0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 34 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 34(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 34(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 34.

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 34 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 68;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 68 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 36;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 613 36 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 226 72;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 226 72 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 453 44;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 453 44 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 906 88;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 906 88 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 813 76;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 813 76 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 627 52;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 627 52 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 255 04;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 255 04 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 510 08;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 510 08 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 020 16;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 020 16 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 040 32;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 040 32 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 080 64;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 788 080 64 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 161 28;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 576 161 28 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 152 322 56;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 152 322 56 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 304 645 12;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 304 645 12 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 609 290 24;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 609 290 24 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 218 580 48;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 218 580 48 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 437 160 96;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 437 160 96 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 874 321 92;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 874 321 92 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 748 643 84;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 748 643 84 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 497 287 68;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 497 287 68 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 994 575 36;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 994 575 36 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 693 989 150 72;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 693 989 150 72 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 387 978 301 44;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 387 978 301 44 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 775 956 602 88;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 775 956 602 88 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 551 913 205 76;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 551 913 205 76 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 103 826 411 52;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 103 826 411 52 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 207 652 823 04;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 207 652 823 04 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 415 305 646 08;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 415 305 646 08 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 830 611 292 16;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 830 611 292 16 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 661 222 584 32;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 661 222 584 32 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 322 445 168 64;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 099 322 445 168 64 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 198 644 890 337 28;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 198 644 890 337 28 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 397 289 780 674 56;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 397 289 780 674 56 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 794 579 561 349 12;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 794 579 561 349 12 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 589 159 122 698 24;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 589 159 122 698 24 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 178 318 245 396 48;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 178 318 245 396 48 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 356 636 490 792 96;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 356 636 490 792 96 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 713 272 981 585 92;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 713 272 981 585 92 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 426 545 963 171 84;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 426 545 963 171 84 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 853 091 926 343 68;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 853 091 926 343 68 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 213 706 183 852 687 36;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 213 706 183 852 687 36 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 427 412 367 705 374 72;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 427 412 367 705 374 72 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 854 824 735 410 749 44;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 854 824 735 410 749 44 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 709 649 470 821 498 88;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 709 649 470 821 498 88 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 419 298 941 642 997 76;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 419 298 941 642 997 76 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 838 597 883 285 995 52;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 838 597 883 285 995 52 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 677 195 766 571 991 04;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 677 195 766 571 991 04 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 354 391 533 143 982 08;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 354 391 533 143 982 08 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 708 783 066 287 964 16;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 