0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 934 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 934 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 869 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 869 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 738 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 738 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 476 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 476 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 894 953 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 894 953 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 789 907 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 789 907 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 579 814 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 579 814 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 159 628 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 159 628 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 319 257 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 319 257 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 638 515 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 638 515 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 277 030 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 277 030 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 554 060 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 554 060 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 108 121 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 108 121 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 216 243 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 216 243 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 432 486 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 432 486 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 584 864 972 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 584 864 972 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 169 729 945 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 169 729 945 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 339 459 891 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 339 459 891 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 678 919 782 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 678 919 782 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 357 839 564 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 357 839 564 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 715 679 129 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 715 679 129 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 431 358 259 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 431 358 259 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 862 716 518 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 862 716 518 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 725 433 036 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 725 433 036 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 450 866 073 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 450 866 073 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 526 901 732 147 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 526 901 732 147 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 053 803 464 294 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 053 803 464 294 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 107 606 928 588 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 107 606 928 588 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 215 213 857 177 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 215 213 857 177 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 430 427 714 355 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 430 427 714 355 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 860 855 428 710 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 860 855 428 710 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 721 710 857 420 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 721 710 857 420 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 443 421 714 841 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 443 421 714 841 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 886 843 429 683 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 390 886 843 429 683 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 773 686 859 366 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 781 773 686 859 366 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 547 373 718 732 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 563 547 373 718 732 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 127 094 747 437 465 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 127 094 747 437 465 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 254 189 494 874 931 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 254 189 494 874 931 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 508 378 989 749 862 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 508 378 989 749 862 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 016 757 979 499 724 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 016 757 979 499 724 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 033 515 958 999 449 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 033 515 958 999 449 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 067 031 917 998 899 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 067 031 917 998 899 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 134 063 835 997 798 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 134 063 835 997 798 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 268 127 671 995 596 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 268 127 671 995 596 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 696 536 255 343 991 193 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 696 536 255 343 991 193 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 393 072 510 687 982 387 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 393 072 510 687 982 387 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 786 145 021 375 964 774 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 786 145 021 375 964 774 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 572 290 042 751 929 548 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 572 290 042 751 929 548 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 144 580 085 503 859 097 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 144 580 085 503 859 097 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 289 160 171 007 718 195 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 289 160 171 007 718 195 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 578 320 342 015 436 390 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 578 320 342 015 436 390 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 156 640 684 030 872 780 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 156 640 684 030 872 780 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 313 281 368 061 745 561 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 313 281 368 061 745 561 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 626 562 736 123 491 123 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 626 562 736 123 491 123 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 253 125 472 246 982 246 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 253 125 472 246 982 246 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 498 506 250 944 493 964 492 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 498 506 250 944 493 964 492 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 997 012 501 888 987 928 985 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 997 012 501 888 987 928 985 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 994 025 003 777 975 857 971 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 994 025 003 777 975 857 971 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 988 050 007 555 951 715 942 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 988 050 007 555 951 715 942 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 976 100 015 111 903 431 884 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 976 100 015 111 903 431 884 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 952 200 030 223 806 863 769 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 952 200 030 223 806 863 769 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 904 400 060 447 613 727 539 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 904 400 060 447 613 727 539 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 808 800 120 895 227 455 078 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 808 800 120 895 227 455 078 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 617 600 241 790 454 910 156 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 617 600 241 790 454 910 156 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 235 200 483 580 909 820 313 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 235 200 483 580 909 820 313 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 998 470 400 967 161 819 640 627 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 998 470 400 967 161 819 640 627 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 996 940 801 934 323 639 281 254 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 996 940 801 934 323 639 281 254 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 993 881 603 868 647 278 562 508 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 993 881 603 868 647 278 562 508 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 987 763 207 737 294 557 125 017 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 987 763 207 737 294 557 125 017 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 975 526 415 474 589 114 250 035 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 975 526 415 474 589 114 250 035 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 951 052 830 949 178 228 500 070 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 951 052 830 949 178 228 500 070 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 902 105 661 898 356 457 000 140 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 902 105 661 898 356 457 000 140 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 804 211 323 796 712 914 000 281 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 804 211 323 796 712 914 000 281 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 608 422 647 593 425 828 000 563 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 608 422 647 593 425 828 000 563 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 216 845 295 186 851 656 001 126 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 216 845 295 186 851 656 001 126 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 998 433 690 590 373 703 312 002 252 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 998 433 690 590 373 703 312 002 252 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 996 867 381 180 747 406 624 004 505 6;
  • 78) 0.397 094 726 562 499 999 999 999 999 996 867 381 180 747 406 624 004 505 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 993 734 762 361 494 813 248 009 011 2;
  • 79) 0.794 189 453 124 999 999 999 999 999 993 734 762 361 494 813 248 009 011 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 987 469 524 722 989 626 496 018 022 4;
  • 80) 0.588 378 906 249 999 999 999 999 999 987 469 524 722 989 626 496 018 022 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 974 939 049 445 979 252 992 036 044 8;
  • 81) 0.176 757 812 499 999 999 999 999 999 974 939 049 445 979 252 992 036 044 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 949 878 098 891 958 505 984 072 089 6;
  • 82) 0.353 515 624 999 999 999 999 999 999 949 878 098 891 958 505 984 072 089 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 899 756 197 783 917 011 968 144 179 2;
  • 83) 0.707 031 249 999 999 999 999 999 999 899 756 197 783 917 011 968 144 179 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 799 512 395 567 834 023 936 288 358 4;
  • 84) 0.414 062 499 999 999 999 999 999 999 799 512 395 567 834 023 936 288 358 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 599 024 791 135 668 047 872 576 716 8;
  • 85) 0.828 124 999 999 999 999 999 999 999 599 024 791 135 668 047 872 576 716 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 198 049 582 271 336 095 745 153 433 6;
  • 86) 0.656 249 999 999 999 999 999 999 999 198 049 582 271 336 095 745 153 433 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 998 396 099 164 542 672 191 490 306 867 2;
  • 87) 0.312 499 999 999 999 999 999 999 998 396 099 164 542 672 191 490 306 867 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 996 792 198 329 085 344 382 980 613 734 4;
  • 88) 0.624 999 999 999 999 999 999 999 996 792 198 329 085 344 382 980 613 734 4 × 2 = 1 + 0.249 999 999 999 999 999 999 999 993 584 396 658 170 688 765 961 227 468 8;
  • 89) 0.249 999 999 999 999 999 999 999 993 584 396 658 170 688 765 961 227 468 8 × 2 = 0 + 0.499 999 999 999 999 999 999 999 987 168 793 316 341 377 531 922 454 937 6;
  • 90) 0.499 999 999 999 999 999 999 999 987 168 793 316 341 377 531 922 454 937 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 974 337 586 632 682 755 063 844 909 875 2;
  • 91) 0.999 999 999 999 999 999 999 999 974 337 586 632 682 755 063 844 909 875 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 948 675 173 265 365 510 127 689 819 750 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 967 3 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