0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 44;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 44 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 88;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 88 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 76;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 76 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 52;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 52 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 903 04;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 903 04 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 806 08;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 806 08 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 612 16;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 612 16 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 224 32;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 224 32 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 448 64;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 448 64 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 897 28;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 897 28 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 794 56;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 794 56 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 589 12;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 589 12 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 575 178 24;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 575 178 24 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 150 356 48;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 150 356 48 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 300 712 96;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 300 712 96 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 601 425 92;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 601 425 92 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 202 851 84;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 202 851 84 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 405 703 68;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 405 703 68 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 811 407 36;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 811 407 36 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 622 814 72;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 622 814 72 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 245 629 44;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 245 629 44 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 491 258 88;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 491 258 88 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 982 517 76;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 692 982 517 76 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 385 965 035 52;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 385 965 035 52 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 771 930 071 04;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 771 930 071 04 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 543 860 142 08;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 543 860 142 08 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 087 720 284 16;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 087 720 284 16 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 175 440 568 32;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 175 440 568 32 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 350 881 136 64;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 350 881 136 64 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 701 762 273 28;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 701 762 273 28 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 403 524 546 56;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 403 524 546 56 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 807 049 093 12;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 807 049 093 12 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 614 098 186 24;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 614 098 186 24 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 395 228 196 372 48;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 395 228 196 372 48 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 790 456 392 744 96;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 790 456 392 744 96 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 580 912 785 489 92;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 580 912 785 489 92 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 161 825 570 979 84;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 161 825 570 979 84 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 323 651 141 959 68;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 323 651 141 959 68 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 647 302 283 919 36;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 647 302 283 919 36 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 294 604 567 838 72;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 294 604 567 838 72 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 589 209 135 677 44;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 589 209 135 677 44 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 213 178 418 271 354 88;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 213 178 418 271 354 88 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 426 356 836 542 709 76;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 426 356 836 542 709 76 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 852 713 673 085 419 52;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 852 713 673 085 419 52 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 705 427 346 170 839 04;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 705 427 346 170 839 04 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 410 854 692 341 678 08;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 410 854 692 341 678 08 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 821 709 384 683 356 16;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 821 709 384 683 356 16 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 643 418 769 366 712 32;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 643 418 769 366 712 32 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 286 837 538 733 424 64;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 286 837 538 733 424 64 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 573 675 077 466 849 28;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 573 675 077 466 849 28 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 147 350 154 933 698 