0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 964;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 669 964 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 928;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 339 928 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 856;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 679 856 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 712;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 359 712 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 719 424;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 719 424 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 438 848;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 438 848 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 877 696;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 877 696 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 755 392;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 755 392 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 510 784;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 510 784 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 021 568;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 021 568 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 043 136;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 043 136 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 086 272;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 086 272 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 172 544;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 172 544 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 345 088;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 840 345 088 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 680 690 176;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 680 690 176 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 361 380 352;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 361 380 352 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 722 760 704;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 722 760 704 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 445 521 408;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 445 521 408 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 891 042 816;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 891 042 816 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 782 085 632;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 782 085 632 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 564 171 264;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 564 171 264 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 128 342 528;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 128 342 528 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 256 685 056;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 256 685 056 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 513 370 112;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 372 513 370 112 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 745 026 740 224;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 745 026 740 224 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 490 053 480 448;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 490 053 480 448 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 980 106 960 896;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 980 106 960 896 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 960 213 921 792;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 960 213 921 792 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 920 427 843 584;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 920 427 843 584 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 840 855 687 168;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 840 855 687 168 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 681 711 374 336;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 681 711 374 336 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 363 422 748 672;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 363 422 748 672 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 174 726 845 497 344;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 174 726 845 497 344 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 349 453 690 994 688;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 349 453 690 994 688 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 698 907 381 989 376;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 698 907 381 989 376 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 397 814 763 978 752;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 397 814 763 978 752 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 795 629 527 957 504;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 795 629 527 957 504 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 591 259 055 915 008;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 591 259 055 915 008 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 182 518 111 830 016;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 182 518 111 830 016 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 365 036 223 660 032;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 365 036 223 660 032 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 730 072 447 320 064;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 730 072 447 320 064 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 460 144 894 640 128;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 460 144 894 640 128 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 170 920 289 789 280 256;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 170 920 289 789 280 256 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 341 840 579 578 560 512;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 341 840 579 578 560 512 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 683 681 159 157 121 024;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 683 681 159 157 121 024 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 367 362 318 314 242 048;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 367 362 318 314 242 048 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 734 724 636 628 484 096;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 734 724 636 628 484 096 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 469 449 273 256 968 192;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 469 449 273 256 968 192 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 938 898 546 513 936 384;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 938 898 546 513 936 384 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 