0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 286 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 286 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 573 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 573 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 147 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 147 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 294 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 294 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 588 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 588 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 177 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 177 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 355 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 355 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 796 710 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 796 710 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 593 420 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 593 420 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 186 841 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 186 841 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 373 683 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 373 683 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 747 366 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 747 366 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 494 732 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 494 732 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 989 465 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 989 465 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 978 931 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 978 931 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 957 862 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 957 862 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 915 724 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 743 915 724 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 487 831 449 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 487 831 449 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 975 662 899 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 975 662 899 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 951 325 798 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 951 325 798 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 902 651 596 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 902 651 596 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 805 303 193 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 805 303 193 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 610 606 387 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 610 606 387 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 221 212 774 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 221 212 774 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 442 425 548 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 442 425 548 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 884 851 097 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 884 851 097 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 769 702 195 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 769 702 195 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 003 539 404 390 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 003 539 404 390 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 007 078 808 780 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 007 078 808 780 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 014 157 617 561 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 014 157 617 561 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 028 315 235 123 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 028 315 235 123 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 056 630 470 246 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 056 630 470 246 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 113 260 940 492 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 113 260 940 492 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 226 521 880 985 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 226 521 880 985 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 453 043 761 971 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 453 043 761 971 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 906 087 523 942 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 906 087 523 942 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 817 812 175 047 884 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 817 812 175 047 884 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 635 624 350 095 769 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 635 624 350 095 769 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 271 248 700 191 539 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 271 248 700 191 539 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 542 497 400 383 078 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 542 497 400 383 078 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 084 994 800 766 156 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 084 994 800 766 156 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 169 989 601 532 313 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 169 989 601 532 313 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 339 979 203 064 627 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 339 979 203 064 627 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 679 958 406 129 254 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 679 958 406 129 254 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 359 916 812 258 508 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 359 916 812 258 508 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 719 833 624 517 017 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 719 833 624 517 017 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 757 439 667 249 034 035 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 757 439 667 249 034 035 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 514 879 334 498 068 070 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 514 879 334 498 068 070 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 029 758 668 996 136 140 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 029 758 668 996 136 140 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 059 517 337 992 272 281 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 059 517 337 992 272 281 