0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 667 48;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 667 48 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 334 96;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 334 96 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 669 92;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 669 92 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 339 84;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 339 84 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 679 68;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 679 68 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 359 36;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 359 36 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 718 72;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 718 72 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 437 44;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 437 44 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 874 88;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 874 88 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 749 76;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 613 749 76 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 227 499 52;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 227 499 52 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 454 999 04;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 454 999 04 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 909 998 08;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 909 998 08 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 819 996 16;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 819 996 16 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 639 992 32;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 639 992 32 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 279 984 64;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 279 984 64 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 559 969 28;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 559 969 28 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 119 938 56;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 119 938 56 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 239 877 12;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 239 877 12 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 084 479 754 24;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 084 479 754 24 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 168 959 508 48;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 168 959 508 48 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 337 919 016 96;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 337 919 016 96 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 675 838 033 92;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 675 838 033 92 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 351 676 067 84;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 351 676 067 84 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 703 352 135 68;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 703 352 135 68 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 406 704 271 36;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 406 704 271 36 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 813 408 542 72;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 813 408 542 72 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 626 817 085 44;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 626 817 085 44 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 253 634 170 88;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 253 634 170 88 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 270 507 268 341 76;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 270 507 268 341 76 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 541 014 536 683 52;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 541 014 536 683 52 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 082 029 073 367 04;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 082 029 073 367 04 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 164 058 146 734 08;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 164 058 146 734 08 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 328 116 293 468 16;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 328 116 293 468 16 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 656 232 586 936 32;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 656 232 586 936 32 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 312 465 173 872 64;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 312 465 173 872 64 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 624 930 347 745 28;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 624 930 347 745 28 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 249 860 695 490 56;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 249 860 695 490 56 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 322 499 721 390 981 12;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 322 499 721 390 981 12 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 644 999 442 781 962 24;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 644 999 442 781 962 24 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 289 998 885 563 924 48;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 289 998 885 563 924 48 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 579 997 771 127 848 96;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 579 997 771 127 848 96 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 159 995 542 255 697 92;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 159 995 542 255 697 92 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 319 991 084 511 395 84;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 319 991 084 511 395 84 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 639 982 169 022 791 68;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 639 982 169 022 791 68 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 279 964 338 045 583 36;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 279 964 338 045 583 36 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 559 928 676 091 166 72;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 559 928 676 091 166 72 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 119 857 352 182 333 44;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 119 857 352 182 333 44 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 239 714 704 364 666 88;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 