56;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 147 350 154 933 698 56 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 282 294 700 309 867 397 12;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 282 294 700 309 867 397 12 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 564 589 400 619 734 794 24;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 564 589 400 619 734 794 24 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 129 178 801 239 469 588 48;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 129 178 801 239 469 588 48 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 258 357 602 478 939 176 96;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 258 357 602 478 939 176 96 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 516 715 204 957 878 353 92;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 516 715 204 957 878 353 92 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 033 430 409 915 756 707 84;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 033 430 409 915 756 707 84 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 066 860 819 831 513 415 68;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 066 860 819 831 513 415 68 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 133 721 639 663 026 831 36;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 133 721 639 663 026 831 36 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 267 443 279 326 053 662 72;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 267 443 279 326 053 662 72 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 534 886 558 652 107 325 44;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 534 886 558 652 107 325 44 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 001 069 773 117 304 214 650 88;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 001 069 773 117 304 214 650 88 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 002 139 546 234 608 429 301 76;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 002 139 546 234 608 429 301 76 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 004 279 092 469 216 858 603 52;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 004 279 092 469 216 858 603 52 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 008 558 184 938 433 717 207 04;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 008 558 184 938 433 717 207 04 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 017 116 369 876 867 434 414 08;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 017 116 369 876 867 434 414 08 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 034 232 739 753 734 868 828 16;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 034 232 739 753 734 868 828 16 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 068 465 479 507 469 737 656 32;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 068 465 479 507 469 737 656 32 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 136 930 959 014 939 475 312 64;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 136 930 959 014 939 475 312 64 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 273 861 918 029 878 950 625 28;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 273 861 918 029 878 950 625 28 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 547 723 836 059 757 901 250 56;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 547 723 836 059 757 901 250 56 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 001 095 447 672 119 515 802 501 12;
  • 73) 0.231 159 210 205 078 125 000 000 000 001 095 447 672 119 515 802 501 12 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 002 190 895 344 239 031 605 002 24;
  • 74) 0.462 318 420 410 156 250 000 000 000 002 190 895 344 239 031 605 002 24 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 004 381 790 688 478 063 210 004 48;
  • 75) 0.924 636 840 820 312 500 000 000 000 004 381 790 688 478 063 210 004 48 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 008 763 581 376 956 126 420 008 96;
  • 76) 0.849 273 681 640 625 000 000 000 000 008 763 581 376 956 126 420 008 96 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 017 527 162 753 912 252 840 017 92;
  • 77) 0.698 547 363 281 250 000 000 000 000 017 527 162 753 912 252 840 017 92 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 035 054 325 507 824 505 680 035 84;
  • 78) 0.397 094 726 562 500 000 000 000 000 035 054 325 507 824 505 680 035 84 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 070 108 651 015 649 011 360 071 68;
  • 79) 0.794 189 453 125 000 000 000 000 000 070 108 651 015 649 011 360 071 68 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 140 217 302 031 298 022 720 143 36;
  • 80) 0.588 378 906 250 000 000 000 000 000 140 217 302 031 298 022 720 143 36 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 280 434 604 062 596 045 440 286 72;
  • 81) 0.176 757 812 500 000 000 000 000 000 280 434 604 062 596 045 440 286 72 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 560 869 208 125 192 090 880 573 44;
  • 82) 0.353 515 625 000 000 000 000 000 000 560 869 208 125 192 090 880 573 44 × 2 = 0 + 0.707 031 250 000 000 000 000 000 001 121 738 416 250 384 181 761 146 88;
  • 83) 0.707 031 250 000 000 000 000 000 001 121 738 416 250 384 181 761 146 88 × 2 = 1 + 0.414 062 500 000 000 000 000 000 002 243 476 832 500 768 363 522 293 76;
  • 84) 0.414 062 500 000 000 000 000 000 002 243 476 832 500 768 363 522 293 76 × 2 = 0 + 0.828 125 000 000 000 000 000 000 004 486 953 665 001 536 727 044 587 52;
  • 85) 0.828 125 000 000 000 000 000 000 004 486 953 665 001 536 727 044 587 52 × 2 = 1 + 0.656 250 000 000 000 000 000 000 008 973 907 330 003 073 454 089 175 04;
  • 86) 0.656 250 000 000 000 000 000 000 008 973 907 330 003 073 454 089 175 04 × 2 = 1 + 0.312 500 000 000 000 000 000 000 017 947 814 660 006 146 908 178 350 08;
  • 87) 0.312 500 000 000 000 000 000 000 017 947 814 660 006 146 908 178 350 08 × 2 = 0 + 0.625 000 000 000 000 000 000 000 035 895 629 320 012 293 816 356 700 16;
  • 88) 0.625 000 000 000 000 000 000 000 035 895 629 320 012 293 816 356 700 16 × 2 = 1 + 0.250 000 000 000 000 000 000 000 071 791 258 640 024 587 632 713 400 32;
  • 89) 0.250 000 000 000 000 000 000 000 071 791 258 640 024 587 632 713 400 32 × 2 = 0 + 0.500 000 000 000 000 000 000 000 143 582 517 280 049 175 265 426 800 64;
  • 90) 0.500 000 000 000 000 000 000 000 143 582 517 280 049 175 265 426 800 64 × 2 = 1 + 0.000 000 000 000 000 000 000 000 287 165 034 560 098 350 530 853 601 28;
  • 91) 0.000 000 000 000 000 000 000 000 287 165 034 560 098 350 530 853 601 28 × 2 = 0 + 0.000 000 000 000 000 000 000 000 574 330 069 120 196 701 061 707 202 56;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 22 converted to 64 bit double precision IEEE 754 binary floating point representation:

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

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100