0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 074;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 074 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 148;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 148 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 296;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 680 296 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 592;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 360 592 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 721 184;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 721 184 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 442 368;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 442 368 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 884 736;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 884 736 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 769 472;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 769 472 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 538 944;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 538 944 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 077 888;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 077 888 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 155 776;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 155 776 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 311 552;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 311 552 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 623 104;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 920 623 104 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 246 208;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 841 246 208 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 682 492 416;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 682 492 416 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 364 984 832;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 364 984 832 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 729 969 664;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 729 969 664 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 459 939 328;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 459 939 328 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 919 878 656;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 919 878 656 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 839 757 312;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 839 757 312 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 679 514 624;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 679 514 624 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 359 029 248;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 359 029 248 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 718 058 496;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 686 718 058 496 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 436 116 992;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 373 436 116 992 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 746 872 233 984;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 746 872 233 984 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 493 744 467 968;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 493 744 467 968 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 987 488 935 936;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 987 488 935 936 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 974 977 871 872;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 974 977 871 872 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 635 949 955 743 744;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 635 949 955 743 744 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 271 899 911 487 488;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 271 899 911 487 488 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 543 799 822 974 976;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 543 799 822 974 976 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 087 599 645 949 952;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 087 599 645 949 952 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 175 199 291 899 904;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 175 199 291 899 904 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 350 398 583 799 808;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 350 398 583 799 808 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 700 797 167 599 616;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 700 797 167 599 616 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 401 594 335 199 232;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 401 594 335 199 232 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 803 188 670 398 464;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 803 188 670 398 464 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 606 377 340 796 928;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 606 377 340 796 928 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 212 754 681 593 856;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 212 754 681 593 856 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 425 509 363 187 712;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 425 509 363 187 712 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 292 851 018 726 375 424;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 292 851 018 726 375 424 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 585 702 037 452 750 848;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 585 702 037 452 750 848 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 171 404 074 905 501 696;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 171 404 074 905 501 696 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 342 808 149 811 003 392;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 342 808 149 811 003 392 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 685 616 299 622 006 784;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 685 616 299 622 006 784 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 371 232 599 244 013 568;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 371 232 599 244 013 568 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 742 465 198 488 027 136;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 742 465 198 488 027 136 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 484 930 396 976 054 272;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 484 930 396 976 054 272 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 810 969 860 793 952 108 544;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 810 969 860 793 952 108 544 