0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 792;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 792 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 584;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 584 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 168;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 168 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 336;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 358 336 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 716 672;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 716 672 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 433 344;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 433 344 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 866 688;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 866 688 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 733 376;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 733 376 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 466 752;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 466 752 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 933 504;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 614 933 504 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 867 008;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 229 867 008 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 734 016;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 459 734 016 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 468 032;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 919 468 032 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 838 936 064;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 838 936 064 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 677 872 128;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 677 872 128 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 355 744 256;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 355 744 256 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 711 488 512;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 711 488 512 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 422 977 024;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 422 977 024 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 845 954 048;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 845 954 048 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 691 908 096;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 691 908 096 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 383 816 192;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 383 816 192 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 767 632 384;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 342 767 632 384 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 535 264 768;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 685 535 264 768 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 070 529 536;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 371 070 529 536 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 141 059 072;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 742 141 059 072 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 484 282 118 144;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 484 282 118 144 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 968 564 236 288;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 968 564 236 288 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 937 128 472 576;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 937 128 472 576 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 874 256 945 152;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 874 256 945 152 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 748 513 890 304;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 748 513 890 304 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 497 027 780 608;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 497 027 780 608 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 086 994 055 561 216;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 086 994 055 561 216 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 173 988 111 122 432;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 173 988 111 122 432 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 347 976 222 244 864;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 347 976 222 244 864 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 695 952 444 489 728;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 695 952 444 489 728 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 391 904 888 979 456;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 391 904 888 979 456 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 783 809 777 958 912;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 783 809 777 958 912 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 567 619 555 917 824;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 567 619 555 917 824 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 135 239 111 835 648;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 135 239 111 835 648 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 270 478 223 671 296;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 270 478 223 671 296 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 540 956 447 342 592;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 540 956 447 342 592 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 081 912 894 685 184;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 081 912 894 685 184 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 163 825 789 370 368;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 163 825 789 370 368 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 340 327 651 578 740 736;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 340 327 651 578 740 736 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 680 655 303 157 481 472;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 680 655 303 157 481 472 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 361 310 606 314 962 944;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 361 310 606 314 962 944 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 722 621 212 629 925 888;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 722 621 212 629 925 888 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 445 242 425 259 851 776;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 445 242 425 259 851 776 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 890 484 850 519 703 552;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 890 484 850 519 703 552 