0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 06 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 06(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 06(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 06.

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 06 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 12;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 12 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 344 24;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 344 24 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 688 48;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 688 48 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 376 96;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 376 96 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 753 92;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 753 92 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 507 84;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 507 84 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 015 68;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 015 68 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 031 36;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 031 36 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 062 72;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 062 72 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 125 44;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 125 44 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 232 250 88;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 232 250 88 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 464 501 76;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 464 501 76 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 929 003 52;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 929 003 52 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 858 007 04;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 858 007 04 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 716 014 08;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 716 014 08 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 432 028 16;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 432 028 16 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 864 056 32;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 864 056 32 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 728 112 64;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 728 112 64 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 456 225 28;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 456 225 28 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 912 450 56;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 912 450 56 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 824 901 12;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 824 901 12 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 649 802 24;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 649 802 24 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 695 299 604 48;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 695 299 604 48 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 390 599 208 96;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 390 599 208 96 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 781 198 417 92;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 781 198 417 92 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 562 396 835 84;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 562 396 835 84 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 124 793 671 68;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 124 793 671 68 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 249 587 343 36;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 249 587 343 36 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 499 174 686 72;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 499 174 686 72 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 998 349 373 44;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 998 349 373 44 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 545 996 698 746 88;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 545 996 698 746 88 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 091 993 397 493 76;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 091 993 397 493 76 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 183 986 794 987 52;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 183 986 794 987 52 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 367 973 589 975 04;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 367 973 589 975 04 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 735 947 179 950 08;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 735 947 179 950 08 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 471 894 359 900 16;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 471 894 359 900 16 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 943 788 719 800 32;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 943 788 719 800 32 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 887 577 439 600 64;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 887 577 439 600 64 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 775 154 879 201 28;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 775 154 879 201 28 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 647 550 309 758 402 56;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 647 550 309 758 402 56 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 295 100 619 516 805 12;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 295 100 619 516 805 12 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 590 201 239 033 610 24;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 590 201 239 033 610 24 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 180 402 478 067 220 48;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 180 402 478 067 220 48 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 360 804 956 134 440 96;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 360 804 956 134 440 96 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 721 609 912 268 881 92;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 721 609 912 268 881 92 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 443 219 824 537 763 84;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 443 219 824 537 763 84 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 886 439 649 075 527 68;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 886 439 649 075 527 68 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 772 879 298 151 055 36;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 772 879 298 151 055 36 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 545 758 596 302 110 