0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 328 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 328 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 656 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 656 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 313 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 313 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 627 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 627 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 254 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 254 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 508 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 508 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 901 017 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 901 017 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 802 035 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 802 035 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 604 070 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 604 070 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 208 140 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 208 140 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 416 281 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 416 281 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 832 563 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 832 563 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 665 126 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 665 126 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 330 252 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 330 252 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 660 505 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 660 505 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 321 011 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 321 011 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 642 022 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 642 022 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 493 284 044 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 493 284 044 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 986 568 089 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 986 568 089 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 973 136 179 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 973 136 179 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 946 272 358 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 946 272 358 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 892 544 716 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 892 544 716 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 785 089 433 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 785 089 433 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 570 178 867 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 570 178 867 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 140 357 734 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 140 357 734 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 280 715 468 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 280 715 468 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 561 430 937 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 561 430 937 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 009 122 861 875 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 009 122 861 875 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 018 245 723 750 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 018 245 723 750 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 036 491 447 500 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 036 491 447 500 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 072 982 895 001 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 072 982 895 001 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 145 965 790 003 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 145 965 790 003 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 291 931 580 006 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 291 931 580 006 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 583 863 160 012 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 583 863 160 012 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 167 726 320 025 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 167 726 320 025 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 410 335 452 640 051 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 410 335 452 640 051 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 820 670 905 280 102 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 820 670 905 280 102 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 641 341 810 560 204 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 641 341 810 560 204 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 282 683 621 120 409 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 282 683 621 120 409 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 565 367 242 240 819 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 565 367 242 240 819 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 130 734 484 481 638 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 130 734 484 481 638 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 261 468 968 963 276 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 261 468 968 963 276 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 522 937 937 926 553 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 522 937 937 926 553 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 045 875 875 853 107 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 045 875 875 853 107 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 690 091 751 751 706 214 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 690 091 751 751 706 214 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 380 183 503 503 412 428 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 380 183 503 503 412 428 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 760 367 007 006 824 857 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 760 367 007 006 824 857 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 520 734 014 013 649 715 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 520 734 014 013 649 715 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 041 468 028 027 299 430 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 041 468 028 027 299 430 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 082 936 056 054 598 860 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 082 936 056 054 598 860 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 165 872 112 109 197 721 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 165 872 112 109 197 721 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 331 744 224 218 395 443 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 331 744 224 218 395 443 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 663 488 448 436 790 886 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 663 488 448 436 790 886 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 326 976 896 873 581 772 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 326 976 896 873 581 772 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 653 953 793 747 163 545 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 653 953 793 747 163 545 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 813 307 907 587 494 327 091 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 813 307 907 587 494 327 091 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 626 615 815 174 988 654 182 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 626 615 815 174 988 654 182 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 253 231 630 349 977 308 364 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 253 231 630 349 977 308 364 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 506 463 260 699 954 616 729 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 506 463 260 699 954 616 729 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 012 926 521 399 909 233 459 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 012 926 521 399 909 233 459 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 025 853 042 799 818 466 918 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 025 853 042 799 818 466 918 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 051 706 085 599 636 933 836 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 051 706 085 599 636 933 836 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 103 412 171 199 273 867 673 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 103 412 171 199 273 867 673 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 206 824 342 398 547 735 347 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 206 824 342 398 547 735 347 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 413 648 684 797 095 470 694 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 413 648 684 797 095 470 694 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 827 297 369 594 190 941 388 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 827 297 369 594 190 941 388 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 001 654 594 739 188 381 882 777 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 001 654 594 739 188 381 882 777 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 003 309 189 478 376 763 765 555 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 003 309 189 478 376 763 765 555 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 006 618 378 956 753 527 531 110 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 006 618 378 956 753 527 531 110 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 013 236 757 913 507 055 062 220 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 013 236 757 913 507 055 062 220 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 026 473 515 827 014 110 124 441 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 026 473 515 827 014 110 124 441 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 052 947 031 654 028 220 248 883 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 052 947 031 654 028 220 248 883 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 105 894 063 308 056 440 497 766 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 105 894 063 308 056 440 497 766 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 211 788 126 616 112 880 995 532 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 211 788 126 616 112 880 995 532 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 423 576 253 232 225 761 991 065 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 423 576 253 232 225 761 991 065 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 847 152 506 464 451 523 982 131 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 847 152 506 464 451 523 982 131 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 001 694 305 012 928 903 047 964 262 4;
  • 78) 0.397 094 726 562 500 000 000 000 001 694 305 012 928 903 047 964 262 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 003 388 610 025 857 806 095 928 524 8;
  • 79) 0.794 189 453 125 000 000 000 000 003 388 610 025 857 806 095 928 524 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 006 777 220 051 715 612 191 857 049 6;
  • 80) 0.588 378 906 250 000 000 000 000 006 777 220 051 715 612 191 857 049 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 013 554 440 103 431 224 383 714 099 2;
  • 81) 0.176 757 812 500 000 000 000 000 013 554 440 103 431 224 383 714 099 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 027 108 880 206 862 448 767 428 198 4;
  • 82) 0.353 515 625 000 000 000 000 000 027 108 880 206 862 448 767 428 198 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 054 217 760 413 724 897 534 856 396 8;
  • 83) 0.707 031 250 000 000 000 000 000 054 217 760 413 724 897 534 856 396 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 108 435 520 827 449 795 069 712 793 6;
  • 84) 0.414 062 500 000 000 000 000 000 108 435 520 827 449 795 069 712 793 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 216 871 041 654 899 590 139 425 587 2;
  • 85) 0.828 125 000 000 000 000 000 000 216 871 041 654 899 590 139 425 587 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 433 742 083 309 799 180 278 851 174 4;
  • 86) 0.656 250 000 000 000 000 000 000 433 742 083 309 799 180 278 851 174 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 867 484 166 619 598 360 557 702 348 8;
  • 87) 0.312 500 000 000 000 000 000 000 867 484 166 619 598 360 557 702 348 8 × 2 = 0 + 0.625 000 000 000 000 000 000 001 734 968 333 239 196 721 115 404 697 6;
  • 88) 0.625 000 000 000 000 000 000 001 734 968 333 239 196 721 115 404 697 6 × 2 = 1 + 0.250 000 000 000 000 000 000 003 469 936 666 478 393 442 230 809 395 2;
  • 89) 0.250 000 000 000 000 000 000 003 469 936 666 478 393 442 230 809 395 2 × 2 = 0 + 0.500 000 000 000 000 000 000 006 939 873 332 956 786 884 461 618 790 4;
  • 90) 0.500 000 000 000 000 000 000 006 939 873 332 956 786 884 461 618 790 4 × 2 = 1 + 0.000 000 000 000 000 000 000 013 879 746 665 913 573 768 923 237 580 8;
  • 91) 0.000 000 000 000 000 000 000 013 879 746 665 913 573 768 923 237 580 8 × 2 = 0 + 0.000 000 000 000 000 000 000 027 759 493 331 827 147 537 846 475 161 6;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 164 2 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