0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

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

Keep track of each remainder.

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


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

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

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

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


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 018 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 018 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 037 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 037 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 074 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 074 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 148 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 148 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 297 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 297 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 792 595 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 792 595 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 585 190 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 585 190 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 170 380 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 170 380 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 340 761 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 340 761 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 681 523 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 681 523 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 363 046 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 363 046 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 726 092 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 726 092 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 452 185 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 452 185 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 904 371 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 904 371 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 808 742 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 808 742 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 587 617 484 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 587 617 484 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 175 234 969 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 175 234 969 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 350 469 939 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 350 469 939 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 700 939 878 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 700 939 878 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 401 879 756 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 401 879 756 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 803 759 513 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 803 759 513 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 607 519 027 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 607 519 027 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 215 038 054 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 215 038 054 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 430 076 108 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 430 076 108 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 860 152 217 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 860 152 217 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 529 720 304 435 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 529 720 304 435 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 059 440 608 870 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 059 440 608 870 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 118 881 217 740 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 118 881 217 740 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 237 762 435 481 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 237 762 435 481 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 475 524 870 963 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 475 524 870 963 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 951 049 741 926 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 951 049 741 926 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 902 099 483 852 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 902 099 483 852 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 804 198 967 705 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 804 198 967 705 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 608 397 935 411 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 608 397 935 411 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 216 795 870 822 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 216 795 870 822 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 433 591 741 644 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 433 591 741 644 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 132 867 183 483 289 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 132 867 183 483 289 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 265 734 366 966 579 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 265 734 366 966 579 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 531 468 733 933 158 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 531 468 733 933 158 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 062 937 467 866 316 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 062 937 467 866 316 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 125 874 935 732 633 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 125 874 935 732 633 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 251 749 871 465 267 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 251 749 871 465 267 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 503 499 742 930 534 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 503 499 742 930 534 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 006 999 485 861 068 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 006 999 485 861 068 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 698 013 998 971 722 137 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 698 013 998 971 722 137 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 396 027 997 943 444 275 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 396 027 997 943 444 275 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 792 055 995 886 888 550 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 792 055 995 886 888 550 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 584 111 991 773 777 100 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 584 111 991 773 777 100 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 168 223 983 547 554 201 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 168 223 983 547 554 201 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 336 447 967 095 108 403 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 336 447 967 095 108 403 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 672 895 934 190 216 806 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 672 895 934 190 216 806 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 345 791 868 380 433 612 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 345 791 868 380 433 612 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 691 583 736 760 867 225 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 691 583 736 760 867 225 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 383 167 473 521 734 451 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 383 167 473 521 734 451 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 766 334 947 043 468 902 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 766 334 947 043 468 902 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 501 532 669 894 086 937 804 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 501 532 669 894 086 937 804 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 003 065 339 788 173 875 609 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 003 065 339 788 173 875 609 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 006 130 679 576 347 751 219 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 006 130 679 576 347 751 219 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 012 261 359 152 695 502 438 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 012 261 359 152 695 502 438 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 024 522 718 305 391 004 876 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 024 522 718 305 391 004 876 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 049 045 436 610 782 009 753 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 049 045 436 610 782 009 753 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 098 090 873 221 564 019 507 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 098 090 873 221 564 019 507 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 196 181 746 443 128 039 014 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 196 181 746 443 128 039 014 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 392 363 492 886 256 078 028 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 392 363 492 886 256 078 028 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 784 726 985 772 512 156 057 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 784 726 985 772 512 156 057 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 001 569 453 971 545 024 312 115 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 001 569 453 971 545 024 312 115 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 003 138 907 943 090 048 624 230 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 003 138 907 943 090 048 624 230 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 006 277 815 886 180 097 248 460 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 006 277 815 886 180 097 248 460 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 012 555 631 772 360 194 496 921 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 012 555 631 772 360 194 496 921 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 025 111 263 544 720 388 993 843 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 025 111 263 544 720 388 993 843 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 050 222 527 089 440 777 987 686 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 050 222 527 089 440 777 987 686 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 100 445 054 178 881 555 975 372 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 100 445 054 178 881 555 975 372 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 200 890 108 357 763 111 950 745 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 200 890 108 357 763 111 950 745 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 401 780 216 715 526 223 901 491 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 401 780 216 715 526 223 901 491 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 803 560 433 431 052 447 802 982 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 803 560 433 431 052 447 802 982 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 001 607 120 866 862 104 895 605 964 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 001 607 120 866 862 104 895 605 964 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 003 214 241 733 724 209 791 211 929 6;
  • 78) 0.397 094 726 562 500 000 000 000 000 003 214 241 733 724 209 791 211 929 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 006 428 483 467 448 419 582 423 859 2;
  • 79) 0.794 189 453 125 000 000 000 000 000 006 428 483 467 448 419 582 423 859 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 012 856 966 934 896 839 164 847 718 4;
  • 80) 0.588 378 906 250 000 000 000 000 000 012 856 966 934 896 839 164 847 718 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 025 713 933 869 793 678 329 695 436 8;
  • 81) 0.176 757 812 500 000 000 000 000 000 025 713 933 869 793 678 329 695 436 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 051 427 867 739 587 356 659 390 873 6;
  • 82) 0.353 515 625 000 000 000 000 000 000 051 427 867 739 587 356 659 390 873 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 102 855 735 479 174 713 318 781 747 2;
  • 83) 0.707 031 250 000 000 000 000 000 000 102 855 735 479 174 713 318 781 747 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 205 711 470 958 349 426 637 563 494 4;
  • 84) 0.414 062 500 000 000 000 000 000 000 205 711 470 958 349 426 637 563 494 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 411 422 941 916 698 853 275 126 988 8;
  • 85) 0.828 125 000 000 000 000 000 000 000 411 422 941 916 698 853 275 126 988 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 822 845 883 833 397 706 550 253 977 6;
  • 86) 0.656 250 000 000 000 000 000 000 000 822 845 883 833 397 706 550 253 977 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 001 645 691 767 666 795 413 100 507 955 2;
  • 87) 0.312 500 000 000 000 000 000 000 001 645 691 767 666 795 413 100 507 955 2 × 2 = 0 + 0.625 000 000 000 000 000 000 000 003 291 383 535 333 590 826 201 015 910 4;
  • 88) 0.625 000 000 000 000 000 000 000 003 291 383 535 333 590 826 201 015 910 4 × 2 = 1 + 0.250 000 000 000 000 000 000 000 006 582 767 070 667 181 652 402 031 820 8;
  • 89) 0.250 000 000 000 000 000 000 000 006 582 767 070 667 181 652 402 031 820 8 × 2 = 0 + 0.500 000 000 000 000 000 000 000 013 165 534 141 334 363 304 804 063 641 6;
  • 90) 0.500 000 000 000 000 000 000 000 013 165 534 141 334 363 304 804 063 641 6 × 2 = 1 + 0.000 000 000 000 000 000 000 000 026 331 068 282 668 726 609 608 127 283 2;
  • 91) 0.000 000 000 000 000 000 000 000 026 331 068 282 668 726 609 608 127 283 2 × 2 = 0 + 0.000 000 000 000 000 000 000 000 052 662 136 565 337 453 219 216 254 566 4;

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


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

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3(10) =

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

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3(10) =

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

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(2) × 2-39


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 009 3 converted to 64 bit double precision IEEE 754 binary floating point representation:

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

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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