0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 38;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 38 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 76;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 76 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 217 52;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 217 52 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 435 04;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 435 04 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 870 08;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 870 08 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 740 16;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 740 16 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 480 32;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 480 32 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 960 64;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 960 64 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 921 28;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 921 28 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 842 56;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 842 56 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 685 12;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 685 12 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 370 24;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 370 24 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 566 740 48;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 566 740 48 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 133 480 96;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 133 480 96 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 266 961 92;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 266 961 92 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 533 923 84;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 533 923 84 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 067 847 68;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 067 847 68 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 135 695 36;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 135 695 36 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 271 390 72;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 271 390 72 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 542 781 44;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 542 781 44 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 085 562 88;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 085 562 88 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 171 125 76;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 171 125 76 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 684 342 251 52;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 684 342 251 52 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 368 684 503 04;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 368 684 503 04 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 737 369 006 08;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 737 369 006 08 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 474 738 012 16;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 474 738 012 16 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 949 476 024 32;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 949 476 024 32 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 898 952 048 64;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 898 952 048 64 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 797 904 097 28;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 797 904 097 28 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 595 808 194 56;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 595 808 194 56 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 191 616 389 12;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 191 616 389 12 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 383 232 778 24;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 383 232 778 24 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 188 766 465 556 48;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 188 766 465 556 48 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 377 532 931 112 96;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 377 532 931 112 96 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 755 065 862 225 92;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 755 065 862 225 92 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 510 131 724 451 84;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 510 131 724 451 84 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 020 263 448 903 68;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 020 263 448 903 68 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 040 526 897 807 36;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 040 526 897 807 36 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 081 053 795 614 72;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 081 053 795 614 72 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 162 107 591 229 44;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 162 107 591 229 44 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 104 324 215 182 458 88;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 104 324 215 182 458 88 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 208 648 430 364 917 76;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 208 648 430 364 917 76 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 417 296 860 729 835 52;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 417 296 860 729 835 52 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 834 593 721 459 671 04;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 834 593 721 459 671 04 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 669 187 442 919 342 08;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 669 187 442 919 342 08 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 338 374 885 838 684 16;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 338 374 885 838 684 16 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 676 749 771 677 368 32;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 676 749 771 677 368 32 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 353 499 543 354 736 64;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 353 499 543 354 736 64 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 706 999 086 709 473 28;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 706 999 086 709 473 28 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 413 998 173 418 946 56;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 413 998 173 418 946 56 