0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 308 46;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 308 46 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 616 92;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 616 92 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 233 84;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 233 84 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 467 68;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 467 68 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 935 36;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 935 36 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 870 72;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 870 72 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 741 44;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 741 44 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 482 88;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 482 88 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 965 76;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 965 76 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 931 52;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 931 52 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 395 863 04;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 395 863 04 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 791 726 08;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 791 726 08 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 583 452 16;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 583 452 16 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 166 904 32;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 166 904 32 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 333 808 64;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 333 808 64 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 667 617 28;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 667 617 28 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 335 234 56;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 335 234 56 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 670 469 12;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 670 469 12 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 340 938 24;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 340 938 24 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 681 876 48;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 681 876 48 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 925 363 752 96;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 925 363 752 96 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 850 727 505 92;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 850 727 505 92 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 701 455 011 84;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 701 455 011 84 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 402 910 023 68;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 402 910 023 68 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 805 820 047 36;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 805 820 047 36 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 611 640 094 72;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 611 640 094 72 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 223 280 189 44;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 223 280 189 44 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 446 560 378 88;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 446 560 378 88 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 893 120 757 76;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 893 120 757 76 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 786 241 515 52;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 786 241 515 52 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 051 572 483 031 04;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 051 572 483 031 04 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 103 144 966 062 08;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 103 144 966 062 08 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 206 289 932 124 16;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 206 289 932 124 16 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 412 579 864 248 32;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 412 579 864 248 32 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 825 159 728 496 64;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 825 159 728 496 64 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 650 319 456 993 28;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 650 319 456 993 28 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 300 638 913 986 56;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 300 638 913 986 56 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 601 277 827 973 12;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 601 277 827 973 12 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 277 202 555 655 946 24;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 277 202 555 655 946 24 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 554 405 111 311 892 48;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 554 405 111 311 892 48 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 108 810 222 623 784 96;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 108 810 222 623 784 96 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 217 620 445 247 569 92;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 217 620 445 247 569 92 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 435 240 890 495 139 84;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 435 240 890 495 139 84 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 870 481 780 990 279 68;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 870 481 780 990 279 68 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 740 963 561 980 559 36;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 740 963 561 980 559 36 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 481 927 123 961 118 72;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 481 927 123 961 118 72 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 963 854 247 922 237 44;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 963 854 247 922 237 44 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 927 708 495 844 474 88;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 927 708 495 844 474 88 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 855 416 991 688 949 76;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 855 416 991 688 949 76 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 071 710 833 983 377 899 52;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 071 