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 119 034 675 984 544 563 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 119 034 675 984 544 563 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 238 069 351 969 089 126 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 238 069 351 969 089 126 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 476 138 703 938 178 252 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 476 138 703 938 178 252 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 952 277 407 876 356 505 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 952 277 407 876 356 505 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 904 554 815 752 713 011 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 904 554 815 752 713 011 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 809 109 631 505 426 022 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 809 109 631 505 426 022 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 623 618 219 263 010 852 044 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 623 618 219 263 010 852 044 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 247 236 438 526 021 704 089 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 247 236 438 526 021 704 089 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 494 472 877 052 043 408 179 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 494 472 877 052 043 408 179 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 988 945 754 104 086 816 358 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 988 945 754 104 086 816 358 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 977 891 508 208 173 632 716 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 977 891 508 208 173 632 716 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 955 783 016 416 347 265 433 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 955 783 016 416 347 265 433 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 911 566 032 832 694 530 867 2;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 911 566 032 832 694 530 867 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 823 132 065 665 389 061 734 4;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 823 132 065 665 389 061 734 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 646 264 131 330 778 123 468 8;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 646 264 131 330 778 123 468 8 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 292 528 262 661 556 246 937 6;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 292 528 262 661 556 246 937 6 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 998 585 056 525 323 112 493 875 2;
  • 68) 0.663 473 725 318 908 691 406 249 999 998 585 056 525 323 112 493 875 2 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 997 170 113 050 646 224 987 750 4;
  • 69) 0.326 947 450 637 817 382 812 499 999 997 170 113 050 646 224 987 750 4 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 994 340 226 101 292 449 975 500 8;
  • 70) 0.653 894 901 275 634 765 624 999 999 994 340 226 101 292 449 975 500 8 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 988 680 452 202 584 899 951 001 6;
  • 71) 0.307 789 802 551 269 531 249 999 999 988 680 452 202 584 899 951 001 6 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 977 360 904 405 169 799 902 003 2;
  • 72) 0.615 579 605 102 539 062 499 999 999 977 360 904 405 169 799 902 003 2 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 954 721 808 810 339 599 804 006 4;
  • 73) 0.231 159 210 205 078 124 999 999 999 954 721 808 810 339 599 804 006 4 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 909 443 617 620 679 199 608 012 8;
  • 74) 0.462 318 420 410 156 249 999 999 999 909 443 617 620 679 199 608 012 8 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 818 887 235 241 358 399 216 025 6;
  • 75) 0.924 636 840 820 312 499 999 999 999 818 887 235 241 358 399 216 025 6 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 637 774 470 482 716 798 432 051 2;
  • 76) 0.849 273 681 640 624 999 999 999 999 637 774 470 482 716 798 432 051 2 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 275 548 940 965 433 596 864 102 4;
  • 77) 0.698 547 363 281 249 999 999 999 999 275 548 940 965 433 596 864 102 4 × 2 = 1 + 0.397 094 726 562 499 999 999 999 998 551 097 881 930 867 193 728 204 8;
  • 78) 0.397 094 726 562 499 999 999 999 998 551 097 881 930 867 193 728 204 8 × 2 = 0 + 0.794 189 453 124 999 999 999 999 997 102 195 763 861 734 387 456 409 6;
  • 79) 0.794 189 453 124 999 999 999 999 997 102 195 763 861 734 387 456 409 6 × 2 = 1 + 0.588 378 906 249 999 999 999 999 994 204 391 527 723 468 774 912 819 2;
  • 80) 0.588 378 906 249 999 999 999 999 994 204 391 527 723 468 774 912 819 2 × 2 = 1 + 0.176 757 812 499 999 999 999 999 988 408 783 055 446 937 549 825 638 4;
  • 81) 0.176 757 812 499 999 999 999 999 988 408 783 055 446 937 549 825 638 4 × 2 = 0 + 0.353 515 624 999 999 999 999 999 976 817 566 110 893 875 099 651 276 8;
  • 82) 0.353 515 624 999 999 999 999 999 976 817 566 110 893 875 099 651 276 8 × 2 = 0 + 0.707 031 249 999 999 999 999 999 953 635 132 221 787 750 199 302 553 6;
  • 83) 0.707 031 249 999 999 999 999 999 953 635 132 221 787 750 199 302 553 6 × 2 = 1 + 0.414 062 499 999 999 999 999 999 907 270 264 443 575 500 398 605 107 2;
  • 84) 0.414 062 499 999 999 999 999 999 907 270 264 443 575 500 398 605 107 2 × 2 = 0 + 0.828 124 999 999 999 999 999 999 814 540 528 887 151 000 797 210 214 4;
  • 85) 0.828 124 999 999 999 999 999 999 814 540 528 887 151 000 797 210 214 4 × 2 = 1 + 0.656 249 999 999 999 999 999 999 629 081 057 774 302 001 594 420 428 8;
  • 86) 0.656 249 999 999 999 999 999 999 629 081 057 774 302 001 594 420 428 8 × 2 = 1 + 0.312 499 999 999 999 999 999 999 258 162 115 548 604 003 188 840 857 6;
  • 87) 0.312 499 999 999 999 999 999 999 258 162 115 548 604 003 188 840 857 6 × 2 = 0 + 0.624 999 999 999 999 999 999 998 516 324 231 097 208 006 377 681 715 2;
  • 88) 0.624 999 999 999 999 999 999 998 516 324 231 097 208 006 377 681 715 2 × 2 = 1 + 0.249 999 999 999 999 999 999 997 032 648 462 194 416 012 755 363 430 4;
  • 89) 0.249 999 999 999 999 999 999 997 032 648 462 194 416 012 755 363 430 4 × 2 = 0 + 0.499 999 999 999 999 999 999 994 065 296 924 388 832 025 510 726 860 8;
  • 90) 0.499 999 999 999 999 999 999 994 065 296 924 388 832 025 510 726 860 8 × 2 = 0 + 0.999 999 999 999 999 999 999 988 130 593 848 777 664 051 021 453 721 6;
  • 91) 0.999 999 999 999 999 999 999 988 130 593 848 777 664 051 021 453 721 6 × 2 = 1 + 0.999 999 999 999 999 999 999 976 261 187 697 555 328 102 042 907 443 2;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 143 4 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