0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 056;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 056 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 112;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 112 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 224;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 224 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 448;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 448 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 896;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 896 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 792;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 792 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 584;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 584 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 168;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 168 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 870 336;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 870 336 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 740 672;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 740 672 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 481 344;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 481 344 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 962 688;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 962 688 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 925 376;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 925 376 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 850 752;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 850 752 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 701 504;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 701 504 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 403 008;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 403 008 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 446 806 016;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 446 806 016 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 893 612 032;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 893 612 032 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 787 224 064;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 787 224 064 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 574 448 128;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 574 448 128 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 148 896 256;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 148 896 256 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 297 792 512;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 297 792 512 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 595 585 024;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 595 585 024 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 191 170 048;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 191 170 048 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 382 340 096;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 382 340 096 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 764 680 192;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 764 680 192 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 529 360 384;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 529 360 384 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 058 720 768;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 058 720 768 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 342 117 441 536;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 342 117 441 536 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 684 234 883 072;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 684 234 883 072 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 368 469 766 144;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 368 469 766 144 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 736 939 532 288;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 736 939 532 288 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 473 879 064 576;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 473 879 064 576 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 947 758 129 152;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 947 758 129 152 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 895 516 258 304;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 895 516 258 304 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 791 032 516 608;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 791 032 516 608 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 582 065 033 216;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 582 065 033 216 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 164 130 066 432;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 164 130 066 432 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 774 328 260 132 864;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 774 328 260 132 864 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 548 656 520 265 728;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 548 656 520 265 728 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 097 313 040 531 456;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 097 313 040 531 456 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 194 626 081 062 912;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 194 626 081 062 912 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 389 252 162 125 824;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 389 252 162 125 824 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 778 504 324 251 648;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 778 504 324 251 648 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 557 008 648 503 296;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 557 008 648 503 296 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 114 017 297 006 592;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 114 017 297 006 592 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 228 034 594 013 184;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 228 034 594 013 184 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 456 069 188 026 368;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 456 069 188 026 368 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 248 912 138 376 052 736;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 248 912 138 376 052 736 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 497 824 276 752 105 472;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 497 824 276 752 105 472 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 995 648 553 504 210 944;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 995 648 553 504 210 944 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 991 297 107 008 421 888;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 991 297 107 008 421 888 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 982 594 214 016 843 776;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 982 594 214 016 843 776 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 965 188 428 033 687 552;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 965 188 428 033 687 552 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 930 376 856 067 375 104;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 930 376 856 067 375 104 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 860 753 712 134 750 208;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 860 753 712 134 750 208 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 721 507 424 269 500 416;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 721 507 424 269 500 416 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 443 014 848 539 000 832;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 443 014 848 539 000 832 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 998 886 029 697 078 001 664;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 998 886 029 697 078 001 664 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 997 772 059 394 156 003 328;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 997 772 059 394 156 003 328 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 995 544 118 788 312 006 656;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 995 544 118 788 312 006 656 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 991 088 237 576 624 013 312;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 991 088 237 576 624 013 312 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 982 176 475 153 248 026 624;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 982 176 475 153 248 026 624 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 964 352 950 306 496 053 248;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 964 352 950 306 496 053 248 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 928 705 900 612 992 106 496;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 928 705 900 612 992 106 496 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 857 411 801 225 984 212 992;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 857 411 801 225 984 212 992 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 714 823 602 451 968 425 984;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 714 823 602 451 968 425 984 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 429 647 204 903 936 851 968;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 429 647 204 903 936 851 968 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 998 859 294 409 807 873 703 936;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 998 859 294 409 807 873 703 936 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 997 718 588 819 615 747 407 872;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 997 718 588 819 615 747 407 872 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 995 437 177 639 231 494 815 744;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 995 437 177 639 231 494 815 744 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 990 874 355 278 462 989 631 488;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 990 874 355 278 462 989 631 488 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 981 748 710 556 925 979 262 976;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 981 748 710 556 925 979 262 976 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 963 497 421 113 851 958 525 952;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 963 497 421 113 851 958 525 952 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 926 994 842 227 703 917 051 904;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 926 994 842 227 703 917 051 904 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 853 989 684 455 407 834 103 808;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 853 989 684 455 407 834 103 808 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 707 979 368 910 815 668 207 616;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 707 979 368 910 815 668 207 616 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 415 958 737 821 631 336 415 232;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 415 958 737 821 631 336 415 232 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 998 831 917 475 643 262 672 830 464;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 998 831 917 475 643 262 672 830 464 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 997 663 834 951 286 525 345 660 928;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 997 663 834 951 286 525 345 660 928 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 995 327 669 902 573 050 691 321 856;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 995 327 669 902 573 050 691 321 856 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 990 655 339 805 146 101 382 643 712;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 990 655 339 805 146 101 382 643 712 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 981 310 679 610 292 202 765 287 424;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 981 310 679 610 292 202 765 287 424 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 962 621 359 220 584 405 530 574 848;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 962 621 359 220 584 405 530 574 848 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 925 242 718 441 168 811 061 149 696;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 925 242 718 441 168 811 061 149 696 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 850 485 436 882 337 622 122 299 392;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 850 485 436 882 337 622 122 299 392 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 700 970 873 764 675 244 244 598 784;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 700 970 873 764 675 244 244 598 784 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 401 941 747 529 350 488 489 197 568;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 401 941 747 529 350 488 489 197 568 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 998 803 883 495 058 700 976 978 395 136;
  • 90) 0.499 999 999 999 999 999 999 999 999 998 803 883 495 058 700 976 978 395 136 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 997 607 766 990 117 401 953 956 790 272;
  • 91) 0.999 999 999 999 999 999 999 999 999 997 607 766 990 117 401 953 956 790 272 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 995 215 533 980 234 803 907 913 580 544;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 028 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