0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 297 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 297 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 595 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 595 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 190 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 190 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 380 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 380 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 761 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 761 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 523 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 523 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 046 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 046 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 092 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 092 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 596 185 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 596 185 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 192 371 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 192 371 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 384 742 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 384 742 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 769 484 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 769 484 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 538 969 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 538 969 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 077 939 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 077 939 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 155 878 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 155 878 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 311 756 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 311 756 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 623 513 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 623 513 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 489 247 027 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 489 247 027 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 978 494 054 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 978 494 054 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 956 988 108 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 956 988 108 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 913 976 217 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 913 976 217 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 827 952 435 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 827 952 435 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 655 904 870 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 655 904 870 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 311 809 740 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 311 809 740 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 623 619 481 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 623 619 481 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 247 238 963 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 247 238 963 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 494 477 926 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 494 477 926 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 988 955 852 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 988 955 852 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 977 911 705 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 977 911 705 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 019 955 823 411 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 019 955 823 411 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 039 911 646 822 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 039 911 646 822 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 079 823 293 644 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 079 823 293 644 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 159 646 587 289 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 159 646 587 289 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 319 293 174 579 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 319 293 174 579 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 638 586 349 158 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 638 586 349 158 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 277 172 698 316 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 277 172 698 316 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 554 345 396 633 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 554 345 396 633 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 637 108 690 793 267 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 637 108 690 793 267 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 274 217 381 586 534 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 274 217 381 586 534 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 548 434 763 173 068 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 548 434 763 173 068 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 096 869 526 346 137 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 096 869 526 346 137 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 193 739 052 692 275 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 193 739 052 692 275 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 387 478 105 384 550 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 387 478 105 384 550 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 774 956 210 769 100 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 774 956 210 769 100 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 549 912 421 538 201 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 549 912 421 538 201 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 099 824 843 076 403 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 099 824 843 076 403 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 199 649 686 152 806 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 199 649 686 152 806 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 516 399 299 372 305 612 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 516 399 299 372 305 612 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 032 798 598 744 611 225 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 032 798 598 744 611 225 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 065 597 197 489 222 451 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 065 597 197 489 222 451 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 131 194 394 978 444 902 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 131 194 394 978 444 902 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 262 388 789 956 889 804 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 262 388 789 956 889 804 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 524 777 579 913 779 609 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 524 777 579 913 779 609 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 049 555 159 827 559 219 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 049 555 159 827 559 219 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 099 110 319 655 118 438 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 099 110 319 655 118 438 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 198 220 639 310 236 876 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 198 220 639 310 236 876 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 396 441 278 620 473 753 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 396 441 278 620 473 753 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 248 792 882 557 240 947 507 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 248 792 882 557 240 947 507 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 497 585 765 114 481 895 014 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 497 585 765 114 481 895 014 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 995 171 530 228 963 790 028 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 995 171 530 228 963 790 028 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 990 343 060 457 927 580 057 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 990 343 060 457 927 580 057 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 980 686 120 915 855 160 115 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 980 686 120 915 855 160 115 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 961 372 241 831 710 320 230 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 961 372 241 831 710 320 230 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 922 744 483 663 420 640 460 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 922 744 483 663 420 640 460 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 845 488 967 326 841 280 921 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 845 488 967 326 841 280 921 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 690 977 934 653 682 561 843 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 690 977 934 653 682 561 843 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 381 955 869 307 365 123 686 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 381 955 869 307 365 123 686 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 998 763 911 738 614 730 247 372 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 998 763 911 738 614 730 247 372 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 997 527 823 477 229 460 494 745 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 997 527 823 477 229 460 494 745 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 995 055 646 954 458 920 989 491 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 995 055 646 954 458 920 989 491 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 990 111 293 908 917 841 978 982 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 990 111 293 908 917 841 978 982 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 980 222 587 817 835 683 957 964 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 980 222 587 817 835 683 957 964 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 960 445 175 635 671 367 915 929 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 960 445 175 635 671 367 915 929 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 920 890 351 271 342 735 831 859 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 920 890 351 271 342 735 831 859 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 841 780 702 542 685 471 663 718 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 841 780 702 542 685 471 663 718 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 683 561 405 085 370 943 327 436 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 683 561 405 085 370 943 327 436 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 367 122 810 170 741 886 654 873 6;
  • 78) 0.397 094 726 562 499 999 999 999 999 367 122 810 170 741 886 654 873 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 998 734 245 620 341 483 773 309 747 2;
  • 79) 0.794 189 453 124 999 999 999 999 998 734 245 620 341 483 773 309 747 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 997 468 491 240 682 967 546 619 494 4;
  • 80) 0.588 378 906 249 999 999 999 999 997 468 491 240 682 967 546 619 494 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 994 936 982 481 365 935 093 238 988 8;
  • 81) 0.176 757 812 499 999 999 999 999 994 936 982 481 365 935 093 238 988 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 989 873 964 962 731 870 186 477 977 6;
  • 82) 0.353 515 624 999 999 999 999 999 989 873 964 962 731 870 186 477 977 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 979 747 929 925 463 740 372 955 955 2;
  • 83) 0.707 031 249 999 999 999 999 999 979 747 929 925 463 740 372 955 955 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 959 495 859 850 927 480 745 911 910 4;
  • 84) 0.414 062 499 999 999 999 999 999 959 495 859 850 927 480 745 911 910 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 918 991 719 701 854 961 491 823 820 8;
  • 85) 0.828 124 999 999 999 999 999 999 918 991 719 701 854 961 491 823 820 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 837 983 439 403 709 922 983 647 641 6;
  • 86) 0.656 249 999 999 999 999 999 999 837 983 439 403 709 922 983 647 641 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 675 966 878 807 419 845 967 295 283 2;
  • 87) 0.312 499 999 999 999 999 999 999 675 966 878 807 419 845 967 295 283 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 351 933 757 614 839 691 934 590 566 4;
  • 88) 0.624 999 999 999 999 999 999 999 351 933 757 614 839 691 934 590 566 4 × 2 = 1 + 0.249 999 999 999 999 999 999 998 703 867 515 229 679 383 869 181 132 8;
  • 89) 0.249 999 999 999 999 999 999 998 703 867 515 229 679 383 869 181 132 8 × 2 = 0 + 0.499 999 999 999 999 999 999 997 407 735 030 459 358 767 738 362 265 6;
  • 90) 0.499 999 999 999 999 999 999 997 407 735 030 459 358 767 738 362 265 6 × 2 = 0 + 0.999 999 999 999 999 999 999 994 815 470 060 918 717 535 476 724 531 2;
  • 91) 0.999 999 999 999 999 999 999 994 815 470 060 918 717 535 476 724 531 2 × 2 = 1 + 0.999 999 999 999 999 999 999 989 630 940 121 837 435 070 953 449 062 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 8 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