0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 974 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 974 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 948 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 948 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 896 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 896 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 793 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 793 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 587 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 587 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 174 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 174 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 348 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 348 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 697 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 697 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 395 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 395 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 658 790 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 658 790 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 317 580 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 317 580 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 635 161 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 635 161 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 270 323 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 270 323 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 540 646 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 540 646 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 081 292 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 081 292 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 162 585 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 162 585 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 325 171 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 325 171 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 650 342 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 650 342 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 300 684 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 300 684 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 378 601 369 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 378 601 369 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 757 202 739 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 757 202 739 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 514 405 478 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 514 405 478 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 028 810 956 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 028 810 956 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 057 621 913 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 057 621 913 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 115 243 827 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 115 243 827 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 230 487 654 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 230 487 654 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 460 975 308 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 460 975 308 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 112 921 950 617 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 112 921 950 617 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 225 843 901 235 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 225 843 901 235 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 451 687 802 470 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 451 687 802 470 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 903 375 604 940 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 903 375 604 940 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 806 751 209 881 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 806 751 209 881 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 613 502 419 763 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 613 502 419 763 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 227 004 839 526 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 227 004 839 526 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 454 009 679 052 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 454 009 679 052 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 908 019 358 105 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 908 019 358 105 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 816 038 716 211 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 816 038 716 211 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 632 077 432 422 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 632 077 432 422 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 264 154 864 844 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 264 154 864 844 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 038 528 309 729 689 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 038 528 309 729 689 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 077 056 619 459 379 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 077 056 619 459 379 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 154 113 238 918 758 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 154 113 238 918 758 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 308 226 477 837 516 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 308 226 477 837 516 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 616 452 955 675 033 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 616 452 955 675 033 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 232 905 911 350 067 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 232 905 911 350 067 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 465 811 822 700 134 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 465 811 822 700 134 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 788 931 623 645 400 268 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 788 931 623 645 400 268 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 577 863 247 290 800 537 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 577 863 247 290 800 537 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 155 726 494 581 601 075 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 155 726 494 581 601 075 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 311 452 989 163 202 150 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 311 452 989 163 202 150 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 622 905 978 326 404 300 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 622 905 978 326 404 300 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 245 811 956 652 808 601 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 245 811 956 652 808 601 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 491 623 913 305 617 203 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 491 623 913 305 617 203 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 983 247 826 611 234 406 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 983 247 826 611 234 406 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 966 495 653 222 468 812 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 966 495 653 222 468 812 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 932 991 306 444 937 625 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 932 991 306 444 937 625 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 865 982 612 889 875 251 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 865 982 612 889 875 251 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 731 965 225 779 750 502 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 731 965 225 779 750 502 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 463 930 451 559 501 004 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 463 930 451 559 501 004 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 998 927 860 903 119 002 009 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 998 927 860 903 119 002 009 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 997 855 721 806 238 004 019 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 997 855 721 806 238 004 019 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 995 711 443 612 476 008 038 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 995 711 443 612 476 008 038 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 991 422 887 224 952 016 076 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 991 422 887 224 952 016 076 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 982 845 774 449 904 032 153 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 982 845 774 449 904 032 153 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 965 691 548 899 808 064 307 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 965 691 548 899 808 064 307 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 931 383 097 799 616 128 614 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 931 383 097 799 616 128 614 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 862 766 195 599 232 257 228 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 862 766 195 599 232 257 228 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 725 532 391 198 464 514 457 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 725 532 391 198 464 514 457 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 451 064 782 396 929 028 915 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 451 064 782 396 929 028 915 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 998 902 129 564 793 858 057 830 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 998 902 129 564 793 858 057 830 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 997 804 259 129 587 716 115 660 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 997 804 259 129 587 716 115 660 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 995 608 518 259 175 432 231 321 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 995 608 518 259 175 432 231 321 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 991 217 036 518 350 864 462 643 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 991 217 036 518 350 864 462 643 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 982 434 073 036 701 728 925 286 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 982 434 073 036 701 728 925 286 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 964 868 146 073 403 457 850 572 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 964 868 146 073 403 457 850 572 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 929 736 292 146 806 915 701 145 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 929 736 292 146 806 915 701 145 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 859 472 584 293 613 831 402 291 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 859 472 584 293 613 831 402 291 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 718 945 168 587 227 662 804 582 4;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 718 945 168 587 227 662 804 582 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 437 890 337 174 455 325 609 164 8;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 437 890 337 174 455 325 609 164 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 998 875 780 674 348 910 651 218 329 6;
  • 81) 0.176 757 812 499 999 999 999 999 999 998 875 780 674 348 910 651 218 329 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 997 751 561 348 697 821 302 436 659 2;
  • 82) 0.353 515 624 999 999 999 999 999 999 997 751 561 348 697 821 302 436 659 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 995 503 122 697 395 642 604 873 318 4;
  • 83) 0.707 031 249 999 999 999 999 999 999 995 503 122 697 395 642 604 873 318 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 991 006 245 394 791 285 209 746 636 8;
  • 84) 0.414 062 499 999 999 999 999 999 999 991 006 245 394 791 285 209 746 636 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 982 012 490 789 582 570 419 493 273 6;
  • 85) 0.828 124 999 999 999 999 999 999 999 982 012 490 789 582 570 419 493 273 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 964 024 981 579 165 140 838 986 547 2;
  • 86) 0.656 249 999 999 999 999 999 999 999 964 024 981 579 165 140 838 986 547 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 928 049 963 158 330 281 677 973 094 4;
  • 87) 0.312 499 999 999 999 999 999 999 999 928 049 963 158 330 281 677 973 094 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 856 099 926 316 660 563 355 946 188 8;
  • 88) 0.624 999 999 999 999 999 999 999 999 856 099 926 316 660 563 355 946 188 8 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 712 199 852 633 321 126 711 892 377 6;
  • 89) 0.249 999 999 999 999 999 999 999 999 712 199 852 633 321 126 711 892 377 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 424 399 705 266 642 253 423 784 755 2;
  • 90) 0.499 999 999 999 999 999 999 999 999 424 399 705 266 642 253 423 784 755 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 998 848 799 410 533 284 506 847 569 510 4;
  • 91) 0.999 999 999 999 999 999 999 999 998 848 799 410 533 284 506 847 569 510 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 997 697 598 821 066 569 013 695 139 020 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 1 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