239 714 704 364 666 88 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 620 479 429 408 729 333 76;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 620 479 429 408 729 333 76 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 240 958 858 817 458 667 52;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 240 958 858 817 458 667 52 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 481 917 717 634 917 335 04;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 481 917 717 634 917 335 04 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 963 835 435 269 834 670 08;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 963 835 435 269 834 670 08 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 927 670 870 539 669 340 16;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 927 670 870 539 669 340 16 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 855 341 741 079 338 680 32;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 855 341 741 079 338 680 32 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 710 683 482 158 677 360 64;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 710 683 482 158 677 360 64 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 421 366 964 317 354 721 28;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 421 366 964 317 354 721 28 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 842 733 928 634 709 442 56;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 842 733 928 634 709 442 56 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 061 685 467 857 269 418 885 12;
  • 60) 0.834 622 944 239 526 987 075 805 664 061 685 467 857 269 418 885 12 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 123 370 935 714 538 837 770 24;
  • 61) 0.669 245 888 479 053 974 151 611 328 123 370 935 714 538 837 770 24 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 246 741 871 429 077 675 540 48;
  • 62) 0.338 491 776 958 107 948 303 222 656 246 741 871 429 077 675 540 48 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 493 483 742 858 155 351 080 96;
  • 63) 0.676 983 553 916 215 896 606 445 312 493 483 742 858 155 351 080 96 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 986 967 485 716 310 702 161 92;
  • 64) 0.353 967 107 832 431 793 212 890 624 986 967 485 716 310 702 161 92 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 973 934 971 432 621 404 323 84;
  • 65) 0.707 934 215 664 863 586 425 781 249 973 934 971 432 621 404 323 84 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 947 869 942 865 242 808 647 68;
  • 66) 0.415 868 431 329 727 172 851 562 499 947 869 942 865 242 808 647 68 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 895 739 885 730 485 617 295 36;
  • 67) 0.831 736 862 659 454 345 703 124 999 895 739 885 730 485 617 295 36 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 791 479 771 460 971 234 590 72;
  • 68) 0.663 473 725 318 908 691 406 249 999 791 479 771 460 971 234 590 72 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 582 959 542 921 942 469 181 44;
  • 69) 0.326 947 450 637 817 382 812 499 999 582 959 542 921 942 469 181 44 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 165 919 085 843 884 938 362 88;
  • 70) 0.653 894 901 275 634 765 624 999 999 165 919 085 843 884 938 362 88 × 2 = 1 + 0.307 789 802 551 269 531 249 999 998 331 838 171 687 769 876 725 76;
  • 71) 0.307 789 802 551 269 531 249 999 998 331 838 171 687 769 876 725 76 × 2 = 0 + 0.615 579 605 102 539 062 499 999 996 663 676 343 375 539 753 451 52;
  • 72) 0.615 579 605 102 539 062 499 999 996 663 676 343 375 539 753 451 52 × 2 = 1 + 0.231 159 210 205 078 124 999 999 993 327 352 686 751 079 506 903 04;
  • 73) 0.231 159 210 205 078 124 999 999 993 327 352 686 751 079 506 903 04 × 2 = 0 + 0.462 318 420 410 156 249 999 999 986 654 705 373 502 159 013 806 08;
  • 74) 0.462 318 420 410 156 249 999 999 986 654 705 373 502 159 013 806 08 × 2 = 0 + 0.924 636 840 820 312 499 999 999 973 309 410 747 004 318 027 612 16;
  • 75) 0.924 636 840 820 312 499 999 999 973 309 410 747 004 318 027 612 16 × 2 = 1 + 0.849 273 681 640 624 999 999 999 946 618 821 494 008 636 055 224 32;
  • 76) 0.849 273 681 640 624 999 999 999 946 618 821 494 008 636 055 224 32 × 2 = 1 + 0.698 547 363 281 249 999 999 999 893 237 642 988 017 272 110 448 64;
  • 77) 0.698 547 363 281 249 999 999 999 893 237 642 988 017 272 110 448 64 × 2 = 1 + 0.397 094 726 562 499 999 999 999 786 475 285 976 034 544 220 897 28;
  • 78) 0.397 094 726 562 499 999 999 999 786 475 285 976 034 544 220 897 28 × 2 = 0 + 0.794 189 453 124 999 999 999 999 572 950 571 952 069 088 441 794 56;
  • 79) 0.794 189 453 124 999 999 999 999 572 950 571 952 069 088 441 794 56 × 2 = 1 + 0.588 378 906 249 999 999 999 999 145 901 143 904 138 176 883 589 12;
  • 80) 0.588 378 906 249 999 999 999 999 145 901 143 904 138 176 883 589 12 × 2 = 1 + 0.176 757 812 499 999 999 999 998 291 802 287 808 276 353 767 178 24;
  • 81) 0.176 757 812 499 999 999 999 998 291 802 287 808 276 353 767 178 24 × 2 = 0 + 0.353 515 624 999 999 999 999 996 583 604 575 616 552 707 534 356 48;
  • 82) 0.353 515 624 999 999 999 999 996 583 604 575 616 552 707 534 356 48 × 2 = 0 + 0.707 031 249 999 999 999 999 993 167 209 151 233 105 415 068 712 96;
  • 83) 0.707 031 249 999 999 999 999 993 167 209 151 233 105 415 068 712 96 × 2 = 1 + 0.414 062 499 999 999 999 999 986 334 418 302 466 210 830 137 425 92;
  • 84) 0.414 062 499 999 999 999 999 986 334 418 302 466 210 830 137 425 92 × 2 = 0 + 0.828 124 999 999 999 999 999 972 668 836 604 932 421 660 274 851 84;
  • 85) 0.828 124 999 999 999 999 999 972 668 836 604 932 421 660 274 851 84 × 2 = 1 + 0.656 249 999 999 999 999 999 945 337 673 209 864 843 320 549 703 68;
  • 86) 0.656 249 999 999 999 999 999 945 337 673 209 864 843 320 549 703 68 × 2 = 1 + 0.312 499 999 999 999 999 999 890 675 346 419 729 686 641 099 407 36;
  • 87) 0.312 499 999 999 999 999 999 890 675 346 419 729 686 641 099 407 36 × 2 = 0 + 0.624 999 999 999 999 999 999 781 350 692 839 459 373 282 198 814 72;
  • 88) 0.624 999 999 999 999 999 999 781 350 692 839 459 373 282 198 814 72 × 2 = 1 + 0.249 999 999 999 999 999 999 562 701 385 678 918 746 564 397 629 44;
  • 89) 0.249 999 999 999 999 999 999 562 701 385 678 918 746 564 397 629 44 × 2 = 0 + 0.499 999 999 999 999 999 999 125 402 771 357 837 493 128 795 258 88;
  • 90) 0.499 999 999 999 999 999 999 125 402 771 357 837 493 128 795 258 88 × 2 = 0 + 0.999 999 999 999 999 999 998 250 805 542 715 674 986 257 590 517 76;
  • 91) 0.999 999 999 999 999 999 998 250 805 542 715 674 986 257 590 517 76 × 2 = 1 + 0.999 999 999 999 999 999 996 501 611 085 431 349 972 515 181 035 52;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 333 74 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