708 783 066 287 964 16 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 417 566 132 575 928 32;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 417 566 132 575 928 32 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 282 835 132 265 151 856 64;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 282 835 132 265 151 856 64 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 565 670 264 530 303 713 28;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 565 670 264 530 303 713 28 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 131 340 529 060 607 426 56;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 131 340 529 060 607 426 56 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 262 681 058 121 214 853 12;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 262 681 058 121 214 853 12 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 525 362 116 242 429 706 24;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 525 362 116 242 429 706 24 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 050 724 232 484 859 412 48;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 050 724 232 484 859 412 48 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 101 448 464 969 718 824 96;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 101 448 464 969 718 824 96 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 202 896 929 939 437 649 92;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 202 896 929 939 437 649 92 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 405 793 859 878 875 299 84;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 405 793 859 878 875 299 84 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 811 587 719 757 750 599 68;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 811 587 719 757 750 599 68 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 001 623 175 439 515 501 199 36;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 001 623 175 439 515 501 199 36 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 003 246 350 879 031 002 398 72;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 003 246 350 879 031 002 398 72 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 006 492 701 758 062 004 797 44;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 006 492 701 758 062 004 797 44 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 012 985 403 516 124 009 594 88;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 012 985 403 516 124 009 594 88 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 025 970 807 032 248 019 189 76;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 025 970 807 032 248 019 189 76 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 051 941 614 064 496 038 379 52;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 051 941 614 064 496 038 379 52 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 103 883 228 128 992 076 759 04;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 103 883 228 128 992 076 759 04 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 207 766 456 257 984 153 518 08;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 207 766 456 257 984 153 518 08 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 415 532 912 515 968 307 036 16;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 415 532 912 515 968 307 036 16 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 831 065 825 031 936 614 072 32;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 831 065 825 031 936 614 072 32 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 001 662 131 650 063 873 228 144 64;
  • 73) 0.231 159 210 205 078 125 000 000 000 001 662 131 650 063 873 228 144 64 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 003 324 263 300 127 746 456 289 28;
  • 74) 0.462 318 420 410 156 250 000 000 000 003 324 263 300 127 746 456 289 28 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 006 648 526 600 255 492 912 578 56;
  • 75) 0.924 636 840 820 312 500 000 000 000 006 648 526 600 255 492 912 578 56 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 013 297 053 200 510 985 825 157 12;
  • 76) 0.849 273 681 640 625 000 000 000 000 013 297 053 200 510 985 825 157 12 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 026 594 106 401 021 971 650 314 24;
  • 77) 0.698 547 363 281 250 000 000 000 000 026 594 106 401 021 971 650 314 24 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 053 188 212 802 043 943 300 628 48;
  • 78) 0.397 094 726 562 500 000 000 000 000 053 188 212 802 043 943 300 628 48 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 106 376 425 604 087 886 601 256 96;
  • 79) 0.794 189 453 125 000 000 000 000 000 106 376 425 604 087 886 601 256 96 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 212 752 851 208 175 773 202 513 92;
  • 80) 0.588 378 906 250 000 000 000 000 000 212 752 851 208 175 773 202 513 92 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 425 505 702 416 351 546 405 027 84;
  • 81) 0.176 757 812 500 000 000 000 000 000 425 505 702 416 351 546 405 027 84 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 851 011 404 832 703 092 810 055 68;
  • 82) 0.353 515 625 000 000 000 000 000 000 851 011 404 832 703 092 810 055 68 × 2 = 0 + 0.707 031 250 000 000 000 000 000 001 702 022 809 665 406 185 620 111 36;
  • 83) 0.707 031 250 000 000 000 000 000 001 702 022 809 665 406 185 620 111 36 × 2 = 1 + 0.414 062 500 000 000 000 000 000 003 404 045 619 330 812 371 240 222 72;
  • 84) 0.414 062 500 000 000 000 000 000 003 404 045 619 330 812 371 240 222 72 × 2 = 0 + 0.828 125 000 000 000 000 000 000 006 808 091 238 661 624 742 480 445 44;
  • 85) 0.828 125 000 000 000 000 000 000 006 808 091 238 661 624 742 480 445 44 × 2 = 1 + 0.656 250 000 000 000 000 000 000 013 616 182 477 323 249 484 960 890 88;
  • 86) 0.656 250 000 000 000 000 000 000 013 616 182 477 323 249 484 960 890 88 × 2 = 1 + 0.312 500 000 000 000 000 000 000 027 232 364 954 646 498 969 921 781 76;
  • 87) 0.312 500 000 000 000 000 000 000 027 232 364 954 646 498 969 921 781 76 × 2 = 0 + 0.625 000 000 000 000 000 000 000 054 464 729 909 292 997 939 843 563 52;
  • 88) 0.625 000 000 000 000 000 000 000 054 464 729 909 292 997 939 843 563 52 × 2 = 1 + 0.250 000 000 000 000 000 000 000 108 929 459 818 585 995 879 687 127 04;
  • 89) 0.250 000 000 000 000 000 000 000 108 929 459 818 585 995 879 687 127 04 × 2 = 0 + 0.500 000 000 000 000 000 000 000 217 858 919 637 171 991 759 374 254 08;
  • 90) 0.500 000 000 000 000 000 000 000 217 858 919 637 171 991 759 374 254 08 × 2 = 1 + 0.000 000 000 000 000 000 000 000 435 717 839 274 343 983 518 748 508 16;
  • 91) 0.000 000 000 000 000 000 000 000 435 717 839 274 343 983 518 748 508 16 × 2 = 0 + 0.000 000 000 000 000 000 000 000 871 435 678 548 687 967 037 497 016 32;

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 34(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 34(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 34(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 34 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