877 797 093 027 872 768;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 877 797 093 027 872 768 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 755 594 186 055 745 536;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 755 594 186 055 745 536 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 511 188 372 111 491 072;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 511 188 372 111 491 072 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 022 376 744 222 982 144;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 022 376 744 222 982 144 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 950 044 753 488 445 964 288;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 950 044 753 488 445 964 288 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 900 089 506 976 891 928 576;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 900 089 506 976 891 928 576 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 800 179 013 953 783 857 152;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 800 179 013 953 783 857 152 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 600 358 027 907 567 714 304;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 600 358 027 907 567 714 304 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 200 716 055 815 135 428 608;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 200 716 055 815 135 428 608 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 401 432 111 630 270 857 216;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 401 432 111 630 270 857 216 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 802 864 223 260 541 714 432;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 802 864 223 260 541 714 432 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 605 728 446 521 083 428 864;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 605 728 446 521 083 428 864 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 211 456 893 042 166 857 728;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 211 456 893 042 166 857 728 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 422 913 786 084 333 715 456;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 422 913 786 084 333 715 456 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 996 845 827 572 168 667 430 912;
  • 65) 0.707 934 215 664 863 586 425 781 249 996 845 827 572 168 667 430 912 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 993 691 655 144 337 334 861 824;
  • 66) 0.415 868 431 329 727 172 851 562 499 993 691 655 144 337 334 861 824 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 987 383 310 288 674 669 723 648;
  • 67) 0.831 736 862 659 454 345 703 124 999 987 383 310 288 674 669 723 648 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 974 766 620 577 349 339 447 296;
  • 68) 0.663 473 725 318 908 691 406 249 999 974 766 620 577 349 339 447 296 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 949 533 241 154 698 678 894 592;
  • 69) 0.326 947 450 637 817 382 812 499 999 949 533 241 154 698 678 894 592 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 899 066 482 309 397 357 789 184;
  • 70) 0.653 894 901 275 634 765 624 999 999 899 066 482 309 397 357 789 184 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 798 132 964 618 794 715 578 368;
  • 71) 0.307 789 802 551 269 531 249 999 999 798 132 964 618 794 715 578 368 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 596 265 929 237 589 431 156 736;
  • 72) 0.615 579 605 102 539 062 499 999 999 596 265 929 237 589 431 156 736 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 192 531 858 475 178 862 313 472;
  • 73) 0.231 159 210 205 078 124 999 999 999 192 531 858 475 178 862 313 472 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 385 063 716 950 357 724 626 944;
  • 74) 0.462 318 420 410 156 249 999 999 998 385 063 716 950 357 724 626 944 × 2 = 0 + 0.924 636 840 820 312 499 999 999 996 770 127 433 900 715 449 253 888;
  • 75) 0.924 636 840 820 312 499 999 999 996 770 127 433 900 715 449 253 888 × 2 = 1 + 0.849 273 681 640 624 999 999 999 993 540 254 867 801 430 898 507 776;
  • 76) 0.849 273 681 640 624 999 999 999 993 540 254 867 801 430 898 507 776 × 2 = 1 + 0.698 547 363 281 249 999 999 999 987 080 509 735 602 861 797 015 552;
  • 77) 0.698 547 363 281 249 999 999 999 987 080 509 735 602 861 797 015 552 × 2 = 1 + 0.397 094 726 562 499 999 999 999 974 161 019 471 205 723 594 031 104;
  • 78) 0.397 094 726 562 499 999 999 999 974 161 019 471 205 723 594 031 104 × 2 = 0 + 0.794 189 453 124 999 999 999 999 948 322 038 942 411 447 188 062 208;
  • 79) 0.794 189 453 124 999 999 999 999 948 322 038 942 411 447 188 062 208 × 2 = 1 + 0.588 378 906 249 999 999 999 999 896 644 077 884 822 894 376 124 416;
  • 80) 0.588 378 906 249 999 999 999 999 896 644 077 884 822 894 376 124 416 × 2 = 1 + 0.176 757 812 499 999 999 999 999 793 288 155 769 645 788 752 248 832;
  • 81) 0.176 757 812 499 999 999 999 999 793 288 155 769 645 788 752 248 832 × 2 = 0 + 0.353 515 624 999 999 999 999 999 586 576 311 539 291 577 504 497 664;
  • 82) 0.353 515 624 999 999 999 999 999 586 576 311 539 291 577 504 497 664 × 2 = 0 + 0.707 031 249 999 999 999 999 999 173 152 623 078 583 155 008 995 328;
  • 83) 0.707 031 249 999 999 999 999 999 173 152 623 078 583 155 008 995 328 × 2 = 1 + 0.414 062 499 999 999 999 999 998 346 305 246 157 166 310 017 990 656;
  • 84) 0.414 062 499 999 999 999 999 998 346 305 246 157 166 310 017 990 656 × 2 = 0 + 0.828 124 999 999 999 999 999 996 692 610 492 314 332 620 035 981 312;
  • 85) 0.828 124 999 999 999 999 999 996 692 610 492 314 332 620 035 981 312 × 2 = 1 + 0.656 249 999 999 999 999 999 993 385 220 984 628 665 240 071 962 624;
  • 86) 0.656 249 999 999 999 999 999 993 385 220 984 628 665 240 071 962 624 × 2 = 1 + 0.312 499 999 999 999 999 999 986 770 441 969 257 330 480 143 925 248;
  • 87) 0.312 499 999 999 999 999 999 986 770 441 969 257 330 480 143 925 248 × 2 = 0 + 0.624 999 999 999 999 999 999 973 540 883 938 514 660 960 287 850 496;
  • 88) 0.624 999 999 999 999 999 999 973 540 883 938 514 660 960 287 850 496 × 2 = 1 + 0.249 999 999 999 999 999 999 947 081 767 877 029 321 920 575 700 992;
  • 89) 0.249 999 999 999 999 999 999 947 081 767 877 029 321 920 575 700 992 × 2 = 0 + 0.499 999 999 999 999 999 999 894 163 535 754 058 643 841 151 401 984;
  • 90) 0.499 999 999 999 999 999 999 894 163 535 754 058 643 841 151 401 984 × 2 = 0 + 0.999 999 999 999 999 999 999 788 327 071 508 117 287 682 302 803 968;
  • 91) 0.999 999 999 999 999 999 999 788 327 071 508 117 287 682 302 803 968 × 2 = 1 + 0.999 999 999 999 999 999 999 576 654 143 016 234 575 364 605 607 936;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 334 982 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