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 780 969 701 039 407 104;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 780 969 701 039 407 104 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 561 939 402 078 814 208;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 561 939 402 078 814 208 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 123 878 804 157 628 416;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 123 878 804 157 628 416 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 974 247 757 608 315 256 832;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 974 247 757 608 315 256 832 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 948 495 515 216 630 513 664;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 948 495 515 216 630 513 664 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 896 991 030 433 261 027 328;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 896 991 030 433 261 027 328 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 793 982 060 866 522 054 656;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 793 982 060 866 522 054 656 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 587 964 121 733 044 109 312;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 587 964 121 733 044 109 312 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 175 928 243 466 088 218 624;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 175 928 243 466 088 218 624 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 351 856 486 932 176 437 248;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 351 856 486 932 176 437 248 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 703 712 973 864 352 874 496;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 703 712 973 864 352 874 496 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 407 425 947 728 705 748 992;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 407 425 947 728 705 748 992 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 498 814 851 895 457 411 497 984;
  • 63) 0.676 983 553 916 215 896 606 445 312 498 814 851 895 457 411 497 984 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 997 629 703 790 914 822 995 968;
  • 64) 0.353 967 107 832 431 793 212 890 624 997 629 703 790 914 822 995 968 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 995 259 407 581 829 645 991 936;
  • 65) 0.707 934 215 664 863 586 425 781 249 995 259 407 581 829 645 991 936 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 990 518 815 163 659 291 983 872;
  • 66) 0.415 868 431 329 727 172 851 562 499 990 518 815 163 659 291 983 872 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 981 037 630 327 318 583 967 744;
  • 67) 0.831 736 862 659 454 345 703 124 999 981 037 630 327 318 583 967 744 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 962 075 260 654 637 167 935 488;
  • 68) 0.663 473 725 318 908 691 406 249 999 962 075 260 654 637 167 935 488 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 924 150 521 309 274 335 870 976;
  • 69) 0.326 947 450 637 817 382 812 499 999 924 150 521 309 274 335 870 976 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 848 301 042 618 548 671 741 952;
  • 70) 0.653 894 901 275 634 765 624 999 999 848 301 042 618 548 671 741 952 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 696 602 085 237 097 343 483 904;
  • 71) 0.307 789 802 551 269 531 249 999 999 696 602 085 237 097 343 483 904 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 393 204 170 474 194 686 967 808;
  • 72) 0.615 579 605 102 539 062 499 999 999 393 204 170 474 194 686 967 808 × 2 = 1 + 0.231 159 210 205 078 124 999 999 998 786 408 340 948 389 373 935 616;
  • 73) 0.231 159 210 205 078 124 999 999 998 786 408 340 948 389 373 935 616 × 2 = 0 + 0.462 318 420 410 156 249 999 999 997 572 816 681 896 778 747 871 232;
  • 74) 0.462 318 420 410 156 249 999 999 997 572 816 681 896 778 747 871 232 × 2 = 0 + 0.924 636 840 820 312 499 999 999 995 145 633 363 793 557 495 742 464;
  • 75) 0.924 636 840 820 312 499 999 999 995 145 633 363 793 557 495 742 464 × 2 = 1 + 0.849 273 681 640 624 999 999 999 990 291 266 727 587 114 991 484 928;
  • 76) 0.849 273 681 640 624 999 999 999 990 291 266 727 587 114 991 484 928 × 2 = 1 + 0.698 547 363 281 249 999 999 999 980 582 533 455 174 229 982 969 856;
  • 77) 0.698 547 363 281 249 999 999 999 980 582 533 455 174 229 982 969 856 × 2 = 1 + 0.397 094 726 562 499 999 999 999 961 165 066 910 348 459 965 939 712;
  • 78) 0.397 094 726 562 499 999 999 999 961 165 066 910 348 459 965 939 712 × 2 = 0 + 0.794 189 453 124 999 999 999 999 922 330 133 820 696 919 931 879 424;
  • 79) 0.794 189 453 124 999 999 999 999 922 330 133 820 696 919 931 879 424 × 2 = 1 + 0.588 378 906 249 999 999 999 999 844 660 267 641 393 839 863 758 848;
  • 80) 0.588 378 906 249 999 999 999 999 844 660 267 641 393 839 863 758 848 × 2 = 1 + 0.176 757 812 499 999 999 999 999 689 320 535 282 787 679 727 517 696;
  • 81) 0.176 757 812 499 999 999 999 999 689 320 535 282 787 679 727 517 696 × 2 = 0 + 0.353 515 624 999 999 999 999 999 378 641 070 565 575 359 455 035 392;
  • 82) 0.353 515 624 999 999 999 999 999 378 641 070 565 575 359 455 035 392 × 2 = 0 + 0.707 031 249 999 999 999 999 998 757 282 141 131 150 718 910 070 784;
  • 83) 0.707 031 249 999 999 999 999 998 757 282 141 131 150 718 910 070 784 × 2 = 1 + 0.414 062 499 999 999 999 999 997 514 564 282 262 301 437 820 141 568;
  • 84) 0.414 062 499 999 999 999 999 997 514 564 282 262 301 437 820 141 568 × 2 = 0 + 0.828 124 999 999 999 999 999 995 029 128 564 524 602 875 640 283 136;
  • 85) 0.828 124 999 999 999 999 999 995 029 128 564 524 602 875 640 283 136 × 2 = 1 + 0.656 249 999 999 999 999 999 990 058 257 129 049 205 751 280 566 272;
  • 86) 0.656 249 999 999 999 999 999 990 058 257 129 049 205 751 280 566 272 × 2 = 1 + 0.312 499 999 999 999 999 999 980 116 514 258 098 411 502 561 132 544;
  • 87) 0.312 499 999 999 999 999 999 980 116 514 258 098 411 502 561 132 544 × 2 = 0 + 0.624 999 999 999 999 999 999 960 233 028 516 196 823 005 122 265 088;
  • 88) 0.624 999 999 999 999 999 999 960 233 028 516 196 823 005 122 265 088 × 2 = 1 + 0.249 999 999 999 999 999 999 920 466 057 032 393 646 010 244 530 176;
  • 89) 0.249 999 999 999 999 999 999 920 466 057 032 393 646 010 244 530 176 × 2 = 0 + 0.499 999 999 999 999 999 999 840 932 114 064 787 292 020 489 060 352;
  • 90) 0.499 999 999 999 999 999 999 840 932 114 064 787 292 020 489 060 352 × 2 = 0 + 0.999 999 999 999 999 999 999 681 864 228 129 574 584 040 978 120 704;
  • 91) 0.999 999 999 999 999 999 999 681 864 228 129 574 584 040 978 120 704 × 2 = 1 + 0.999 999 999 999 999 999 999 363 728 456 259 149 168 081 956 241 408;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 896 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