0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 061 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 061 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 123 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 123 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 247 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 247 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 494 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 494 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 988 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 988 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 977 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 977 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 955 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 955 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 910 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 910 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 820 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 820 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 743 641 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 743 641 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 487 283 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 487 283 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 974 566 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 974 566 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 949 132 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 949 132 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 898 265 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 898 265 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 796 531 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 796 531 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 593 062 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 593 062 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 186 124 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 186 124 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 372 249 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 372 249 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 744 499 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 744 499 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 577 488 998 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 577 488 998 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 154 977 996 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 154 977 996 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 309 955 993 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 309 955 993 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 619 911 987 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 619 911 987 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 239 823 974 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 239 823 974 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 479 647 948 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 479 647 948 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 959 295 897 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 959 295 897 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 918 591 795 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 918 591 795 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 837 183 590 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 837 183 590 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 674 367 180 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 674 367 180 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 687 348 734 361 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 687 348 734 361 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 374 697 468 723 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 374 697 468 723 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 749 394 937 446 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 749 394 937 446 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 498 789 874 892 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 498 789 874 892 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 997 579 749 785 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 997 579 749 785 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 995 159 499 571 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 995 159 499 571 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 990 318 999 142 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 990 318 999 142 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 980 637 998 284 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 980 637 998 284 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 961 275 996 569 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 961 275 996 569 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 922 551 993 139 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 922 551 993 139 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 551 845 103 986 278 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 551 845 103 986 278 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 103 690 207 972 556 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 103 690 207 972 556 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 207 380 415 945 113 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 207 380 415 945 113 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 414 760 831 890 227 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 414 760 831 890 227 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 829 521 663 780 454 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 829 521 663 780 454 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 659 043 327 560 908 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 659 043 327 560 908 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 318 086 655 121 817 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 318 086 655 121 817 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 636 173 310 243 635 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 636 173 310 243 635 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 272 346 620 487 270 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 272 346 620 487 270 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 544 693 240 974 540 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 544 693 240 974 540 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 089 386 481 949 081 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 089 386 481 949 081 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 002 178 772 963 898 163 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 002 178 772 963 898 163 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 004 357 545 927 796 326 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 004 357 545 927 796 326 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 008 715 091 855 592 652 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 008 715 091 855 592 652 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 017 430 183 711 185 305 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 017 430 183 711 185 305 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 034 860 367 422 370 611 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 034 860 367 422 370 611 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 069 720 734 844 741 222 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 069 720 734 844 741 222 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 139 441 469 689 482 444 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 139 441 469 689 482 444 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 278 882 939 378 964 889 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 278 882 939 378 964 889 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 557 765 878 757 929 779 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 557 765 878 757 929 779 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 001 115 531 757 515 859 558 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 001 115 531 757 515 859 558 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 002 231 063 515 031 719 116 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 002 231 063 515 031 719 116 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 004 462 127 030 063 438 233 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 004 462 127 030 063 438 233 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 008 924 254 060 126 876 467 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 008 924 254 060 126 876 467 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 017 848 508 120 253 752 934 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 017 848 508 120 253 752 934 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 035 697 016 240 507 505 868 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 035 697 016 240 507 505 868 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 071 394 032 481 015 011 737 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 071 394 032 481 015 011 737 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 142 788 064 962 030 023 475 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 142 788 064 962 030 023 475 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 285 576 129 924 060 046 950 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 285 576 129 924 060 046 950 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 571 152 259 848 120 093 900 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 571 152 259 848 120 093 900 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 001 142 304 519 696 240 187 801 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 001 142 304 519 696 240 187 801 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 002 284 609 039 392 480 375 603 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 002 284 609 039 392 480 375 603 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 004 569 218 078 784 960 751 206 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 004 569 218 078 784 960 751 206 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 009 138 436 157 569 921 502 412 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 009 138 436 157 569 921 502 412 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 018 276 872 315 139 843 004 825 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 018 276 872 315 139 843 004 825 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 036 553 744 630 279 686 009 651 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 036 553 744 630 279 686 009 651 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 073 107 489 260 559 372 019 302 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 073 107 489 260 559 372 019 302 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 146 214 978 521 118 744 038 604 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 146 214 978 521 118 744 038 604 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 292 429 957 042 237 488 077 209 6;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 292 429 957 042 237 488 077 209 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 584 859 914 084 474 976 154 419 2;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 584 859 914 084 474 976 154 419 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 001 169 719 828 168 949 952 308 838 4;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 001 169 719 828 168 949 952 308 838 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 002 339 439 656 337 899 904 617 676 8;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 002 339 439 656 337 899 904 617 676 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 004 678 879 312 675 799 809 235 353 6;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 004 678 879 312 675 799 809 235 353 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 009 357 758 625 351 599 618 470 707 2;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 009 357 758 625 351 599 618 470 707 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 018 715 517 250 703 199 236 941 414 4;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 018 715 517 250 703 199 236 941 414 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 037 431 034 501 406 398 473 882 828 8;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 037 431 034 501 406 398 473 882 828 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 074 862 069 002 812 796 947 765 657 6;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 074 862 069 002 812 796 947 765 657 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 149 724 138 005 625 593 895 531 315 2;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 149 724 138 005 625 593 895 531 315 2 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 299 448 276 011 251 187 791 062 630 4;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 299 448 276 011 251 187 791 062 630 4 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 598 896 552 022 502 375 582 125 260 8;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 598 896 552 022 502 375 582 125 260 8 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 001 197 793 104 045 004 751 164 250 521 6;
  • 91) 0.000 000 000 000 000 000 000 000 000 001 197 793 104 045 004 751 164 250 521 6 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 002 395 586 208 090 009 502 328 501 043 2;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(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 1010


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 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


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 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 030 9 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 1010

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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