0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 42 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 42(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 42(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 42.

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 42 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 84;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 84 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 68;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 68 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 227 36;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 227 36 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 454 72;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 454 72 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 909 44;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 909 44 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 818 88;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 818 88 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 637 76;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 637 76 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 275 52;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 275 52 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 551 04;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 551 04 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 102 08;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 102 08 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 204 16;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 204 16 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 408 32;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 408 32 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 816 64;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 816 64 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 153 633 28;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 153 633 28 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 307 266 56;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 307 266 56 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 614 533 12;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 614 533 12 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 229 066 24;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 229 066 24 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 458 132 48;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 458 132 48 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 916 264 96;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 916 264 96 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 832 529 92;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 832 529 92 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 665 059 84;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 665 059 84 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 847 330 119 68;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 847 330 119 68 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 694 660 239 36;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 694 660 239 36 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 389 320 478 72;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 389 320 478 72 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 778 640 957 44;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 778 640 957 44 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 557 281 914 88;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 557 281 914 88 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 114 563 829 76;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 114 563 829 76 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 229 127 659 52;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 229 127 659 52 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 458 255 319 04;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 458 255 319 04 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 916 510 638 08;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 916 510 638 08 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 833 021 276 16;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 833 021 276 16 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 666 042 552 32;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 666 042 552 32 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 199 332 085 104 64;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 199 332 085 104 64 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 398 664 170 209 28;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 398 664 170 209 28 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 797 328 340 418 56;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 797 328 340 418 56 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 594 656 680 837 12;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 594 656 680 837 12 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 189 313 361 674 24;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 189 313 361 674 24 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 378 626 723 348 48;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 378 626 723 348 48 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 757 253 446 696 96;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 757 253 446 696 96 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 514 506 893 393 92;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 514 506 893 393 92 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 107 029 013 786 787 84;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 107 029 013 786 787 84 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 214 058 027 573 575 68;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 214 058 027 573 575 68 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 428 116 055 147 151 36;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 428 116 055 147 151 36 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 856 232 110 294 302 72;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 856 232 110 294 302 72 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 712 464 220 588 605 44;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 712 464 220 588 605 44 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 424 928 441 177 210 88;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 424 928 441 177 210 88 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 849 856 882 354 421 76;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 849 856 882 354 421 76 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 699 713 764 708 843 52;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 699 713 764 708 843 52 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 399 427 529 417 687 04;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 399 427 529 417 687 04 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 798 855 058 835 374 08;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 