0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 86 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 86(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 86(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 86.

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 86 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 72;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 72 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 347 44;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 347 44 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 694 88;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 694 88 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 389 76;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 389 76 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 779 52;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 779 52 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 559 04;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 559 04 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 118 08;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 118 08 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 236 16;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 236 16 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 472 32;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 472 32 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 944 64;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 944 64 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 889 28;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 889 28 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 467 778 56;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 467 778 56 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 935 557 12;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 935 557 12 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 871 114 24;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 871 114 24 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 742 228 48;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 742 228 48 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 484 456 96;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 484 456 96 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 968 913 92;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 968 913 92 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 937 827 84;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 937 827 84 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 875 655 68;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 875 655 68 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 751 311 36;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 751 311 36 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 175 502 622 72;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 175 502 622 72 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 351 005 245 44;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 351 005 245 44 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 702 010 490 88;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 702 010 490 88 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 404 020 981 76;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 404 020 981 76 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 808 041 963 52;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 808 041 963 52 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 616 083 927 04;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 616 083 927 04 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 232 167 854 08;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 232 167 854 08 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 464 335 708 16;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 464 335 708 16 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 928 671 416 32;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 928 671 416 32 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 273 857 342 832 64;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 273 857 342 832 64 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 547 714 685 665 28;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 547 714 685 665 28 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 095 429 371 330 56;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 095 429 371 330 56 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 190 858 742 661 12;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 190 858 742 661 12 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 381 717 485 322 24;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 381 717 485 322 24 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 763 434 970 644 48;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 763 434 970 644 48 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 526 869 941 288 96;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 526 869 941 288 96 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 331 053 739 882 577 92;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 331 053 739 882 577 92 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 662 107 479 765 155 84;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 662 107 479 765 155 84 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 324 214 959 530 311 68;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 324 214 959 530 311 68 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 648 429 919 060 623 36;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 648 429 919 060 623 36 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 296 859 838 121 246 72;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 296 859 838 121 246 72 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 593 719 676 242 493 44;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 593 719 676 242 493 44 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 187 439 352 484 986 88;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 187 439 352 484 986 88 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 374 878 704 969 973 76;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 374 878 704 969 973 76 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 749 757 409 939 947 52;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 749 757 409 939 947 52 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 499 514 819 879 895 04;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 499 514 819 879 895 04 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 999 029 639 759 790 08;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 999 029 639 759 790 08 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 998 059 279 519 580 16;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 998 059 279 519 580 16 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 996 118 559 039 