0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 312 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 312 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 625 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 625 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 250 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 250 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 500 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 500 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 001 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 001 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 003 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 003 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 006 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 006 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 012 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 012 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 600 025 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 600 025 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 200 051 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 200 051 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 400 102 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 400 102 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 800 204 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 800 204 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 600 409 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 600 409 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 200 819 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 200 819 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 401 638 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 401 638 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 803 276 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 803 276 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 606 553 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 606 553 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 213 107 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 213 107 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 982 426 214 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 982 426 214 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 964 852 428 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 964 852 428 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 929 704 857 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 929 704 857 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 859 409 715 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 859 409 715 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 718 819 430 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 718 819 430 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 437 638 860 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 437 638 860 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 875 277 721 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 875 277 721 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 750 555 443 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 750 555 443 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 501 110 886 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 501 110 886 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 002 221 772 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 002 221 772 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 014 004 443 545 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 014 004 443 545 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 028 008 887 091 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 028 008 887 091 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 056 017 774 182 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 056 017 774 182 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 112 035 548 364 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 112 035 548 364 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 224 071 096 729 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 224 071 096 729 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 448 142 193 459 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 448 142 193 459 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 896 284 386 918 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 896 284 386 918 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 792 568 773 836 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 792 568 773 836 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 585 137 547 673 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 585 137 547 673 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 639 170 275 095 347 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 639 170 275 095 347 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 278 340 550 190 694 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 278 340 550 190 694 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 556 681 100 381 388 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 556 681 100 381 388 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 113 362 200 762 777 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 113 362 200 762 777 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 226 724 401 525 555 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 226 724 401 525 555 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 453 448 803 051 110 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 453 448 803 051 110 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 906 897 606 102 220 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 906 897 606 102 220 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 813 795 212 204 441 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 813 795 212 204 441 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 627 590 424 408 883 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 627 590 424 408 883 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 255 180 848 817 766 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 255 180 848 817 766 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 518 510 361 697 635 532 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 518 510 361 697 635 532 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 037 020 723 395 271 065 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 037 020 723 395 271 065 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 074 041 446 790 542 131 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 074 041 446 790 542 131 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 148 082 893 581 084 262 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 148 082 893 581 084 262 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 296 165 787 162 168 524 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 296 165 787 162 168 524 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 592 331 574 324 337 049 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 592 331 574 324 337 049 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 184 663 148 648 674 099 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 184 663 148 648 674 099 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 369 326 297 297 348 198 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 369 326 297 297 348 198 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 738 652 594 594 696 396 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 738 652 594 594 696 396 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 477 305 189 189 392 793 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 477 305 189 189 392 793 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 954 610 378 378 785 587 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 954 610 378 378 785 587 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 501 909 220 756 757 571 174 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 501 909 220 756 757 571 174 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 003 818 441 513 515 142 348 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 003 818 441 513 515 142 348 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 007 636 883 027 030 284 697 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 007 636 883 027 030 284 697 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 015 273 766 054 060 569 395 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 015 273 766 054 060 569 395 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 030 547 532 108 121 138 790 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 030 547 532 108 121 138 790 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 061 095 064 216 242 277 580 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 061 095 064 216 242 277 580 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 122 190 128 432 484 555 161 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 122 190 128 432 484 555 161 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 244 380 256 864 969 110 323 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 244 380 256 864 969 110 323 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 488 760 513 729 938 220 646 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 488 760 513 729 938 220 646 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 977 521 027 459 876 441 292 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 977 521 027 459 876 441 292 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 001 955 042 054 919 752 882 585 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 001 955 042 054 919 752 882 585 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 003 910 084 109 839 505 765 171 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 003 910 084 109 839 505 765 171 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 007 820 168 219 679 011 530 342 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 007 820 168 219 679 011 530 342 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 015 640 336 439 358 023 060 684 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 015 640 336 439 358 023 060 684 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 031 280 672 878 716 046 121 369 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 031 280 672 878 716 046 121 369 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 062 561 345 757 432 092 242 739 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 062 561 345 757 432 092 242 739 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 125 122 691 514 864 184 485 478 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 125 122 691 514 864 184 485 478 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 250 245 383 029 728 368 970 956 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 250 245 383 029 728 368 970 956 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 500 490 766 059 456 737 941 913 6;
  • 78) 0.397 094 726 562 500 000 000 000 000 500 490 766 059 456 737 941 913 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 001 000 981 532 118 913 475 883 827 2;
  • 79) 0.794 189 453 125 000 000 000 000 001 000 981 532 118 913 475 883 827 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 002 001 963 064 237 826 951 767 654 4;
  • 80) 0.588 378 906 250 000 000 000 000 002 001 963 064 237 826 951 767 654 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 004 003 926 128 475 653 903 535 308 8;
  • 81) 0.176 757 812 500 000 000 000 000 004 003 926 128 475 653 903 535 308 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 008 007 852 256 951 307 807 070 617 6;
  • 82) 0.353 515 625 000 000 000 000 000 008 007 852 256 951 307 807 070 617 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 016 015 704 513 902 615 614 141 235 2;
  • 83) 0.707 031 250 000 000 000 000 000 016 015 704 513 902 615 614 141 235 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 032 031 409 027 805 231 228 282 470 4;
  • 84) 0.414 062 500 000 000 000 000 000 032 031 409 027 805 231 228 282 470 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 064 062 818 055 610 462 456 564 940 8;
  • 85) 0.828 125 000 000 000 000 000 000 064 062 818 055 610 462 456 564 940 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 128 125 636 111 220 924 913 129 881 6;
  • 86) 0.656 250 000 000 000 000 000 000 128 125 636 111 220 924 913 129 881 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 256 251 272 222 441 849 826 259 763 2;
  • 87) 0.312 500 000 000 000 000 000 000 256 251 272 222 441 849 826 259 763 2 × 2 = 0 + 0.625 000 000 000 000 000 000 000 512 502 544 444 883 699 652 519 526 4;
  • 88) 0.625 000 000 000 000 000 000 000 512 502 544 444 883 699 652 519 526 4 × 2 = 1 + 0.250 000 000 000 000 000 000 001 025 005 088 889 767 399 305 039 052 8;
  • 89) 0.250 000 000 000 000 000 000 001 025 005 088 889 767 399 305 039 052 8 × 2 = 0 + 0.500 000 000 000 000 000 000 002 050 010 177 779 534 798 610 078 105 6;
  • 90) 0.500 000 000 000 000 000 000 002 050 010 177 779 534 798 610 078 105 6 × 2 = 1 + 0.000 000 000 000 000 000 000 004 100 020 355 559 069 597 220 156 211 2;
  • 91) 0.000 000 000 000 000 000 000 004 100 020 355 559 069 597 220 156 211 2 × 2 = 0 + 0.000 000 000 000 000 000 000 008 200 040 711 118 139 194 440 312 422 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3(10) =

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 8 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Your Greatest Competitor, Your Most Powerful Ally. Gemini Notebook YouTube Video of 7.54 minutes

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