0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 289 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 289 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 579 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 579 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 158 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 158 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 316 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 316 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 633 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 633 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 267 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 267 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 534 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 534 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 068 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 068 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 594 137 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 594 137 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 188 275 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 188 275 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 376 550 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 376 550 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 753 100 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 753 100 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 506 201 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 506 201 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 012 403 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 012 403 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 024 806 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 024 806 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 049 612 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 049 612 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 099 225 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 099 225 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 198 451 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 198 451 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 976 396 902 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 976 396 902 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 952 793 804 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 952 793 804 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 905 587 609 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 905 587 609 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 811 175 219 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 811 175 219 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 622 350 438 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 622 350 438 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 244 700 876 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 244 700 876 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 489 401 753 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 489 401 753 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 978 803 507 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 978 803 507 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 957 607 014 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 001 957 607 014 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 003 915 214 028 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 003 915 214 028 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 007 830 428 057 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 007 830 428 057 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 015 660 856 115 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 015 660 856 115 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 031 321 712 230 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 031 321 712 230 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 062 643 424 460 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 062 643 424 460 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 125 286 848 921 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 125 286 848 921 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 250 573 697 843 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 250 573 697 843 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 501 147 395 686 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 501 147 395 686 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 002 294 791 372 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 002 294 791 372 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 004 589 582 745 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 004 589 582 745 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 636 009 179 165 491 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 636 009 179 165 491 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 272 018 358 330 982 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 272 018 358 330 982 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 544 036 716 661 964 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 544 036 716 661 964 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 088 073 433 323 929 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 088 073 433 323 929 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 176 146 866 647 859 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 176 146 866 647 859 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 352 293 733 295 718 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 352 293 733 295 718 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 704 587 466 591 436 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 704 587 466 591 436 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 409 174 933 182 873 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 409 174 933 182 873 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 818 349 866 365 747 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 818 349 866 365 747 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 757 636 699 732 731 494 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 757 636 699 732 731 494 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 515 273 399 465 462 988 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 515 273 399 465 462 988 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 030 546 798 930 925 977 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 030 546 798 930 925 977 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 061 093 597 861 851 955 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 061 093 597 861 851 955 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 122 187 195 723 703 910 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 122 187 195 723 703 910 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 244 374 391 447 407 820 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 244 374 391 447 407 820 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 488 748 782 894 815 641 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 488 748 782 894 815 641 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 977 497 565 789 631 283 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 977 497 565 789 631 283 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 954 995 131 579 262 566 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 954 995 131 579 262 566 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 909 990 263 158 525 132 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 909 990 263 158 525 132 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 623 819 980 526 317 050 265 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 623 819 980 526 317 050 265 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 247 639 961 052 634 100 531 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 247 639 961 052 634 100 531 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 495 279 922 105 268 201 062 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 495 279 922 105 268 201 062 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 990 559 844 210 536 402 124 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 990 559 844 210 536 402 124 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 981 119 688 421 072 804 249 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 981 119 688 421 072 804 249 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 962 239 376 842 145 608 499 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 962 239 376 842 145 608 499 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 924 478 753 684 291 216 998 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 924 478 753 684 291 216 998 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 848 957 507 368 582 433 996 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 848 957 507 368 582 433 996 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 697 915 014 737 164 867 993 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 697 915 014 737 164 867 993 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 395 830 029 474 329 735 987 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 395 830 029 474 329 735 987 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 998 791 660 058 948 659 471 974 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 998 791 660 058 948 659 471 974 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 997 583 320 117 897 318 943 948 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 997 583 320 117 897 318 943 948 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 995 166 640 235 794 637 887 897 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 995 166 640 235 794 637 887 897 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 990 333 280 471 589 275 775 795 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 990 333 280 471 589 275 775 795 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 980 666 560 943 178 551 551 590 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 980 666 560 943 178 551 551 590 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 961 333 121 886 357 103 103 180 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 961 333 121 886 357 103 103 180 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 922 666 243 772 714 206 206 361 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 922 666 243 772 714 206 206 361 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 845 332 487 545 428 412 412 723 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 845 332 487 545 428 412 412 723 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 690 664 975 090 856 824 825 446 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 690 664 975 090 856 824 825 446 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 381 329 950 181 713 649 650 892 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 381 329 950 181 713 649 650 892 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 998 762 659 900 363 427 299 301 785 6;
  • 78) 0.397 094 726 562 499 999 999 999 998 762 659 900 363 427 299 301 785 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 997 525 319 800 726 854 598 603 571 2;
  • 79) 0.794 189 453 124 999 999 999 999 997 525 319 800 726 854 598 603 571 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 995 050 639 601 453 709 197 207 142 4;
  • 80) 0.588 378 906 249 999 999 999 999 995 050 639 601 453 709 197 207 142 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 990 101 279 202 907 418 394 414 284 8;
  • 81) 0.176 757 812 499 999 999 999 999 990 101 279 202 907 418 394 414 284 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 980 202 558 405 814 836 788 828 569 6;
  • 82) 0.353 515 624 999 999 999 999 999 980 202 558 405 814 836 788 828 569 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 960 405 116 811 629 673 577 657 139 2;
  • 83) 0.707 031 249 999 999 999 999 999 960 405 116 811 629 673 577 657 139 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 920 810 233 623 259 347 155 314 278 4;
  • 84) 0.414 062 499 999 999 999 999 999 920 810 233 623 259 347 155 314 278 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 841 620 467 246 518 694 310 628 556 8;
  • 85) 0.828 124 999 999 999 999 999 999 841 620 467 246 518 694 310 628 556 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 683 240 934 493 037 388 621 257 113 6;
  • 86) 0.656 249 999 999 999 999 999 999 683 240 934 493 037 388 621 257 113 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 366 481 868 986 074 777 242 514 227 2;
  • 87) 0.312 499 999 999 999 999 999 999 366 481 868 986 074 777 242 514 227 2 × 2 = 0 + 0.624 999 999 999 999 999 999 998 732 963 737 972 149 554 485 028 454 4;
  • 88) 0.624 999 999 999 999 999 999 998 732 963 737 972 149 554 485 028 454 4 × 2 = 1 + 0.249 999 999 999 999 999 999 997 465 927 475 944 299 108 970 056 908 8;
  • 89) 0.249 999 999 999 999 999 999 997 465 927 475 944 299 108 970 056 908 8 × 2 = 0 + 0.499 999 999 999 999 999 999 994 931 854 951 888 598 217 940 113 817 6;
  • 90) 0.499 999 999 999 999 999 999 994 931 854 951 888 598 217 940 113 817 6 × 2 = 0 + 0.999 999 999 999 999 999 999 989 863 709 903 777 196 435 880 227 635 2;
  • 91) 0.999 999 999 999 999 999 999 989 863 709 903 777 196 435 880 227 635 2 × 2 = 1 + 0.999 999 999 999 999 999 999 979 727 419 807 554 392 871 760 455 270 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 144 8 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