798 855 058 835 374 08 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 597 710 117 670 748 16;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 597 710 117 670 748 16 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 283 195 420 235 341 496 32;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 283 195 420 235 341 496 32 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 566 390 840 470 682 992 64;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 566 390 840 470 682 992 64 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 132 781 680 941 365 985 28;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 132 781 680 941 365 985 28 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 265 563 361 882 731 970 56;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 265 563 361 882 731 970 56 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 531 126 723 765 463 941 12;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 531 126 723 765 463 941 12 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 062 253 447 530 927 882 24;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 062 253 447 530 927 882 24 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 124 506 895 061 855 764 48;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 124 506 895 061 855 764 48 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 249 013 790 123 711 528 96;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 249 013 790 123 711 528 96 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 498 027 580 247 423 057 92;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 498 027 580 247 423 057 92 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 996 055 160 494 846 115 84;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 996 055 160 494 846 115 84 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 001 992 110 320 989 692 231 68;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 001 992 110 320 989 692 231 68 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 003 984 220 641 979 384 463 36;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 003 984 220 641 979 384 463 36 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 007 968 441 283 958 768 926 72;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 007 968 441 283 958 768 926 72 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 015 936 882 567 917 537 853 44;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 015 936 882 567 917 537 853 44 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 031 873 765 135 835 075 706 88;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 031 873 765 135 835 075 706 88 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 063 747 530 271 670 151 413 76;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 063 747 530 271 670 151 413 76 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 127 495 060 543 340 302 827 52;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 127 495 060 543 340 302 827 52 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 254 990 121 086 680 605 655 04;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 254 990 121 086 680 605 655 04 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 509 980 242 173 361 211 310 08;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 509 980 242 173 361 211 310 08 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 001 019 960 484 346 722 422 620 16;
  • 72) 0.615 579 605 102 539 062 500 000 000 001 019 960 484 346 722 422 620 16 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 002 039 920 968 693 444 845 240 32;
  • 73) 0.231 159 210 205 078 125 000 000 000 002 039 920 968 693 444 845 240 32 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 004 079 841 937 386 889 690 480 64;
  • 74) 0.462 318 420 410 156 250 000 000 000 004 079 841 937 386 889 690 480 64 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 008 159 683 874 773 779 380 961 28;
  • 75) 0.924 636 840 820 312 500 000 000 000 008 159 683 874 773 779 380 961 28 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 016 319 367 749 547 558 761 922 56;
  • 76) 0.849 273 681 640 625 000 000 000 000 016 319 367 749 547 558 761 922 56 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 032 638 735 499 095 117 523 845 12;
  • 77) 0.698 547 363 281 250 000 000 000 000 032 638 735 499 095 117 523 845 12 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 065 277 470 998 190 235 047 690 24;
  • 78) 0.397 094 726 562 500 000 000 000 000 065 277 470 998 190 235 047 690 24 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 130 554 941 996 380 470 095 380 48;
  • 79) 0.794 189 453 125 000 000 000 000 000 130 554 941 996 380 470 095 380 48 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 261 109 883 992 760 940 190 760 96;
  • 80) 0.588 378 906 250 000 000 000 000 000 261 109 883 992 760 940 190 760 96 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 522 219 767 985 521 880 381 521 92;
  • 81) 0.176 757 812 500 000 000 000 000 000 522 219 767 985 521 880 381 521 92 × 2 = 0 + 0.353 515 625 000 000 000 000 000 001 044 439 535 971 043 760 763 043 84;
  • 82) 0.353 515 625 000 000 000 000 000 001 044 439 535 971 043 760 763 043 84 × 2 = 0 + 0.707 031 250 000 000 000 000 000 002 088 879 071 942 087 521 526 087 68;
  • 83) 0.707 031 250 000 000 000 000 000 002 088 879 071 942 087 521 526 087 68 × 2 = 1 + 0.414 062 500 000 000 000 000 000 004 177 758 143 884 175 043 052 175 36;
  • 84) 0.414 062 500 000 000 000 000 000 004 177 758 143 884 175 043 052 175 36 × 2 = 0 + 0.828 125 000 000 000 000 000 000 008 355 516 287 768 350 086 104 350 72;
  • 85) 0.828 125 000 000 000 000 000 000 008 355 516 287 768 350 086 104 350 72 × 2 = 1 + 0.656 250 000 000 000 000 000 000 016 711 032 575 536 700 172 208 701 44;
  • 86) 0.656 250 000 000 000 000 000 000 016 711 032 575 536 700 172 208 701 44 × 2 = 1 + 0.312 500 000 000 000 000 000 000 033 422 065 151 073 400 344 417 402 88;
  • 87) 0.312 500 000 000 000 000 000 000 033 422 065 151 073 400 344 417 402 88 × 2 = 0 + 0.625 000 000 000 000 000 000 000 066 844 130 302 146 800 688 834 805 76;
  • 88) 0.625 000 000 000 000 000 000 000 066 844 130 302 146 800 688 834 805 76 × 2 = 1 + 0.250 000 000 000 000 000 000 000 133 688 260 604 293 601 377 669 611 52;
  • 89) 0.250 000 000 000 000 000 000 000 133 688 260 604 293 601 377 669 611 52 × 2 = 0 + 0.500 000 000 000 000 000 000 000 267 376 521 208 587 202 755 339 223 04;
  • 90) 0.500 000 000 000 000 000 000 000 267 376 521 208 587 202 755 339 223 04 × 2 = 1 + 0.000 000 000 000 000 000 000 000 534 753 042 417 174 405 510 678 446 08;
  • 91) 0.000 000 000 000 000 000 000 000 534 753 042 417 174 405 510 678 446 08 × 2 = 0 + 0.000 000 000 000 000 000 000 001 069 506 084 834 348 811 021 356 892 16;

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 42(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 42(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 42(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 42 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