0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 995 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 995 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 991 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 991 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 983 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 983 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 966 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 966 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 932 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 932 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 865 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 865 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 731 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 731 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 167 462 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 167 462 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 334 924 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 334 924 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 669 849 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 669 849 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 339 699 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 339 699 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 679 398 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 679 398 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 358 796 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 358 796 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 717 593 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 717 593 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 435 187 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 435 187 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 870 374 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 870 374 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 740 748 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 740 748 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 347 481 497 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 347 481 497 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 694 962 995 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 694 962 995 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 389 925 990 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 389 925 990 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 779 851 980 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 779 851 980 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 559 703 961 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 559 703 961 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 119 407 923 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 119 407 923 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 238 815 846 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 238 815 846 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 477 631 692 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 477 631 692 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 955 263 385 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 955 263 385 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 910 526 771 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 910 526 771 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 115 821 053 542 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 115 821 053 542 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 231 642 107 084 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 231 642 107 084 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 463 284 214 169 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 463 284 214 169 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 926 568 428 339 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 926 568 428 339 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 853 136 856 678 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 853 136 856 678 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 706 273 713 356 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 706 273 713 356 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 412 547 426 713 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 412 547 426 713 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 825 094 853 427 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 825 094 853 427 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 650 189 706 854 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 650 189 706 854 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 131 300 379 413 708 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 131 300 379 413 708 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 262 600 758 827 417 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 262 600 758 827 417 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 525 201 517 654 835 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 525 201 517 654 835 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 050 403 035 309 670 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 050 403 035 309 670 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 100 806 070 619 340 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 100 806 070 619 340 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 201 612 141 238 681 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 201 612 141 238 681 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 403 224 282 477 363 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 403 224 282 477 363 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 806 448 564 954 726 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 806 448 564 954 726 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 612 897 129 909 452 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 612 897 129 909 452 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 395 225 794 259 818 905 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 395 225 794 259 818 905 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 790 451 588 519 637 811 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 790 451 588 519 637 811 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 580 903 177 039 275 622 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 580 903 177 039 275 622 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 161 806 354 078 551 244 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 161 806 354 078 551 244 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 323 612 708 157 102 489 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 323 612 708 157 102 489 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 647 225 416 314 204 979 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 647 225 416 314 204 979 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 294 450 832 628 409 958 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 294 450 832 628 409 958 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 588 901 665 256 819 916 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 588 901 665 256 819 916 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 177 803 330 513 639 833 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 177 803 330 513 639 833 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 355 606 661 027 279 667 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 355 606 661 027 279 667 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 711 213 322 054 559 334 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 711 213 322 054 559 334 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 001 422 426 644 109 118 668 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 001 422 426 644 109 118 668 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 002 844 853 288 218 237 337 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 002 844 853 288 218 237 337 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 005 689 706 576 436 474 675 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 005 689 706 576 436 474 675 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 011 379 413 152 872 949 350 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 011 379 413 152 872 949 350 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 022 758 826 305 745 898 700 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 022 758 826 305 745 898 700 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 045 517 652 611 491 797 401 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 045 517 652 611 491 797 401 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 091 035 305 222 983 594 803 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 091 035 305 222 983 594 803 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 182 070 610 445 967 189 606 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 182 070 610 445 967 189 606 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 364 141 220 891 934 379 212 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 364 141 220 891 934 379 212 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 728 282 441 783 868 758 425 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 728 282 441 783 868 758 425 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 001 456 564 883 567 737 516 851 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 001 456 564 883 567 737 516 851 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 002 913 129 767 135 475 033 702 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 002 913 129 767 135 475 033 702 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 005 826 259 534 270 950 067 404 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 005 826 259 534 270 950 067 404 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 011 652 519 068 541 900 134 809 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 011 652 519 068 541 900 134 809 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 023 305 038 137 083 800 269 619 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 023 305 038 137 083 800 269 619 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 046 610 076 274 167 600 539 238 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 046 610 076 274 167 600 539 238 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 093 220 152 548 335 201 078 476 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 093 220 152 548 335 201 078 476 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 186 440 305 096 670 402 156 953 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 186 440 305 096 670 402 156 953 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 372 880 610 193 340 804 313 907 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 372 880 610 193 340 804 313 907 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 745 761 220 386 681 608 627 814 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 745 761 220 386 681 608 627 814 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 001 491 522 440 773 363 217 255 628 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 001 491 522 440 773 363 217 255 628 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 002 983 044 881 546 726 434 511 257 6;
  • 79) 0.794 189 453 125 000 000 000 000 000 002 983 044 881 546 726 434 511 257 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 005 966 089 763 093 452 869 022 515 2;
  • 80) 0.588 378 906 250 000 000 000 000 000 005 966 089 763 093 452 869 022 515 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 011 932 179 526 186 905 738 045 030 4;
  • 81) 0.176 757 812 500 000 000 000 000 000 011 932 179 526 186 905 738 045 030 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 023 864 359 052 373 811 476 090 060 8;
  • 82) 0.353 515 625 000 000 000 000 000 000 023 864 359 052 373 811 476 090 060 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 047 728 718 104 747 622 952 180 121 6;
  • 83) 0.707 031 250 000 000 000 000 000 000 047 728 718 104 747 622 952 180 121 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 095 457 436 209 495 245 904 360 243 2;
  • 84) 0.414 062 500 000 000 000 000 000 000 095 457 436 209 495 245 904 360 243 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 190 914 872 418 990 491 808 720 486 4;
  • 85) 0.828 125 000 000 000 000 000 000 000 190 914 872 418 990 491 808 720 486 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 381 829 744 837 980 983 617 440 972 8;
  • 86) 0.656 250 000 000 000 000 000 000 000 381 829 744 837 980 983 617 440 972 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 763 659 489 675 961 967 234 881 945 6;
  • 87) 0.312 500 000 000 000 000 000 000 000 763 659 489 675 961 967 234 881 945 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 001 527 318 979 351 923 934 469 763 891 2;
  • 88) 0.625 000 000 000 000 000 000 000 001 527 318 979 351 923 934 469 763 891 2 × 2 = 1 + 0.250 000 000 000 000 000 000 000 003 054 637 958 703 847 868 939 527 782 4;
  • 89) 0.250 000 000 000 000 000 000 000 003 054 637 958 703 847 868 939 527 782 4 × 2 = 0 + 0.500 000 000 000 000 000 000 000 006 109 275 917 407 695 737 879 055 564 8;
  • 90) 0.500 000 000 000 000 000 000 000 006 109 275 917 407 695 737 879 055 564 8 × 2 = 1 + 0.000 000 000 000 000 000 000 000 012 218 551 834 815 391 475 758 111 129 6;
  • 91) 0.000 000 000 000 000 000 000 000 012 218 551 834 815 391 475 758 111 129 6 × 2 = 0 + 0.000 000 000 000 000 000 000 000 024 437 103 669 630 782 951 516 222 259 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 997 9 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