0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 9;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 9 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 345 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 345 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 691 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 691 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 383 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 383 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 766 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 766 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 532 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 532 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 065 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 065 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 131 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 131 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 262 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 262 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 524 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 524 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 049 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 049 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 466 099 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 466 099 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 932 198 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 932 198 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 864 396 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 864 396 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 728 793 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 728 793 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 457 587 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 457 587 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 915 174 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 915 174 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 830 348 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 830 348 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 660 697 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 660 697 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 321 395 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 321 395 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 174 642 790 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 174 642 790 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 349 285 580 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 349 285 580 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 698 571 161 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 698 571 161 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 397 142 323 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 397 142 323 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 794 284 646 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 794 284 646 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 588 569 292 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 588 569 292 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 177 138 585 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 177 138 585 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 354 277 171 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 354 277 171 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 708 554 342 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 708 554 342 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 273 417 108 684 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 273 417 108 684 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 546 834 217 369 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 546 834 217 369 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 093 668 434 739 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 093 668 434 739 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 187 336 869 478 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 187 336 869 478 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 374 673 738 956 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 374 673 738 956 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 749 347 477 913 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 749 347 477 913 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 498 694 955 827 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 498 694 955 827 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 997 389 911 654 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 997 389 911 654 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 994 779 823 308 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 994 779 823 308 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 989 559 646 617 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 989 559 646 617 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 647 979 119 293 235 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 647 979 119 293 235 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 295 958 238 586 470 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 295 958 238 586 470 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 591 916 477 172 940 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 591 916 477 172 940 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 183 832 954 345 881 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 183 832 954 345 881 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 367 665 908 691 763 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 367 665 908 691 763 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 735 331 817 383 526 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 735 331 817 383 526 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 470 663 634 767 052 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 470 663 634 767 052 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 941 327 269 534 105 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 941 327 269 534 105 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 882 654 539 068 211 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 882 654 539 068 211 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 765 309 078 136 422 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 765 309 078 136 422 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 530 618 156 272 844 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 530 618 156 272 844 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 247 061 236 312 545 689 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 247 061 236 312 545 689 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 494 122 472 625 091 379 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 494 122 472 625 091 379 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 988 244 945 250 182 758 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 988 244 945 250 182 758 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 976 489 890 500 365 516 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 976 489 890 500 365 516 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 952 979 781 000 731 033 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 952 979 781 000 731 033 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 905 959 562 001 462 067 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 905 959 562 001 462 067 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 811 919 124 002 924 134 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 811 919 124 002 924 134 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 623 838 248 005 848 268 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 623 838 248 005 848 268 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 247 676 496 011 696 537 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 247 676 496 011 696 537 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 495 352 992 023 393 075 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 495 352 992 023 393 075 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 252 990 705 984 046 786 150 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 252 990 705 984 046 786 150 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 505 981 411 968 093 572 300 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 505 981 411 968 093 572 300 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 011 962 823 936 187 144 601 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 011 962 823 936 187 144 601 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 023 925 647 872 374 289 203 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 023 925 647 872 374 289 203 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 047 851 295 744 748 578 406 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 047 851 295 744 748 578 406 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 095 702 591 489 497 156 812 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 095 702 591 489 497 156 812 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 191 405 182 978 994 313 625 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 191 405 182 978 994 313 625 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 382 810 365 957 988 627 251 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 382 810 365 957 988 627 251 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 765 620 731 915 977 254 502 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 765 620 731 915 977 254 502 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 001 531 241 463 831 954 509 004 8;
  • 71) 0.307 789 802 551 269 531 250 000 001 531 241 463 831 954 509 004 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 003 062 482 927 663 909 018 009 6;
  • 72) 0.615 579 605 102 539 062 500 000 003 062 482 927 663 909 018 009 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 006 124 965 855 327 818 036 019 2;
  • 73) 0.231 159 210 205 078 125 000 000 006 124 965 855 327 818 036 019 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 012 249 931 710 655 636 072 038 4;
  • 74) 0.462 318 420 410 156 250 000 000 012 249 931 710 655 636 072 038 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 024 499 863 421 311 272 144 076 8;
  • 75) 0.924 636 840 820 312 500 000 000 024 499 863 421 311 272 144 076 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 048 999 726 842 622 544 288 153 6;
  • 76) 0.849 273 681 640 625 000 000 000 048 999 726 842 622 544 288 153 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 097 999 453 685 245 088 576 307 2;
  • 77) 0.698 547 363 281 250 000 000 000 097 999 453 685 245 088 576 307 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 195 998 907 370 490 177 152 614 4;
  • 78) 0.397 094 726 562 500 000 000 000 195 998 907 370 490 177 152 614 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 391 997 814 740 980 354 305 228 8;
  • 79) 0.794 189 453 125 000 000 000 000 391 997 814 740 980 354 305 228 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 783 995 629 481 960 708 610 457 6;
  • 80) 0.588 378 906 250 000 000 000 000 783 995 629 481 960 708 610 457 6 × 2 = 1 + 0.176 757 812 500 000 000 000 001 567 991 258 963 921 417 220 915 2;
  • 81) 0.176 757 812 500 000 000 000 001 567 991 258 963 921 417 220 915 2 × 2 = 0 + 0.353 515 625 000 000 000 000 003 135 982 517 927 842 834 441 830 4;
  • 82) 0.353 515 625 000 000 000 000 003 135 982 517 927 842 834 441 830 4 × 2 = 0 + 0.707 031 250 000 000 000 000 006 271 965 035 855 685 668 883 660 8;
  • 83) 0.707 031 250 000 000 000 000 006 271 965 035 855 685 668 883 660 8 × 2 = 1 + 0.414 062 500 000 000 000 000 012 543 930 071 711 371 337 767 321 6;
  • 84) 0.414 062 500 000 000 000 000 012 543 930 071 711 371 337 767 321 6 × 2 = 0 + 0.828 125 000 000 000 000 000 025 087 860 143 422 742 675 534 643 2;
  • 85) 0.828 125 000 000 000 000 000 025 087 860 143 422 742 675 534 643 2 × 2 = 1 + 0.656 250 000 000 000 000 000 050 175 720 286 845 485 351 069 286 4;
  • 86) 0.656 250 000 000 000 000 000 050 175 720 286 845 485 351 069 286 4 × 2 = 1 + 0.312 500 000 000 000 000 000 100 351 440 573 690 970 702 138 572 8;
  • 87) 0.312 500 000 000 000 000 000 100 351 440 573 690 970 702 138 572 8 × 2 = 0 + 0.625 000 000 000 000 000 000 200 702 881 147 381 941 404 277 145 6;
  • 88) 0.625 000 000 000 000 000 000 200 702 881 147 381 941 404 277 145 6 × 2 = 1 + 0.250 000 000 000 000 000 000 401 405 762 294 763 882 808 554 291 2;
  • 89) 0.250 000 000 000 000 000 000 401 405 762 294 763 882 808 554 291 2 × 2 = 0 + 0.500 000 000 000 000 000 000 802 811 524 589 527 765 617 108 582 4;
  • 90) 0.500 000 000 000 000 000 000 802 811 524 589 527 765 617 108 582 4 × 2 = 1 + 0.000 000 000 000 000 000 001 605 623 049 179 055 531 234 217 164 8;
  • 91) 0.000 000 000 000 000 000 001 605 623 049 179 055 531 234 217 164 8 × 2 = 0 + 0.000 000 000 000 000 000 003 211 246 098 358 111 062 468 434 329 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 45 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