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 138 827 996 346 837 893 12;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 138 827 996 346 837 893 12 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 277 655 992 693 675 786 24;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 277 655 992 693 675 786 24 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 555 311 985 387 351 572 48;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 555 311 985 387 351 572 48 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 110 623 970 774 703 144 96;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 110 623 970 774 703 144 96 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 221 247 941 549 406 289 92;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 221 247 941 549 406 289 92 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 442 495 883 098 812 579 84;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 442 495 883 098 812 579 84 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 884 991 766 197 625 159 68;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 884 991 766 197 625 159 68 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 769 983 532 395 250 319 36;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 769 983 532 395 250 319 36 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 539 967 064 790 500 638 72;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 539 967 064 790 500 638 72 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 079 934 129 581 001 277 44;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 079 934 129 581 001 277 44 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 998 159 868 259 162 002 554 88;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 998 159 868 259 162 002 554 88 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 996 319 736 518 324 005 109 76;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 996 319 736 518 324 005 109 76 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 992 639 473 036 648 010 219 52;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 992 639 473 036 648 010 219 52 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 985 278 946 073 296 020 439 04;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 985 278 946 073 296 020 439 04 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 970 557 892 146 592 040 878 08;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 970 557 892 146 592 040 878 08 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 941 115 784 293 184 081 756 16;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 941 115 784 293 184 081 756 16 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 882 231 568 586 368 163 512 32;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 882 231 568 586 368 163 512 32 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 764 463 137 172 736 327 024 64;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 764 463 137 172 736 327 024 64 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 528 926 274 345 472 654 049 28;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 528 926 274 345 472 654 049 28 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 057 852 548 690 945 308 098 56;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 057 852 548 690 945 308 098 56 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 998 115 705 097 381 890 616 197 12;
  • 72) 0.615 579 605 102 539 062 499 999 999 998 115 705 097 381 890 616 197 12 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 996 231 410 194 763 781 232 394 24;
  • 73) 0.231 159 210 205 078 124 999 999 999 996 231 410 194 763 781 232 394 24 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 992 462 820 389 527 562 464 788 48;
  • 74) 0.462 318 420 410 156 249 999 999 999 992 462 820 389 527 562 464 788 48 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 984 925 640 779 055 124 929 576 96;
  • 75) 0.924 636 840 820 312 499 999 999 999 984 925 640 779 055 124 929 576 96 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 969 851 281 558 110 249 859 153 92;
  • 76) 0.849 273 681 640 624 999 999 999 999 969 851 281 558 110 249 859 153 92 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 939 702 563 116 220 499 718 307 84;
  • 77) 0.698 547 363 281 249 999 999 999 999 939 702 563 116 220 499 718 307 84 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 879 405 126 232 440 999 436 615 68;
  • 78) 0.397 094 726 562 499 999 999 999 999 879 405 126 232 440 999 436 615 68 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 758 810 252 464 881 998 873 231 36;
  • 79) 0.794 189 453 124 999 999 999 999 999 758 810 252 464 881 998 873 231 36 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 517 620 504 929 763 997 746 462 72;
  • 80) 0.588 378 906 249 999 999 999 999 999 517 620 504 929 763 997 746 462 72 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 035 241 009 859 527 995 492 925 44;
  • 81) 0.176 757 812 499 999 999 999 999 999 035 241 009 859 527 995 492 925 44 × 2 = 0 + 0.353 515 624 999 999 999 999 999 998 070 482 019 719 055 990 985 850 88;
  • 82) 0.353 515 624 999 999 999 999 999 998 070 482 019 719 055 990 985 850 88 × 2 = 0 + 0.707 031 249 999 999 999 999 999 996 140 964 039 438 111 981 971 701 76;
  • 83) 0.707 031 249 999 999 999 999 999 996 140 964 039 438 111 981 971 701 76 × 2 = 1 + 0.414 062 499 999 999 999 999 999 992 281 928 078 876 223 963 943 403 52;
  • 84) 0.414 062 499 999 999 999 999 999 992 281 928 078 876 223 963 943 403 52 × 2 = 0 + 0.828 124 999 999 999 999 999 999 984 563 856 157 752 447 927 886 807 04;
  • 85) 0.828 124 999 999 999 999 999 999 984 563 856 157 752 447 927 886 807 04 × 2 = 1 + 0.656 249 999 999 999 999 999 999 969 127 712 315 504 895 855 773 614 08;
  • 86) 0.656 249 999 999 999 999 999 999 969 127 712 315 504 895 855 773 614 08 × 2 = 1 + 0.312 499 999 999 999 999 999 999 938 255 424 631 009 791 711 547 228 16;
  • 87) 0.312 499 999 999 999 999 999 999 938 255 424 631 009 791 711 547 228 16 × 2 = 0 + 0.624 999 999 999 999 999 999 999 876 510 849 262 019 583 423 094 456 32;
  • 88) 0.624 999 999 999 999 999 999 999 876 510 849 262 019 583 423 094 456 32 × 2 = 1 + 0.249 999 999 999 999 999 999 999 753 021 698 524 039 166 846 188 912 64;
  • 89) 0.249 999 999 999 999 999 999 999 753 021 698 524 039 166 846 188 912 64 × 2 = 0 + 0.499 999 999 999 999 999 999 999 506 043 397 048 078 333 692 377 825 28;
  • 90) 0.499 999 999 999 999 999 999 999 506 043 397 048 078 333 692 377 825 28 × 2 = 0 + 0.999 999 999 999 999 999 999 999 012 086 794 096 156 667 384 755 650 56;
  • 91) 0.999 999 999 999 999 999 999 999 012 086 794 096 156 667 384 755 650 56 × 2 = 1 + 0.999 999 999 999 999 999 999 998 024 173 588 192 313 334 769 511 301 12;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 19 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