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 621 939 721 587 904 217 088;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 621 939 721 587 904 217 088 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 243 879 443 175 808 434 176;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 243 879 443 175 808 434 176 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 487 758 886 351 616 868 352;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 487 758 886 351 616 868 352 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 975 517 772 703 233 736 704;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 975 517 772 703 233 736 704 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 951 035 545 406 467 473 408;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 951 035 545 406 467 473 408 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 902 071 090 812 934 946 816;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 902 071 090 812 934 946 816 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 804 142 181 625 869 893 632;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 804 142 181 625 869 893 632 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 608 284 363 251 739 787 264;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 608 284 363 251 739 787 264 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 216 568 726 503 479 574 528;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 216 568 726 503 479 574 528 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 433 137 453 006 959 149 056;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 433 137 453 006 959 149 056 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 866 274 906 013 918 298 112;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 866 274 906 013 918 298 112 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 732 549 812 027 836 596 224;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 732 549 812 027 836 596 224 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 465 099 624 055 673 192 448;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 465 099 624 055 673 192 448 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 998 930 199 248 111 346 384 896;
  • 64) 0.353 967 107 832 431 793 212 890 624 998 930 199 248 111 346 384 896 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 997 860 398 496 222 692 769 792;
  • 65) 0.707 934 215 664 863 586 425 781 249 997 860 398 496 222 692 769 792 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 995 720 796 992 445 385 539 584;
  • 66) 0.415 868 431 329 727 172 851 562 499 995 720 796 992 445 385 539 584 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 991 441 593 984 890 771 079 168;
  • 67) 0.831 736 862 659 454 345 703 124 999 991 441 593 984 890 771 079 168 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 982 883 187 969 781 542 158 336;
  • 68) 0.663 473 725 318 908 691 406 249 999 982 883 187 969 781 542 158 336 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 965 766 375 939 563 084 316 672;
  • 69) 0.326 947 450 637 817 382 812 499 999 965 766 375 939 563 084 316 672 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 931 532 751 879 126 168 633 344;
  • 70) 0.653 894 901 275 634 765 624 999 999 931 532 751 879 126 168 633 344 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 863 065 503 758 252 337 266 688;
  • 71) 0.307 789 802 551 269 531 249 999 999 863 065 503 758 252 337 266 688 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 726 131 007 516 504 674 533 376;
  • 72) 0.615 579 605 102 539 062 499 999 999 726 131 007 516 504 674 533 376 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 452 262 015 033 009 349 066 752;
  • 73) 0.231 159 210 205 078 124 999 999 999 452 262 015 033 009 349 066 752 × 2 = 0 + 0.462 318 420 410 156 249 999 999 998 904 524 030 066 018 698 133 504;
  • 74) 0.462 318 420 410 156 249 999 999 998 904 524 030 066 018 698 133 504 × 2 = 0 + 0.924 636 840 820 312 499 999 999 997 809 048 060 132 037 396 267 008;
  • 75) 0.924 636 840 820 312 499 999 999 997 809 048 060 132 037 396 267 008 × 2 = 1 + 0.849 273 681 640 624 999 999 999 995 618 096 120 264 074 792 534 016;
  • 76) 0.849 273 681 640 624 999 999 999 995 618 096 120 264 074 792 534 016 × 2 = 1 + 0.698 547 363 281 249 999 999 999 991 236 192 240 528 149 585 068 032;
  • 77) 0.698 547 363 281 249 999 999 999 991 236 192 240 528 149 585 068 032 × 2 = 1 + 0.397 094 726 562 499 999 999 999 982 472 384 481 056 299 170 136 064;
  • 78) 0.397 094 726 562 499 999 999 999 982 472 384 481 056 299 170 136 064 × 2 = 0 + 0.794 189 453 124 999 999 999 999 964 944 768 962 112 598 340 272 128;
  • 79) 0.794 189 453 124 999 999 999 999 964 944 768 962 112 598 340 272 128 × 2 = 1 + 0.588 378 906 249 999 999 999 999 929 889 537 924 225 196 680 544 256;
  • 80) 0.588 378 906 249 999 999 999 999 929 889 537 924 225 196 680 544 256 × 2 = 1 + 0.176 757 812 499 999 999 999 999 859 779 075 848 450 393 361 088 512;
  • 81) 0.176 757 812 499 999 999 999 999 859 779 075 848 450 393 361 088 512 × 2 = 0 + 0.353 515 624 999 999 999 999 999 719 558 151 696 900 786 722 177 024;
  • 82) 0.353 515 624 999 999 999 999 999 719 558 151 696 900 786 722 177 024 × 2 = 0 + 0.707 031 249 999 999 999 999 999 439 116 303 393 801 573 444 354 048;
  • 83) 0.707 031 249 999 999 999 999 999 439 116 303 393 801 573 444 354 048 × 2 = 1 + 0.414 062 499 999 999 999 999 998 878 232 606 787 603 146 888 708 096;
  • 84) 0.414 062 499 999 999 999 999 998 878 232 606 787 603 146 888 708 096 × 2 = 0 + 0.828 124 999 999 999 999 999 997 756 465 213 575 206 293 777 416 192;
  • 85) 0.828 124 999 999 999 999 999 997 756 465 213 575 206 293 777 416 192 × 2 = 1 + 0.656 249 999 999 999 999 999 995 512 930 427 150 412 587 554 832 384;
  • 86) 0.656 249 999 999 999 999 999 995 512 930 427 150 412 587 554 832 384 × 2 = 1 + 0.312 499 999 999 999 999 999 991 025 860 854 300 825 175 109 664 768;
  • 87) 0.312 499 999 999 999 999 999 991 025 860 854 300 825 175 109 664 768 × 2 = 0 + 0.624 999 999 999 999 999 999 982 051 721 708 601 650 350 219 329 536;
  • 88) 0.624 999 999 999 999 999 999 982 051 721 708 601 650 350 219 329 536 × 2 = 1 + 0.249 999 999 999 999 999 999 964 103 443 417 203 300 700 438 659 072;
  • 89) 0.249 999 999 999 999 999 999 964 103 443 417 203 300 700 438 659 072 × 2 = 0 + 0.499 999 999 999 999 999 999 928 206 886 834 406 601 400 877 318 144;
  • 90) 0.499 999 999 999 999 999 999 928 206 886 834 406 601 400 877 318 144 × 2 = 0 + 0.999 999 999 999 999 999 999 856 413 773 668 813 202 801 754 636 288;
  • 91) 0.999 999 999 999 999 999 999 856 413 773 668 813 202 801 754 636 288 × 2 = 1 + 0.999 999 999 999 999 999 999 712 827 547 337 626 405 603 509 272 576;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 037 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