710 833 983 377 899 52 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 143 421 667 966 755 799 04;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 143 421 667 966 755 799 04 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 286 843 335 933 511 598 08;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 286 843 335 933 511 598 08 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 573 686 671 867 023 196 16;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 573 686 671 867 023 196 16 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 147 373 343 734 046 392 32;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 147 373 343 734 046 392 32 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 294 746 687 468 092 784 64;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 294 746 687 468 092 784 64 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 589 493 374 936 185 569 28;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 589 493 374 936 185 569 28 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 178 986 749 872 371 138 56;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 178 986 749 872 371 138 56 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 357 973 499 744 742 277 12;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 357 973 499 744 742 277 12 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 715 946 999 489 484 554 24;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 715 946 999 489 484 554 24 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 001 431 893 998 978 969 108 48;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 001 431 893 998 978 969 108 48 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 002 863 787 997 957 938 216 96;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 002 863 787 997 957 938 216 96 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 005 727 575 995 915 876 433 92;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 005 727 575 995 915 876 433 92 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 011 455 151 991 831 752 867 84;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 011 455 151 991 831 752 867 84 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 022 910 303 983 663 505 735 68;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 022 910 303 983 663 505 735 68 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 045 820 607 967 327 011 471 36;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 045 820 607 967 327 011 471 36 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 091 641 215 934 654 022 942 72;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 091 641 215 934 654 022 942 72 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 183 282 431 869 308 045 885 44;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 183 282 431 869 308 045 885 44 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 366 564 863 738 616 091 770 88;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 366 564 863 738 616 091 770 88 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 733 129 727 477 232 183 541 76;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 733 129 727 477 232 183 541 76 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 001 466 259 454 954 464 367 083 52;
  • 71) 0.307 789 802 551 269 531 250 000 000 001 466 259 454 954 464 367 083 52 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 002 932 518 909 908 928 734 167 04;
  • 72) 0.615 579 605 102 539 062 500 000 000 002 932 518 909 908 928 734 167 04 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 005 865 037 819 817 857 468 334 08;
  • 73) 0.231 159 210 205 078 125 000 000 000 005 865 037 819 817 857 468 334 08 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 011 730 075 639 635 714 936 668 16;
  • 74) 0.462 318 420 410 156 250 000 000 000 011 730 075 639 635 714 936 668 16 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 023 460 151 279 271 429 873 336 32;
  • 75) 0.924 636 840 820 312 500 000 000 000 023 460 151 279 271 429 873 336 32 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 046 920 302 558 542 859 746 672 64;
  • 76) 0.849 273 681 640 625 000 000 000 000 046 920 302 558 542 859 746 672 64 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 093 840 605 117 085 719 493 345 28;
  • 77) 0.698 547 363 281 250 000 000 000 000 093 840 605 117 085 719 493 345 28 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 187 681 210 234 171 438 986 690 56;
  • 78) 0.397 094 726 562 500 000 000 000 000 187 681 210 234 171 438 986 690 56 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 375 362 420 468 342 877 973 381 12;
  • 79) 0.794 189 453 125 000 000 000 000 000 375 362 420 468 342 877 973 381 12 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 750 724 840 936 685 755 946 762 24;
  • 80) 0.588 378 906 250 000 000 000 000 000 750 724 840 936 685 755 946 762 24 × 2 = 1 + 0.176 757 812 500 000 000 000 000 001 501 449 681 873 371 511 893 524 48;
  • 81) 0.176 757 812 500 000 000 000 000 001 501 449 681 873 371 511 893 524 48 × 2 = 0 + 0.353 515 625 000 000 000 000 000 003 002 899 363 746 743 023 787 048 96;
  • 82) 0.353 515 625 000 000 000 000 000 003 002 899 363 746 743 023 787 048 96 × 2 = 0 + 0.707 031 250 000 000 000 000 000 006 005 798 727 493 486 047 574 097 92;
  • 83) 0.707 031 250 000 000 000 000 000 006 005 798 727 493 486 047 574 097 92 × 2 = 1 + 0.414 062 500 000 000 000 000 000 012 011 597 454 986 972 095 148 195 84;
  • 84) 0.414 062 500 000 000 000 000 000 012 011 597 454 986 972 095 148 195 84 × 2 = 0 + 0.828 125 000 000 000 000 000 000 024 023 194 909 973 944 190 296 391 68;
  • 85) 0.828 125 000 000 000 000 000 000 024 023 194 909 973 944 190 296 391 68 × 2 = 1 + 0.656 250 000 000 000 000 000 000 048 046 389 819 947 888 380 592 783 36;
  • 86) 0.656 250 000 000 000 000 000 000 048 046 389 819 947 888 380 592 783 36 × 2 = 1 + 0.312 500 000 000 000 000 000 000 096 092 779 639 895 776 761 185 566 72;
  • 87) 0.312 500 000 000 000 000 000 000 096 092 779 639 895 776 761 185 566 72 × 2 = 0 + 0.625 000 000 000 000 000 000 000 192 185 559 279 791 553 522 371 133 44;
  • 88) 0.625 000 000 000 000 000 000 000 192 185 559 279 791 553 522 371 133 44 × 2 = 1 + 0.250 000 000 000 000 000 000 000 384 371 118 559 583 107 044 742 266 88;
  • 89) 0.250 000 000 000 000 000 000 000 384 371 118 559 583 107 044 742 266 88 × 2 = 0 + 0.500 000 000 000 000 000 000 000 768 742 237 119 166 214 089 484 533 76;
  • 90) 0.500 000 000 000 000 000 000 000 768 742 237 119 166 214 089 484 533 76 × 2 = 1 + 0.000 000 000 000 000 000 000 001 537 484 474 238 332 428 178 969 067 52;
  • 91) 0.000 000 000 000 000 000 000 001 537 484 474 238 332 428 178 969 067 52 × 2 = 0 + 0.000 000 000 000 000 000 000 003 074 968 948 476 664 856 357 938 135 04;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 154 23 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