160 32;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 996 118 559 039 160 32 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 992 237 118 078 320 64;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 992 237 118 078 320 64 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 247 984 474 236 156 641 28;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 247 984 474 236 156 641 28 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 495 968 948 472 313 282 56;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 495 968 948 472 313 282 56 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 991 937 896 944 626 565 12;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 991 937 896 944 626 565 12 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 983 875 793 889 253 130 24;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 983 875 793 889 253 130 24 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 967 751 587 778 506 260 48;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 967 751 587 778 506 260 48 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 935 503 175 557 012 520 96;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 935 503 175 557 012 520 96 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 871 006 351 114 025 041 92;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 871 006 351 114 025 041 92 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 742 012 702 228 050 083 84;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 742 012 702 228 050 083 84 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 484 025 404 456 100 167 68;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 484 025 404 456 100 167 68 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 968 050 808 912 200 335 36;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 968 050 808 912 200 335 36 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 253 936 101 617 824 400 670 72;
  • 62) 0.338 491 776 958 107 948 303 222 656 253 936 101 617 824 400 670 72 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 507 872 203 235 648 801 341 44;
  • 63) 0.676 983 553 916 215 896 606 445 312 507 872 203 235 648 801 341 44 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 015 744 406 471 297 602 682 88;
  • 64) 0.353 967 107 832 431 793 212 890 625 015 744 406 471 297 602 682 88 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 031 488 812 942 595 205 365 76;
  • 65) 0.707 934 215 664 863 586 425 781 250 031 488 812 942 595 205 365 76 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 062 977 625 885 190 410 731 52;
  • 66) 0.415 868 431 329 727 172 851 562 500 062 977 625 885 190 410 731 52 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 125 955 251 770 380 821 463 04;
  • 67) 0.831 736 862 659 454 345 703 125 000 125 955 251 770 380 821 463 04 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 251 910 503 540 761 642 926 08;
  • 68) 0.663 473 725 318 908 691 406 250 000 251 910 503 540 761 642 926 08 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 503 821 007 081 523 285 852 16;
  • 69) 0.326 947 450 637 817 382 812 500 000 503 821 007 081 523 285 852 16 × 2 = 0 + 0.653 894 901 275 634 765 625 000 001 007 642 014 163 046 571 704 32;
  • 70) 0.653 894 901 275 634 765 625 000 001 007 642 014 163 046 571 704 32 × 2 = 1 + 0.307 789 802 551 269 531 250 000 002 015 284 028 326 093 143 408 64;
  • 71) 0.307 789 802 551 269 531 250 000 002 015 284 028 326 093 143 408 64 × 2 = 0 + 0.615 579 605 102 539 062 500 000 004 030 568 056 652 186 286 817 28;
  • 72) 0.615 579 605 102 539 062 500 000 004 030 568 056 652 186 286 817 28 × 2 = 1 + 0.231 159 210 205 078 125 000 000 008 061 136 113 304 372 573 634 56;
  • 73) 0.231 159 210 205 078 125 000 000 008 061 136 113 304 372 573 634 56 × 2 = 0 + 0.462 318 420 410 156 250 000 000 016 122 272 226 608 745 147 269 12;
  • 74) 0.462 318 420 410 156 250 000 000 016 122 272 226 608 745 147 269 12 × 2 = 0 + 0.924 636 840 820 312 500 000 000 032 244 544 453 217 490 294 538 24;
  • 75) 0.924 636 840 820 312 500 000 000 032 244 544 453 217 490 294 538 24 × 2 = 1 + 0.849 273 681 640 625 000 000 000 064 489 088 906 434 980 589 076 48;
  • 76) 0.849 273 681 640 625 000 000 000 064 489 088 906 434 980 589 076 48 × 2 = 1 + 0.698 547 363 281 250 000 000 000 128 978 177 812 869 961 178 152 96;
  • 77) 0.698 547 363 281 250 000 000 000 128 978 177 812 869 961 178 152 96 × 2 = 1 + 0.397 094 726 562 500 000 000 000 257 956 355 625 739 922 356 305 92;
  • 78) 0.397 094 726 562 500 000 000 000 257 956 355 625 739 922 356 305 92 × 2 = 0 + 0.794 189 453 125 000 000 000 000 515 912 711 251 479 844 712 611 84;
  • 79) 0.794 189 453 125 000 000 000 000 515 912 711 251 479 844 712 611 84 × 2 = 1 + 0.588 378 906 250 000 000 000 001 031 825 422 502 959 689 425 223 68;
  • 80) 0.588 378 906 250 000 000 000 001 031 825 422 502 959 689 425 223 68 × 2 = 1 + 0.176 757 812 500 000 000 000 002 063 650 845 005 919 378 850 447 36;
  • 81) 0.176 757 812 500 000 000 000 002 063 650 845 005 919 378 850 447 36 × 2 = 0 + 0.353 515 625 000 000 000 000 004 127 301 690 011 838 757 700 894 72;
  • 82) 0.353 515 625 000 000 000 000 004 127 301 690 011 838 757 700 894 72 × 2 = 0 + 0.707 031 250 000 000 000 000 008 254 603 380 023 677 515 401 789 44;
  • 83) 0.707 031 250 000 000 000 000 008 254 603 380 023 677 515 401 789 44 × 2 = 1 + 0.414 062 500 000 000 000 000 016 509 206 760 047 355 030 803 578 88;
  • 84) 0.414 062 500 000 000 000 000 016 509 206 760 047 355 030 803 578 88 × 2 = 0 + 0.828 125 000 000 000 000 000 033 018 413 520 094 710 061 607 157 76;
  • 85) 0.828 125 000 000 000 000 000 033 018 413 520 094 710 061 607 157 76 × 2 = 1 + 0.656 250 000 000 000 000 000 066 036 827 040 189 420 123 214 315 52;
  • 86) 0.656 250 000 000 000 000 000 066 036 827 040 189 420 123 214 315 52 × 2 = 1 + 0.312 500 000 000 000 000 000 132 073 654 080 378 840 246 428 631 04;
  • 87) 0.312 500 000 000 000 000 000 132 073 654 080 378 840 246 428 631 04 × 2 = 0 + 0.625 000 000 000 000 000 000 264 147 308 160 757 680 492 857 262 08;
  • 88) 0.625 000 000 000 000 000 000 264 147 308 160 757 680 492 857 262 08 × 2 = 1 + 0.250 000 000 000 000 000 000 528 294 616 321 515 360 985 714 524 16;
  • 89) 0.250 000 000 000 000 000 000 528 294 616 321 515 360 985 714 524 16 × 2 = 0 + 0.500 000 000 000 000 000 001 056 589 232 643 030 721 971 429 048 32;
  • 90) 0.500 000 000 000 000 000 001 056 589 232 643 030 721 971 429 048 32 × 2 = 1 + 0.000 000 000 000 000 000 002 113 178 465 286 061 443 942 858 096 64;
  • 91) 0.000 000 000 000 000 000 002 113 178 465 286 061 443 942 858 096 64 × 2 = 0 + 0.000 000 000 000 000 000 004 226 356 930 572 122 887 885 716 193 28;

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 86(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 86(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 86(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 86 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