0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 324 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 324 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 648 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 648 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 296 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 296 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 593 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 593 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 187 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 187 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 374 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 374 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 748 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 748 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 497 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 497 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 995 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 995 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 205 990 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 205 990 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 411 980 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 411 980 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 823 961 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 823 961 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 647 923 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 647 923 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 295 846 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 295 846 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 591 692 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 591 692 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 183 385 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 183 385 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 366 771 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 366 771 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 733 542 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 733 542 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 985 467 084 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 985 467 084 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 970 934 169 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 970 934 169 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 941 868 339 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 941 868 339 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 883 736 678 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 883 736 678 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 767 473 356 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 767 473 356 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 534 946 713 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 534 946 713 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 069 893 427 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 069 893 427 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 139 786 854 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 139 786 854 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 279 573 708 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 279 573 708 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 559 147 417 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 559 147 417 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 017 118 294 835 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 017 118 294 835 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 034 236 589 670 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 034 236 589 670 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 068 473 179 340 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 068 473 179 340 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 136 946 358 681 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 136 946 358 681 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 273 892 717 363 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 273 892 717 363 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 547 785 434 726 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 547 785 434 726 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 095 570 869 452 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 095 570 869 452 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 410 191 141 738 905 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 410 191 141 738 905 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 820 382 283 477 811 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 820 382 283 477 811 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 640 764 566 955 622 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 640 764 566 955 622 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 281 529 133 911 244 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 281 529 133 911 244 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 563 058 267 822 489 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 563 058 267 822 489 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 126 116 535 644 979 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 126 116 535 644 979 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 252 233 071 289 958 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 252 233 071 289 958 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 504 466 142 579 916 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 504 466 142 579 916 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 008 932 285 159 833 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 008 932 285 159 833 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 690 017 864 570 319 667 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 690 017 864 570 319 667 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 380 035 729 140 639 334 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 380 035 729 140 639 334 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 760 071 458 281 278 668 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 760 071 458 281 278 668 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 520 142 916 562 557 337 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 520 142 916 562 557 337 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 040 285 833 125 114 675 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 040 285 833 125 114 675 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 080 571 666 250 229 350 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 080 571 666 250 229 350 