0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 320 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 320 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 641 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 641 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 283 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 283 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 566 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 566 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 132 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 132 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 265 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 265 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 531 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 531 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 062 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 062 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 124 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 124 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 204 249 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 204 249 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 408 499 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 408 499 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 816 998 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 816 998 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 633 996 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 633 996 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 267 993 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 267 993 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 535 987 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 535 987 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 071 974 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 071 974 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 143 948 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 143 948 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 287 897 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 287 897 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 575 795 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 575 795 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 969 151 590 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 969 151 590 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 938 303 180 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 938 303 180 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 876 606 361 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 876 606 361 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 753 212 723 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 753 212 723 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 506 425 446 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 506 425 446 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 012 850 892 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 012 850 892 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 025 701 785 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 025 701 785 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 051 403 571 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 051 403 571 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 102 807 142 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 102 807 142 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 016 205 614 284 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 016 205 614 284 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 032 411 228 569 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 032 411 228 569 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 064 822 457 139 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 064 822 457 139 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 129 644 914 278 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 129 644 914 278 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 259 289 828 556 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 259 289 828 556 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 518 579 657 113 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 518 579 657 113 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 037 159 314 227 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 037 159 314 227 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 410 074 318 628 454 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 410 074 318 628 454 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 820 148 637 256 908 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 820 148 637 256 908 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 640 297 274 513 817 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 640 297 274 513 817 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 280 594 549 027 635 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 280 594 549 027 635 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 561 189 098 055 270 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 561 189 098 055 270 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 122 378 196 110 540 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 122 378 196 110 540 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 244 756 392 221 081 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 244 756 392 221 081 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 489 512 784 442 163 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 489 512 784 442 163 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 979 025 568 884 326 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 979 025 568 884 326 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 958 051 137 768 652 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 958 051 137 768 652 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 916 102 275 537 305 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 916 102 275 537 305 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 832 204 551 074 611 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 832 204 551 074 611 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 519 664 409 102 149 222 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 519 664 409 102 149 222 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 039 328 818 204 298 444 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 039 328 818 204 298 444 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 078 657 636 408 596 889 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 078 657 636 408 596 889 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 157 315 272 817 193 779 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 157 315 272 817 193 779 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 314 630 545 634 387 558 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 314 630 545 634 387 558 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 629 261 091 268 775 116 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 629 261 091 268 775 116 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 258 522 182 537 550 233 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 258 522 182 537 550 233 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 517 044 365 075 100 467 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 517 044 365 075 100 467 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 813 034 088 730 150 200 934 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 813 034 088 730 150 200 934 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 626 068 177 460 300 401 868 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 626 068 177 460 300 401 868 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 252 136 354 920 600 803 737 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 252 136 354 920 600 803 737 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 504 272 709 841 201 607 475 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 504 272 709 841 201 607 475 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 008 545 419 682 403 214 950 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 008 545 419 682 403 214 950 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 017 090 839 364 806 429 900 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 017 090 839 364 806 429 900 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 034 181 678 729 612 859 801 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 034 181 678 729 612 859 801 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 068 363 357 459 225 719 603 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 068 363 357 459 225 719 603 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 136 726 714 918 451 439 206 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 136 726 714 918 451 439 206 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 273 453 429 836 902 878 412 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 273 453 429 836 902 878 412 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 546 906 859 673 805 756 825 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 546 906 859 673 805 756 825 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 001 093 813 719 347 611 513 651 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 001 093 813 719 347 611 513 651 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 002 187 627 438 695 223 027 302 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 002 187 627 438 695 223 027 302 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 004 375 254 877 390 446 054 604 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 004 375 254 877 390 446 054 604 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 008 750 509 754 780 892 109 209 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 008 750 509 754 780 892 109 209 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 017 501 019 509 561 784 218 419 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 017 501 019 509 561 784 218 419 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 035 002 039 019 123 568 436 838 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 035 002 039 019 123 568 436 838 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 070 004 078 038 247 136 873 676 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 070 004 078 038 247 136 873 676 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 140 008 156 076 494 273 747 353 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 140 008 156 076 494 273 747 353 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 280 016 312 152 988 547 494 707 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 280 016 312 152 988 547 494 707 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 560 032 624 305 977 094 989 414 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 560 032 624 305 977 094 989 414 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 001 120 065 248 611 954 189 978 828 8;
  • 78) 0.397 094 726 562 500 000 000 000 001 120 065 248 611 954 189 978 828 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 002 240 130 497 223 908 379 957 657 6;
  • 79) 0.794 189 453 125 000 000 000 000 002 240 130 497 223 908 379 957 657 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 004 480 260 994 447 816 759 915 315 2;
  • 80) 0.588 378 906 250 000 000 000 000 004 480 260 994 447 816 759 915 315 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 008 960 521 988 895 633 519 830 630 4;
  • 81) 0.176 757 812 500 000 000 000 000 008 960 521 988 895 633 519 830 630 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 017 921 043 977 791 267 039 661 260 8;
  • 82) 0.353 515 625 000 000 000 000 000 017 921 043 977 791 267 039 661 260 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 035 842 087 955 582 534 079 322 521 6;
  • 83) 0.707 031 250 000 000 000 000 000 035 842 087 955 582 534 079 322 521 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 071 684 175 911 165 068 158 645 043 2;
  • 84) 0.414 062 500 000 000 000 000 000 071 684 175 911 165 068 158 645 043 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 143 368 351 822 330 136 317 290 086 4;
  • 85) 0.828 125 000 000 000 000 000 000 143 368 351 822 330 136 317 290 086 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 286 736 703 644 660 272 634 580 172 8;
  • 86) 0.656 250 000 000 000 000 000 000 286 736 703 644 660 272 634 580 172 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 573 473 407 289 320 545 269 160 345 6;
  • 87) 0.312 500 000 000 000 000 000 000 573 473 407 289 320 545 269 160 345 6 × 2 = 0 + 0.625 000 000 000 000 000 000 001 146 946 814 578 641 090 538 320 691 2;
  • 88) 0.625 000 000 000 000 000 000 001 146 946 814 578 641 090 538 320 691 2 × 2 = 1 + 0.250 000 000 000 000 000 000 002 293 893 629 157 282 181 076 641 382 4;
  • 89) 0.250 000 000 000 000 000 000 002 293 893 629 157 282 181 076 641 382 4 × 2 = 0 + 0.500 000 000 000 000 000 000 004 587 787 258 314 564 362 153 282 764 8;
  • 90) 0.500 000 000 000 000 000 000 004 587 787 258 314 564 362 153 282 764 8 × 2 = 1 + 0.000 000 000 000 000 000 000 009 175 574 516 629 128 724 306 565 529 6;
  • 91) 0.000 000 000 000 000 000 000 009 175 574 516 629 128 724 306 565 529 6 × 2 = 0 + 0.000 000 000 000 000 000 000 018 351 149 033 258 257 448 613 131 059 2;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 4 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