0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 078;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 078 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 156;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 156 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 312;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 312 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 624;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 624 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 248;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 248 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 496;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 496 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 992;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 992 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 937 984;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 937 984 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 875 968;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 875 968 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 751 936;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 751 936 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 503 872;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 503 872 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 007 744;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 007 744 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 015 488;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 015 488 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 556 030 976;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 556 030 976 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 112 061 952;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 112 061 952 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 224 123 904;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 224 123 904 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 448 247 808;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 448 247 808 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 896 495 616;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 896 495 616 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 792 991 232;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 792 991 232 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 585 982 464;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 585 982 464 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 171 964 928;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 171 964 928 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 343 929 856;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 343 929 856 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 687 859 712;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 687 859 712 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 375 719 424;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 375 719 424 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 751 438 848;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 751 438 848 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 502 877 696;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 502 877 696 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 587 005 755 392;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 587 005 755 392 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 174 011 510 784;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 174 011 510 784 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 348 023 021 568;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 348 023 021 568 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 696 046 043 136;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 696 046 043 136 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 392 092 086 272;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 392 092 086 272 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 784 184 172 544;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 784 184 172 544 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 568 368 345 088;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 568 368 345 088 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 136 736 690 176;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 136 736 690 176 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 273 473 380 352;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 273 473 380 352 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 546 946 760 704;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 546 946 760 704 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 945 093 893 521 408;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 945 093 893 521 408 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 890 187 787 042 816;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 890 187 787 042 816 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 780 375 574 085 632;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 780 375 574 085 632 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 560 751 148 171 264;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 560 751 148 171 264 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 121 502 296 342 528;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 121 502 296 342 528 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 243 004 592 685 056;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 243 004 592 685 056 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 486 009 185 370 112;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 486 009 185 370 112 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 972 018 370 740 224;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 972 018 370 740 224 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 944 036 741 480 448;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 944 036 741 480 448 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 888 073 482 960 896;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 888 073 482 960 896 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 063 776 146 965 921 792;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 063 776 146 965 921 792 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 127 552 293 931 843 584;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 127 552 293 931 843 584 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 255 104 587 863 687 168;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 255 104 587 863 687 168 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 510 209 175 727 374 336;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 510 209 175 727 374 336 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 020 418 351 454 748 672;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 020 418 351 454 748 672 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 040 836 702 909 497 344;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 040 836 702 909 497 344 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 081 673 405 818 994 688;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 081 673 405 818 994 688 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 163 346 811 637 989 376;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 163 346 811 637 989 376 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 326 693 623 275 978 752;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 326 693 623 275 978 752 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 653 387 246 551 957 504;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 653 387 246 551 957 504 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 001 306 774 493 103 915 008;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 001 306 774 493 103 915 008 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 002 613 548 986 207 830 016;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 002 613 548 986 207 830 016 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 005 227 097 972 415 660 032;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 005 227 097 972 415 660 032 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 010 454 195 944 831 320 064;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 010 454 195 944 831 320 064 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 020 908 391 889 662 640 128;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 020 908 391 889 662 640 128 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 041 816 783 779 325 280 256;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 041 816 783 779 325 280 256 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 083 633 567 558 650 560 512;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 083 633 567 558 650 560 512 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 167 267 135 117 301 121 024;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 167 267 135 117 301 121 024 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 334 534 270 234 602 242 048;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 334 534 270 234 602 242 048 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 669 068 540 469 204 484 096;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 669 068 540 469 204 484 096 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 001 338 137 080 938 408 968 192;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 001 338 137 080 938 408 968 192 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 002 676 274 161 876 817 936 384;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 002 676 274 161 876 817 936 384 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 005 352 548 323 753 635 872 768;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 005 352 548 323 753 635 872 768 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 010 705 096 647 507 271 745 536;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 010 705 096 647 507 271 745 536 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 021 410 193 295 014 543 491 072;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 021 410 193 295 014 543 491 072 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 042 820 386 590 029 086 982 144;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 042 820 386 590 029 086 982 144 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 085 640 773 180 058 173 964 288;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 085 640 773 180 058 173 964 288 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 171 281 546 360 116 347 928 576;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 171 281 546 360 116 347 928 576 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 342 563 092 720 232 695 857 152;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 342 563 092 720 232 695 857 152 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 685 126 185 440 465 391 714 304;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 685 126 185 440 465 391 714 304 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 001 370 252 370 880 930 783 428 608;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 001 370 252 370 880 930 783 428 608 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 002 740 504 741 761 861 566 857 216;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 002 740 504 741 761 861 566 857 216 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 005 481 009 483 523 723 133 714 432;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 005 481 009 483 523 723 133 714 432 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 010 962 018 967 047 446 267 428 864;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 010 962 018 967 047 446 267 428 864 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 021 924 037 934 094 892 534 857 728;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 021 924 037 934 094 892 534 857 728 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 043 848 075 868 189 785 069 715 456;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 043 848 075 868 189 785 069 715 456 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 087 696 151 736 379 570 139 430 912;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 087 696 151 736 379 570 139 430 912 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 175 392 303 472 759 140 278 861 824;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 175 392 303 472 759 140 278 861 824 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 350 784 606 945 518 280 557 723 648;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 350 784 606 945 518 280 557 723 648 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 701 569 213 891 036 561 115 447 296;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 701 569 213 891 036 561 115 447 296 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 001 403 138 427 782 073 122 230 894 592;
  • 88) 0.625 000 000 000 000 000 000 000 000 001 403 138 427 782 073 122 230 894 592 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 002 806 276 855 564 146 244 461 789 184;
  • 89) 0.250 000 000 000 000 000 000 000 000 002 806 276 855 564 146 244 461 789 184 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 005 612 553 711 128 292 488 923 578 368;
  • 90) 0.500 000 000 000 000 000 000 000 000 005 612 553 711 128 292 488 923 578 368 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 011 225 107 422 256 584 977 847 156 736;
  • 91) 0.000 000 000 000 000 000 000 000 000 011 225 107 422 256 584 977 847 156 736 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 022 450 214 844 513 169 955 694 313 472;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 039 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