 858 24 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 563 165 716 48;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 563 165 716 48 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 101 126 331 432 96;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 101 126 331 432 96 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 202 252 662 865 92;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 202 252 662 865 92 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 404 505 325 731 84;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 404 505 325 731 84 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 809 010 651 463 68;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 809 010 651 463 68 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 618 021 302 927 36;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 618 021 302 927 36 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 236 042 605 854 72;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 236 042 605 854 72 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 472 085 211 709 44;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 472 085 211 709 44 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 944 170 423 418 88;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 944 170 423 418 88 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 888 340 846 837 76;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 888 340 846 837 76 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 107 776 681 693 675 52;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 107 776 681 693 675 52 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 215 553 363 387 351 04;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 215 553 363 387 351 04 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 431 106 726 774 702 08;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 431 106 726 774 702 08 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 862 213 453 549 404 16;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 862 213 453 549 404 16 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 724 426 907 098 808 32;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 724 426 907 098 808 32 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 448 853 814 197 616 64;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 448 853 814 197 616 64 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 897 707 628 395 233 28;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 897 707 628 395 233 28 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 795 415 256 790 466 56;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 795 415 256 790 466 56 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 590 830 513 580 933 12;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 590 830 513 580 933 12 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 071 181 661 027 161 866 24;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 071 181 661 027 161 866 24 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 142 363 322 054 323 732 48;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 142 363 322 054 323 732 48 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 284 726 644 108 647 464 96;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 284 726 644 108 647 464 96 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 569 453 288 217 294 929 92;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 569 453 288 217 294 929 92 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 138 906 576 434 589 859 84;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 138 906 576 434 589 859 84 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 277 813 152 869 179 719 68;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 277 813 152 869 179 719 68 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 555 626 305 738 359 439 36;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 555 626 305 738 359 439 36 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 111 252 611 476 718 878 72;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 111 252 611 476 718 878 72 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 222 505 222 953 437 757 44;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 222 505 222 953 437 757 44 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 445 010 445 906 875 514 88;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 445 010 445 906 875 514 88 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 890 020 891 813 751 029 76;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 890 020 891 813 751 029 76 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 001 780 041 783 627 502 059 52;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 001 780 041 783 627 502 059 52 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 003 560 083 567 255 004 119 04;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 003 560 083 567 255 004 119 04 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 007 120 167 134 510 008 238 08;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 007 120 167 134 510 008 238 08 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 014 240 334 269 020 016 476 16;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 014 240 334 269 020 016 476 16 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 028 480 668 538 040 032 952 32;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 028 480 668 538 040 032 952 32 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 056 961 337 076 080 065 904 64;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 056 961 337 076 080 065 904 64 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 113 922 674 152 160 131 809 28;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 113 922 674 152 160 131 809 28 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 227 845 348 304 320 263 618 56;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 227 845 348 304 320 263 618 56 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 455 690 696 608 640 527 237 12;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 455 690 696 608 640 527 237 12 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 911 381 393 217 281 054 474 24;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 911 381 393 217 281 054 474 24 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 001 822 762 786 434 562 108 948 48;
  • 72) 0.615 579 605 102 539 062 500 000 000 001 822 762 786 434 562 108 948 48 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 003 645 525 572 869 124 217 896 96;
  • 73) 0.231 159 210 205 078 125 000 000 000 003 645 525 572 869 124 217 896 96 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 007 291 051 145 738 248 435 793 92;
  • 74) 0.462 318 420 410 156 250 000 000 000 007 291 051 145 738 248 435 793 92 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 014 582 102 291 476 496 871 587 84;
  • 75) 0.924 636 840 820 312 500 000 000 000 014 582 102 291 476 496 871 587 84 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 029 164 204 582 952 993 743 175 68;
  • 76) 0.849 273 681 640 625 000 000 000 000 029 164 204 582 952 993 743 175 68 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 058 328 409 165 905 987 486 351 36;
  • 77) 0.698 547 363 281 250 000 000 000 000 058 328 409 165 905 987 486 351 36 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 116 656 818 331 811 974 972 702 72;
  • 78) 0.397 094 726 562 500 000 000 000 000 116 656 818 331 811 974 972 702 72 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 233 313 636 663 623 949 945 405 44;
  • 79) 0.794 189 453 125 000 000 000 000 000 233 313 636 663 623 949 945 405 44 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 466 627 273 327 247 899 890 810 88;
  • 80) 0.588 378 906 250 000 000 000 000 000 466 627 273 327 247 899 890 810 88 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 933 254 546 654 495 799 781 621 76;
  • 81) 0.176 757 812 500 000 000 000 000 000 933 254 546 654 495 799 781 621 76 × 2 = 0 + 0.353 515 625 000 000 000 000 000 001 866 509 093 308 991 599 563 243 52;
  • 82) 0.353 515 625 000 000 000 000 000 001 866 509 093 308 991 599 563 243 52 × 2 = 0 + 0.707 031 250 000 000 000 000 000 003 733 018 186 617 983 199 126 487 04;
  • 83) 0.707 031 250 000 000 000 000 000 003 733 018 186 617 983 199 126 487 04 × 2 = 1 + 0.414 062 500 000 000 000 000 000 007 466 036 373 235 966 398 252 974 08;
  • 84) 0.414 062 500 000 000 000 000 000 007 466 036 373 235 966 398 252 974 08 × 2 = 0 + 0.828 125 000 000 000 000 000 000 014 932 072 746 471 932 796 505 948 16;
  • 85) 0.828 125 000 000 000 000 000 000 014 932 072 746 471 932 796 505 948 16 × 2 = 1 + 0.656 250 000 000 000 000 000 000 029 864 145 492 943 865 593 011 896 32;
  • 86) 0.656 250 000 000 000 000 000 000 029 864 145 492 943 865 593 011 896 32 × 2 = 1 + 0.312 500 000 000 000 000 000 000 059 728 290 985 887 731 186 023 792 64;
  • 87) 0.312 500 000 000 000 000 000 000 059 728 290 985 887 731 186 023 792 64 × 2 = 0 + 0.625 000 000 000 000 000 000 000 119 456 581 971 775 462 372 047 585 28;
  • 88) 0.625 000 000 000 000 000 000 000 119 456 581 971 775 462 372 047 585 28 × 2 = 1 + 0.250 000 000 000 000 000 000 000 238 913 163 943 550 924 744 095 170 56;
  • 89) 0.250 000 000 000 000 000 000 000 238 913 163 943 550 924 744 095 170 56 × 2 = 0 + 0.500 000 000 000 000 000 000 000 477 826 327 887 101 849 488 190 341 12;
  • 90) 0.500 000 000 000 000 000 000 000 477 826 327 887 101 849 488 190 341 12 × 2 = 1 + 0.000 000 000 000 000 000 000 000 955 652 655 774 203 698 976 380 682 24;
  • 91) 0.000 000 000 000 000 000 000 000 955 652 655 774 203 698 976 380 682 24 × 2 = 0 + 0.000 000 000 000 000 000 000 001 911 305 311 548 407 397 952 761 364 48;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 76 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