0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 96 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 96(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 96(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 96.

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 96 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 92;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 92 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 951 84;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 951 84 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 903 68;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 903 68 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 807 36;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 807 36 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 614 72;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 614 72 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 229 44;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 229 44 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 458 88;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 458 88 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 917 76;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 917 76 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 835 52;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 835 52 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 671 04;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 671 04 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 342 08;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 342 08 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 684 16;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 684 16 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 368 32;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 368 32 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 554 736 64;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 554 736 64 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 109 473 28;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 109 473 28 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 218 946 56;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 218 946 56 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 437 893 12;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 437 893 12 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 875 786 24;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 875 786 24 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 751 572 48;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 751 572 48 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 503 144 96;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 503 144 96 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 006 289 92;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 006 289 92 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 012 579 84;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 012 579 84 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 025 159 68;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 025 159 68 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 072 050 319 36;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 072 050 319 36 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 144 100 638 72;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 144 100 638 72 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 288 201 277 44;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 288 201 277 44 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 576 402 554 88;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 576 402 554 88 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 152 805 109 76;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 152 805 109 76 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 305 610 219 52;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 305 610 219 52 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 611 220 439 04;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 611 220 439 04 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 222 440 878 08;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 222 440 878 08 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 444 881 756 16;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 444 881 756 16 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 620 889 763 512 32;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 620 889 763 512 32 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 241 779 527 024 64;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 241 779 527 024 64 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 483 559 054 049 28;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 483 559 054 049 28 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 967 118 108 098 56;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 967 118 108 098 56 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 934 236 216 197 12;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 934 236 216 197 12 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 868 472 432 394 24;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 868 472 432 394 24 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 736 944 864 788 48;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 736 944 864 788 48 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 473 889 729 576 96;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 473 889 729 576 96 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 947 779 459 153 92;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 947 779 459 153 92 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 157 895 558 918 307 84;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 157 895 558 918 307 84 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 315 791 117 836 615 68;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 315 791 117 836 615 68 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 631 582 235 673 231 36;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 631 582 235 673 231 36 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 263 164 471 346 462 72;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 263 164 471 346 462 72 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 526 328 942 692 925 44;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 526 328 942 692 925 44 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 052 657 885 385 850 88;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 052 657 885 385 850 88 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 105 315 770 771 701 76;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 105 315 770 771 701 76 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 210 631 541 543 403 52;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 210 631 541 543 403 52 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 421 263 083 086 807 04;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 421 263 083 086 807 04 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 842 526 166 173 614 08;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 842 526 166 173 614 08 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 685 052 332 347 228 16;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 685 052 332 347 228 16 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 370 104 664 694 456 32;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 370 104 664 694 456 32 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 998 740 209 329 388 912 64;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 998 740 209 329 388 912 64 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 997 480 418 658 777 825 28;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 997 480 418 658 777 825 28 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 994 960 837 317 555 650 56;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 994 960 837 317 555 650 56 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 989 921 674 635 111 301 12;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 989 921 674 635 111 301 12 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 979 843 349 270 222 602 24;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 979 843 349 270 222 602 24 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 959 686 698 540 445 204 48;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 959 686 698 540 445 204 48 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 919 373 397 080 890 408 96;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 919 373 397 080 890 408 96 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 838 746 794 161 780 817 92;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 838 746 794 161 780 817 92 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 677 493 588 323 561 635 84;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 677 493 588 323 561 635 84 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 354 987 176 647 123 271 68;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 354 987 176 647 123 271 68 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 998 709 974 353 294 246 543 36;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 998 709 974 353 294 246 543 36 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 997 419 948 706 588 493 086 72;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 997 419 948 706 588 493 086 72 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 994 839 897 413 176 986 173 44;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 994 839 897 413 176 986 173 44 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 989 679 794 826 353 972 346 88;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 989 679 794 826 353 972 346 88 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 979 359 589 652 707 944 693 76;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 979 359 589 652 707 944 693 76 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 958 719 179 305 415 889 387 52;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 958 719 179 305 415 889 387 52 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 917 438 358 610 831 778 775 04;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 917 438 358 610 831 778 775 04 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 834 876 717 221 663 557 550 08;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 834 876 717 221 663 557 550 08 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 669 753 434 443 327 115 100 16;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 669 753 434 443 327 115 100 16 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 339 506 868 886 654 230 200 32;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 339 506 868 886 654 230 200 32 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 998 679 013 737 773 308 460 400 64;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 998 679 013 737 773 308 460 400 64 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 997 358 027 475 546 616 920 801 28;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 997 358 027 475 546 616 920 801 28 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 994 716 054 951 093 233 841 602 56;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 994 716 054 951 093 233 841 602 56 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 989 432 109 902 186 467 683 205 12;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 989 432 109 902 186 467 683 205 12 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 978 864 219 804 372 935 366 410 24;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 978 864 219 804 372 935 366 410 24 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 957 728 439 608 745 870 732 820 48;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 957 728 439 608 745 870 732 820 48 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 915 456 879 217 491 741 465 640 96;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 915 456 879 217 491 741 465 640 96 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 830 913 758 434 983 482 931 281 92;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 830 913 758 434 983 482 931 281 92 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 661 827 516 869 966 965 862 563 84;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 661 827 516 869 966 965 862 563 84 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 323 655 033 739 933 931 725 127 68;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 323 655 033 739 933 931 725 127 68 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 998 647 310 067 479 867 863 450 255 36;
  • 85) 0.828 124 999 999 999 999 999 999 999 998 647 310 067 479 867 863 450 255 36 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 997 294 620 134 959 735 726 900 510 72;
  • 86) 0.656 249 999 999 999 999 999 999 999 997 294 620 134 959 735 726 900 510 72 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 994 589 240 269 919 471 453 801 021 44;
  • 87) 0.312 499 999 999 999 999 999 999 999 994 589 240 269 919 471 453 801 021 44 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 989 178 480 539 838 942 907 602 042 88;
  • 88) 0.624 999 999 999 999 999 999 999 999 989 178 480 539 838 942 907 602 042 88 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 978 356 961 079 677 885 815 204 085 76;
  • 89) 0.249 999 999 999 999 999 999 999 999 978 356 961 079 677 885 815 204 085 76 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 956 713 922 159 355 771 630 408 171 52;
  • 90) 0.499 999 999 999 999 999 999 999 999 956 713 922 159 355 771 630 408 171 52 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 913 427 844 318 711 543 260 816 343 04;
  • 91) 0.999 999 999 999 999 999 999 999 999 913 427 844 318 711 543 260 816 343 04 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 826 855 688 637 423 086 521 632 686 08;

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 96(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 96(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 96(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 96 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