0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

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

Keep track of each remainder.

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


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

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

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

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


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 980 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 980 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 960 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 960 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 920 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 920 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 841 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 841 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 683 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 683 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 366 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 366 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 732 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 732 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 465 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 465 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 330 931 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 330 931 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 661 862 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 661 862 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 323 724 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 323 724 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 647 449 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 647 449 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 294 899 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 294 899 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 589 798 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 589 798 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 179 596 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 179 596 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 359 193 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 359 193 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 718 387 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 718 387 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 436 774 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 436 774 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 690 873 548 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 690 873 548 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 381 747 097 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 381 747 097 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 763 494 195 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 763 494 195 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 526 988 390 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 526 988 390 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 053 976 780 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 053 976 780 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 107 953 561 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 107 953 561 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 215 907 123 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 215 907 123 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 431 814 246 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 431 814 246 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 863 628 492 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 863 628 492 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 727 256 985 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 727 256 985 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 227 454 513 971 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 227 454 513 971 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 454 909 027 942 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 454 909 027 942 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 909 818 055 884 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 909 818 055 884 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 819 636 111 769 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 819 636 111 769 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 639 272 223 539 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 639 272 223 539 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 278 544 447 078 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 278 544 447 078 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 557 088 894 156 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 557 088 894 156 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 114 177 788 313 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 114 177 788 313 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 228 355 576 627 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 228 355 576 627 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 456 711 153 254 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 456 711 153 254 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 520 913 422 306 508 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 520 913 422 306 508 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 041 826 844 613 017 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 041 826 844 613 017 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 083 653 689 226 035 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 083 653 689 226 035 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 167 307 378 452 070 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 167 307 378 452 070 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 334 614 756 904 140 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 334 614 756 904 140 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 669 229 513 808 281 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 669 229 513 808 281 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 338 459 027 616 563 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 338 459 027 616 563 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 676 918 055 233 126 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 676 918 055 233 126 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 353 836 110 466 252 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 353 836 110 466 252 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 707 672 220 932 505 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 707 672 220 932 505 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 157 415 344 441 865 011 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 157 415 344 441 865 011 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 314 830 688 883 730 022 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 314 830 688 883 730 022 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 629 661 377 767 460 044 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 629 661 377 767 460 044 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 259 322 755 534 920 089 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 259 322 755 534 920 089 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 518 645 511 069 840 179 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 518 645 511 069 840 179 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 037 291 022 139 680 358 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 037 291 022 139 680 358 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 074 582 044 279 360 716 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 074 582 044 279 360 716 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 149 164 088 558 721 433 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 149 164 088 558 721 433 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 298 328 177 117 442 867 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 298 328 177 117 442 867 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 596 656 354 234 885 734 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 596 656 354 234 885 734 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 001 193 312 708 469 771 468 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 001 193 312 708 469 771 468 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 002 386 625 416 939 542 937 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 002 386 625 416 939 542 937 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 004 773 250 833 879 085 875 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 004 773 250 833 879 085 875 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 009 546 501 667 758 171 750 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 009 546 501 667 758 171 750 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 019 093 003 335 516 343 500 8;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 019 093 003 335 516 343 500 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 038 186 006 671 032 687 001 6;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 038 186 006 671 032 687 001 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 076 372 013 342 065 374 003 2;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 076 372 013 342 065 374 003 2 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 152 744 026 684 130 748 006 4;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 152 744 026 684 130 748 006 4 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 305 488 053 368 261 496 012 8;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 305 488 053 368 261 496 012 8 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 610 976 106 736 522 992 025 6;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 610 976 106 736 522 992 025 6 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 001 221 952 213 473 045 984 051 2;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 001 221 952 213 473 045 984 051 2 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 002 443 904 426 946 091 968 102 4;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 002 443 904 426 946 091 968 102 4 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 004 887 808 853 892 183 936 204 8;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 004 887 808 853 892 183 936 204 8 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 009 775 617 707 784 367 872 409 6;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 009 775 617 707 784 367 872 409 6 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 019 551 235 415 568 735 744 819 2;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 019 551 235 415 568 735 744 819 2 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 039 102 470 831 137 471 489 638 4;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 039 102 470 831 137 471 489 638 4 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 078 204 941 662 274 942 979 276 8;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 078 204 941 662 274 942 979 276 8 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 156 409 883 324 549 885 958 553 6;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 156 409 883 324 549 885 958 553 6 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 312 819 766 649 099 771 917 107 2;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 312 819 766 649 099 771 917 107 2 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 625 639 533 298 199 543 834 214 4;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 625 639 533 298 199 543 834 214 4 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 001 251 279 066 596 399 087 668 428 8;
  • 80) 0.588 378 906 250 000 000 000 000 000 001 251 279 066 596 399 087 668 428 8 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 002 502 558 133 192 798 175 336 857 6;
  • 81) 0.176 757 812 500 000 000 000 000 000 002 502 558 133 192 798 175 336 857 6 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 005 005 116 266 385 596 350 673 715 2;
  • 82) 0.353 515 625 000 000 000 000 000 000 005 005 116 266 385 596 350 673 715 2 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 010 010 232 532 771 192 701 347 430 4;
  • 83) 0.707 031 250 000 000 000 000 000 000 010 010 232 532 771 192 701 347 430 4 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 020 020 465 065 542 385 402 694 860 8;
  • 84) 0.414 062 500 000 000 000 000 000 000 020 020 465 065 542 385 402 694 860 8 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 040 040 930 131 084 770 805 389 721 6;
  • 85) 0.828 125 000 000 000 000 000 000 000 040 040 930 131 084 770 805 389 721 6 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 080 081 860 262 169 541 610 779 443 2;
  • 86) 0.656 250 000 000 000 000 000 000 000 080 081 860 262 169 541 610 779 443 2 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 160 163 720 524 339 083 221 558 886 4;
  • 87) 0.312 500 000 000 000 000 000 000 000 160 163 720 524 339 083 221 558 886 4 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 320 327 441 048 678 166 443 117 772 8;
  • 88) 0.625 000 000 000 000 000 000 000 000 320 327 441 048 678 166 443 117 772 8 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 640 654 882 097 356 332 886 235 545 6;
  • 89) 0.250 000 000 000 000 000 000 000 000 640 654 882 097 356 332 886 235 545 6 × 2 = 0 + 0.500 000 000 000 000 000 000 000 001 281 309 764 194 712 665 772 471 091 2;
  • 90) 0.500 000 000 000 000 000 000 000 001 281 309 764 194 712 665 772 471 091 2 × 2 = 1 + 0.000 000 000 000 000 000 000 000 002 562 619 528 389 425 331 544 942 182 4;
  • 91) 0.000 000 000 000 000 000 000 000 002 562 619 528 389 425 331 544 942 182 4 × 2 = 0 + 0.000 000 000 000 000 000 000 000 005 125 239 056 778 850 663 089 884 364 8;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 990 1 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