72;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 545 758 596 302 110 72 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 091 517 192 604 221 44;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 091 517 192 604 221 44 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 246 183 034 385 208 442 88;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 246 183 034 385 208 442 88 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 492 366 068 770 416 885 76;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 492 366 068 770 416 885 76 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 984 732 137 540 833 771 52;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 984 732 137 540 833 771 52 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 969 464 275 081 667 543 04;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 969 464 275 081 667 543 04 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 938 928 550 163 335 086 08;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 938 928 550 163 335 086 08 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 877 857 100 326 670 172 16;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 877 857 100 326 670 172 16 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 755 714 200 653 340 344 32;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 755 714 200 653 340 344 32 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 511 428 401 306 680 688 64;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 511 428 401 306 680 688 64 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 022 856 802 613 361 377 28;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 022 856 802 613 361 377 28 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 045 713 605 226 722 754 56;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 045 713 605 226 722 754 56 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 252 091 427 210 453 445 509 12;
  • 62) 0.338 491 776 958 107 948 303 222 656 252 091 427 210 453 445 509 12 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 504 182 854 420 906 891 018 24;
  • 63) 0.676 983 553 916 215 896 606 445 312 504 182 854 420 906 891 018 24 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 008 365 708 841 813 782 036 48;
  • 64) 0.353 967 107 832 431 793 212 890 625 008 365 708 841 813 782 036 48 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 016 731 417 683 627 564 072 96;
  • 65) 0.707 934 215 664 863 586 425 781 250 016 731 417 683 627 564 072 96 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 033 462 835 367 255 128 145 92;
  • 66) 0.415 868 431 329 727 172 851 562 500 033 462 835 367 255 128 145 92 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 066 925 670 734 510 256 291 84;
  • 67) 0.831 736 862 659 454 345 703 125 000 066 925 670 734 510 256 291 84 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 133 851 341 469 020 512 583 68;
  • 68) 0.663 473 725 318 908 691 406 250 000 133 851 341 469 020 512 583 68 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 267 702 682 938 041 025 167 36;
  • 69) 0.326 947 450 637 817 382 812 500 000 267 702 682 938 041 025 167 36 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 535 405 365 876 082 050 334 72;
  • 70) 0.653 894 901 275 634 765 625 000 000 535 405 365 876 082 050 334 72 × 2 = 1 + 0.307 789 802 551 269 531 250 000 001 070 810 731 752 164 100 669 44;
  • 71) 0.307 789 802 551 269 531 250 000 001 070 810 731 752 164 100 669 44 × 2 = 0 + 0.615 579 605 102 539 062 500 000 002 141 621 463 504 328 201 338 88;
  • 72) 0.615 579 605 102 539 062 500 000 002 141 621 463 504 328 201 338 88 × 2 = 1 + 0.231 159 210 205 078 125 000 000 004 283 242 927 008 656 402 677 76;
  • 73) 0.231 159 210 205 078 125 000 000 004 283 242 927 008 656 402 677 76 × 2 = 0 + 0.462 318 420 410 156 250 000 000 008 566 485 854 017 312 805 355 52;
  • 74) 0.462 318 420 410 156 250 000 000 008 566 485 854 017 312 805 355 52 × 2 = 0 + 0.924 636 840 820 312 500 000 000 017 132 971 708 034 625 610 711 04;
  • 75) 0.924 636 840 820 312 500 000 000 017 132 971 708 034 625 610 711 04 × 2 = 1 + 0.849 273 681 640 625 000 000 000 034 265 943 416 069 251 221 422 08;
  • 76) 0.849 273 681 640 625 000 000 000 034 265 943 416 069 251 221 422 08 × 2 = 1 + 0.698 547 363 281 250 000 000 000 068 531 886 832 138 502 442 844 16;
  • 77) 0.698 547 363 281 250 000 000 000 068 531 886 832 138 502 442 844 16 × 2 = 1 + 0.397 094 726 562 500 000 000 000 137 063 773 664 277 004 885 688 32;
  • 78) 0.397 094 726 562 500 000 000 000 137 063 773 664 277 004 885 688 32 × 2 = 0 + 0.794 189 453 125 000 000 000 000 274 127 547 328 554 009 771 376 64;
  • 79) 0.794 189 453 125 000 000 000 000 274 127 547 328 554 009 771 376 64 × 2 = 1 + 0.588 378 906 250 000 000 000 000 548 255 094 657 108 019 542 753 28;
  • 80) 0.588 378 906 250 000 000 000 000 548 255 094 657 108 019 542 753 28 × 2 = 1 + 0.176 757 812 500 000 000 000 001 096 510 189 314 216 039 085 506 56;
  • 81) 0.176 757 812 500 000 000 000 001 096 510 189 314 216 039 085 506 56 × 2 = 0 + 0.353 515 625 000 000 000 000 002 193 020 378 628 432 078 171 013 12;
  • 82) 0.353 515 625 000 000 000 000 002 193 020 378 628 432 078 171 013 12 × 2 = 0 + 0.707 031 250 000 000 000 000 004 386 040 757 256 864 156 342 026 24;
  • 83) 0.707 031 250 000 000 000 000 004 386 040 757 256 864 156 342 026 24 × 2 = 1 + 0.414 062 500 000 000 000 000 008 772 081 514 513 728 312 684 052 48;
  • 84) 0.414 062 500 000 000 000 000 008 772 081 514 513 728 312 684 052 48 × 2 = 0 + 0.828 125 000 000 000 000 000 017 544 163 029 027 456 625 368 104 96;
  • 85) 0.828 125 000 000 000 000 000 017 544 163 029 027 456 625 368 104 96 × 2 = 1 + 0.656 250 000 000 000 000 000 035 088 326 058 054 913 250 736 209 92;
  • 86) 0.656 250 000 000 000 000 000 035 088 326 058 054 913 250 736 209 92 × 2 = 1 + 0.312 500 000 000 000 000 000 070 176 652 116 109 826 501 472 419 84;
  • 87) 0.312 500 000 000 000 000 000 070 176 652 116 109 826 501 472 419 84 × 2 = 0 + 0.625 000 000 000 000 000 000 140 353 304 232 219 653 002 944 839 68;
  • 88) 0.625 000 000 000 000 000 000 140 353 304 232 219 653 002 944 839 68 × 2 = 1 + 0.250 000 000 000 000 000 000 280 706 608 464 439 306 005 889 679 36;
  • 89) 0.250 000 000 000 000 000 000 280 706 608 464 439 306 005 889 679 36 × 2 = 0 + 0.500 000 000 000 000 000 000 561 413 216 928 878 612 011 779 358 72;
  • 90) 0.500 000 000 000 000 000 000 561 413 216 928 878 612 011 779 358 72 × 2 = 1 + 0.000 000 000 000 000 000 001 122 826 433 857 757 224 023 558 717 44;
  • 91) 0.000 000 000 000 000 000 001 122 826 433 857 757 224 023 558 717 44 × 2 = 0 + 0.000 000 000 000 000 000 002 245 652 867 715 514 448 047 117 434 88;

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 06(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 06(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 06(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 06 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