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 161 143 332 500 458 700 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 161 143 332 500 458 700 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 322 286 665 000 917 401 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 322 286 665 000 917 401 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 644 573 330 001 834 803 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 644 573 330 001 834 803 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 289 146 660 003 669 606 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 289 146 660 003 669 606 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 578 293 320 007 339 212 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 578 293 320 007 339 212 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 813 156 586 640 014 678 425 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 813 156 586 640 014 678 425 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 626 313 173 280 029 356 851 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 626 313 173 280 029 356 851 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 252 626 346 560 058 713 702 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 252 626 346 560 058 713 702 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 505 252 693 120 117 427 404 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 505 252 693 120 117 427 404 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 010 505 386 240 234 854 809 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 010 505 386 240 234 854 809 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 021 010 772 480 469 709 619 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 021 010 772 480 469 709 619 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 042 021 544 960 939 419 238 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 042 021 544 960 939 419 238 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 084 043 089 921 878 838 476 8;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 084 043 089 921 878 838 476 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 168 086 179 843 757 676 953 6;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 168 086 179 843 757 676 953 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 336 172 359 687 515 353 907 2;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 336 172 359 687 515 353 907 2 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 672 344 719 375 030 707 814 4;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 672 344 719 375 030 707 814 4 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 001 344 689 438 750 061 415 628 8;
  • 68) 0.663 473 725 318 908 691 406 250 000 001 344 689 438 750 061 415 628 8 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 002 689 378 877 500 122 831 257 6;
  • 69) 0.326 947 450 637 817 382 812 500 000 002 689 378 877 500 122 831 257 6 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 005 378 757 755 000 245 662 515 2;
  • 70) 0.653 894 901 275 634 765 625 000 000 005 378 757 755 000 245 662 515 2 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 010 757 515 510 000 491 325 030 4;
  • 71) 0.307 789 802 551 269 531 250 000 000 010 757 515 510 000 491 325 030 4 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 021 515 031 020 000 982 650 060 8;
  • 72) 0.615 579 605 102 539 062 500 000 000 021 515 031 020 000 982 650 060 8 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 043 030 062 040 001 965 300 121 6;
  • 73) 0.231 159 210 205 078 125 000 000 000 043 030 062 040 001 965 300 121 6 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 086 060 124 080 003 930 600 243 2;
  • 74) 0.462 318 420 410 156 250 000 000 000 086 060 124 080 003 930 600 243 2 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 172 120 248 160 007 861 200 486 4;
  • 75) 0.924 636 840 820 312 500 000 000 000 172 120 248 160 007 861 200 486 4 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 344 240 496 320 015 722 400 972 8;
  • 76) 0.849 273 681 640 625 000 000 000 000 344 240 496 320 015 722 400 972 8 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 688 480 992 640 031 444 801 945 6;
  • 77) 0.698 547 363 281 250 000 000 000 000 688 480 992 640 031 444 801 945 6 × 2 = 1 + 0.397 094 726 562 500 000 000 000 001 376 961 985 280 062 889 603 891 2;
  • 78) 0.397 094 726 562 500 000 000 000 001 376 961 985 280 062 889 603 891 2 × 2 = 0 + 0.794 189 453 125 000 000 000 000 002 753 923 970 560 125 779 207 782 4;
  • 79) 0.794 189 453 125 000 000 000 000 002 753 923 970 560 125 779 207 782 4 × 2 = 1 + 0.588 378 906 250 000 000 000 000 005 507 847 941 120 251 558 415 564 8;
  • 80) 0.588 378 906 250 000 000 000 000 005 507 847 941 120 251 558 415 564 8 × 2 = 1 + 0.176 757 812 500 000 000 000 000 011 015 695 882 240 503 116 831 129 6;
  • 81) 0.176 757 812 500 000 000 000 000 011 015 695 882 240 503 116 831 129 6 × 2 = 0 + 0.353 515 625 000 000 000 000 000 022 031 391 764 481 006 233 662 259 2;
  • 82) 0.353 515 625 000 000 000 000 000 022 031 391 764 481 006 233 662 259 2 × 2 = 0 + 0.707 031 250 000 000 000 000 000 044 062 783 528 962 012 467 324 518 4;
  • 83) 0.707 031 250 000 000 000 000 000 044 062 783 528 962 012 467 324 518 4 × 2 = 1 + 0.414 062 500 000 000 000 000 000 088 125 567 057 924 024 934 649 036 8;
  • 84) 0.414 062 500 000 000 000 000 000 088 125 567 057 924 024 934 649 036 8 × 2 = 0 + 0.828 125 000 000 000 000 000 000 176 251 134 115 848 049 869 298 073 6;
  • 85) 0.828 125 000 000 000 000 000 000 176 251 134 115 848 049 869 298 073 6 × 2 = 1 + 0.656 250 000 000 000 000 000 000 352 502 268 231 696 099 738 596 147 2;
  • 86) 0.656 250 000 000 000 000 000 000 352 502 268 231 696 099 738 596 147 2 × 2 = 1 + 0.312 500 000 000 000 000 000 000 705 004 536 463 392 199 477 192 294 4;
  • 87) 0.312 500 000 000 000 000 000 000 705 004 536 463 392 199 477 192 294 4 × 2 = 0 + 0.625 000 000 000 000 000 000 001 410 009 072 926 784 398 954 384 588 8;
  • 88) 0.625 000 000 000 000 000 000 001 410 009 072 926 784 398 954 384 588 8 × 2 = 1 + 0.250 000 000 000 000 000 000 002 820 018 145 853 568 797 908 769 177 6;
  • 89) 0.250 000 000 000 000 000 000 002 820 018 145 853 568 797 908 769 177 6 × 2 = 0 + 0.500 000 000 000 000 000 000 005 640 036 291 707 137 595 817 538 355 2;
  • 90) 0.500 000 000 000 000 000 000 005 640 036 291 707 137 595 817 538 355 2 × 2 = 1 + 0.000 000 000 000 000 000 000 011 280 072 583 414 275 191 635 076 710 4;
  • 91) 0.000 000 000 000 000 000 000 011 280 072 583 414 275 191 635 076 710 4 × 2 = 0 + 0.000 000 000 000 000 000 000 022 560 145 166 828 550 383 270 153 420 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 162 1 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