0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 024 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 024 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 049 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 049 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 098 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 098 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 196 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 196 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 393 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 393 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 792 787 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 792 787 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 585 574 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 585 574 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 171 148 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 171 148 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 342 297 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 342 297 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 684 595 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 684 595 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 369 190 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 369 190 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 738 380 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 738 380 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 476 761 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 476 761 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 953 523 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 953 523 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 907 046 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 907 046 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 587 814 092 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 587 814 092 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 175 628 185 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 175 628 185 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 351 256 371 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 351 256 371 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 702 512 742 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 702 512 742 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 405 025 484 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 405 025 484 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 810 050 969 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 810 050 969 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 620 101 939 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 620 101 939 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 240 203 878 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 240 203 878 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 480 407 756 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 480 407 756 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 960 815 513 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 960 815 513 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 529 921 631 027 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 529 921 631 027 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 059 843 262 054 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 059 843 262 054 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 119 686 524 108 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 119 686 524 108 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 239 373 048 217 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 239 373 048 217 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 478 746 096 435 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 478 746 096 435 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 957 492 192 870 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 957 492 192 870 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 914 984 385 740 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 914 984 385 740 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 829 968 771 481 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 829 968 771 481 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 659 937 542 963 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 659 937 542 963 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 319 875 085 926 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 319 875 085 926 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 639 750 171 852 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 639 750 171 852 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 133 279 500 343 705 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 133 279 500 343 705 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 266 559 000 687 411 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 266 559 000 687 411 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 533 118 001 374 822 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 533 118 001 374 822 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 066 236 002 749 644 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 066 236 002 749 644 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 132 472 005 499 289 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 132 472 005 499 289 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 264 944 010 998 579 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 264 944 010 998 579 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 529 888 021 997 158 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 529 888 021 997 158 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 059 776 043 994 316 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 059 776 043 994 316 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 698 119 552 087 988 633 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 698 119 552 087 988 633 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 396 239 104 175 977 267 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 396 239 104 175 977 267 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 792 478 208 351 954 534 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 792 478 208 351 954 534 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 584 956 416 703 909 068 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 584 956 416 703 909 068 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 169 912 833 407 818 137 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 169 912 833 407 818 137 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 339 825 666 815 636 275 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 339 825 666 815 636 275 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 679 651 333 631 272 550 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 679 651 333 631 272 550 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 359 302 667 262 545 100 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 359 302 667 262 545 100 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 718 605 334 525 090 201 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 718 605 334 525 090 201 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 437 210 669 050 180 403 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 437 210 669 050 180 403 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 874 421 338 100 360 806 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 874 421 338 100 360 806 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 501 748 842 676 200 721 612 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 501 748 842 676 200 721 612 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 003 497 685 352 401 443 225 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 003 497 685 352 401 443 225 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 006 995 370 704 802 886 451 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 006 995 370 704 802 886 451 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 013 990 741 409 605 772 902 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 013 990 741 409 605 772 902 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 027 981 482 819 211 545 804 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 027 981 482 819 211 545 804 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 055 962 965 638 423 091 609 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 055 962 965 638 423 091 609 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 111 925 931 276 846 183 219 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 111 925 931 276 846 183 219 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 223 851 862 553 692 366 438 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 223 851 862 553 692 366 438 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 447 703 725 107 384 732 876 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 447 703 725 107 384 732 876 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 895 407 450 214 769 465 753 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 895 407 450 214 769 465 753 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 001 790 814 900 429 538 931 507 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 001 790 814 900 429 538 931 507 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 003 581 629 800 859 077 863 014 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 003 581 629 800 859 077 863 014 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 007 163 259 601 718 155 726 028 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 007 163 259 601 718 155 726 028 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 014 326 519 203 436 311 452 057 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 014 326 519 203 436 311 452 057 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 028 653 038 406 872 622 904 115 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 028 653 038 406 872 622 904 115 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 057 306 076 813 745 245 808 230 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 057 306 076 813 745 245 808 230 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 114 612 153 627 490 491 616 460 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 114 612 153 627 490 491 616 460 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 229 224 307 254 980 983 232 921 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 229 224 307 254 980 983 232 921 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 458 448 614 509 961 966 465 843 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 458 448 614 509 961 966 465 843 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 916 897 229 019 923 932 931 686 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 916 897 229 019 923 932 931 686 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 001 833 794 458 039 847 865 863 372 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 001 833 794 458 039 847 865 863 372 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 003 667 588 916 079 695 731 726 745 6;
  • 78) 0.397 094 726 562 500 000 000 000 000 003 667 588 916 079 695 731 726 745 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 007 335 177 832 159 391 463 453 491 2;
  • 79) 0.794 189 453 125 000 000 000 000 000 007 335 177 832 159 391 463 453 491 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 014 670 355 664 318 782 926 906 982 4;
  • 80) 0.588 378 906 250 000 000 000 000 000 014 670 355 664 318 782 926 906 982 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 029 340 711 328 637 565 853 813 964 8;
  • 81) 0.176 757 812 500 000 000 000 000 000 029 340 711 328 637 565 853 813 964 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 058 681 422 657 275 131 707 627 929 6;
  • 82) 0.353 515 625 000 000 000 000 000 000 058 681 422 657 275 131 707 627 929 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 117 362 845 314 550 263 415 255 859 2;
  • 83) 0.707 031 250 000 000 000 000 000 000 117 362 845 314 550 263 415 255 859 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 234 725 690 629 100 526 830 511 718 4;
  • 84) 0.414 062 500 000 000 000 000 000 000 234 725 690 629 100 526 830 511 718 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 469 451 381 258 201 053 661 023 436 8;
  • 85) 0.828 125 000 000 000 000 000 000 000 469 451 381 258 201 053 661 023 436 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 938 902 762 516 402 107 322 046 873 6;
  • 86) 0.656 250 000 000 000 000 000 000 000 938 902 762 516 402 107 322 046 873 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 001 877 805 525 032 804 214 644 093 747 2;
  • 87) 0.312 500 000 000 000 000 000 000 001 877 805 525 032 804 214 644 093 747 2 × 2 = 0 + 0.625 000 000 000 000 000 000 000 003 755 611 050 065 608 429 288 187 494 4;
  • 88) 0.625 000 000 000 000 000 000 000 003 755 611 050 065 608 429 288 187 494 4 × 2 = 1 + 0.250 000 000 000 000 000 000 000 007 511 222 100 131 216 858 576 374 988 8;
  • 89) 0.250 000 000 000 000 000 000 000 007 511 222 100 131 216 858 576 374 988 8 × 2 = 0 + 0.500 000 000 000 000 000 000 000 015 022 444 200 262 433 717 152 749 977 6;
  • 90) 0.500 000 000 000 000 000 000 000 015 022 444 200 262 433 717 152 749 977 6 × 2 = 1 + 0.000 000 000 000 000 000 000 000 030 044 888 400 524 867 434 305 499 955 2;
  • 91) 0.000 000 000 000 000 000 000 000 030 044 888 400 524 867 434 305 499 955 2 × 2 = 0 + 0.000 000 000 000 000 000 000 000 060 089 776 801 049 734 868 610 999 910 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 012 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