0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 388;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 388 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 776;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 776 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 552;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 552 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 811 104;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 811 104 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 622 208;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 622 208 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 244 416;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 244 416 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 488 832;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 488 832 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 977 664;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 977 664 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 955 328;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 955 328 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 910 656;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 910 656 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 821 312;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 821 312 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 642 624;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 642 624 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 279 285 248;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 279 285 248 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 558 570 496;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 558 570 496 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 117 140 992;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 117 140 992 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 234 281 984;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 234 281 984 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 468 563 968;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 468 563 968 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 937 127 936;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 937 127 936 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 874 255 872;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 874 255 872 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 748 511 744;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 748 511 744 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 497 023 488;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 497 023 488 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 994 046 976;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 994 046 976 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 988 093 952;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 988 093 952 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 976 187 904;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 976 187 904 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 151 952 375 808;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 151 952 375 808 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 303 904 751 616;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 303 904 751 616 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 607 809 503 232;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 607 809 503 232 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 215 619 006 464;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 215 619 006 464 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 431 238 012 928;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 431 238 012 928 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 862 476 025 856;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 862 476 025 856 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 724 952 051 712;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 724 952 051 712 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 449 904 103 424;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 449 904 103 424 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 899 808 206 848;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 899 808 206 848 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 245 799 616 413 696;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 245 799 616 413 696 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 491 599 232 827 392;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 491 599 232 827 392 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 983 198 465 654 784;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 983 198 465 654 784 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 966 396 931 309 568;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 966 396 931 309 568 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 932 793 862 619 136;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 932 793 862 619 136 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 865 587 725 238 272;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 865 587 725 238 272 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 731 175 450 476 544;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 731 175 450 476 544 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 462 350 900 953 088;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 462 350 900 953 088 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 924 701 801 906 176;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 924 701 801 906 176 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 849 403 603 812 352;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 849 403 603 812 352 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 635 698 807 207 624 704;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 635 698 807 207 624 704 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 271 397 614 415 249 408;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 271 397 614 415 249 408 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 542 795 228 830 498 816;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 542 795 228 830 498 816 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 085 590 457 660 997 632;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 085 590 457 660 997 632 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 171 180 915 321 995 264;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 171 180 915 321 995 264 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 342 361 830 643 990 528;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 342 361 830 643 990 528 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 684 723 661 287 981 056;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 684 723 661 287 981 056 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 369 447 322 575 962 112;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 369 447 322 575 962 112 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 738 894 645 151 924 224;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 738 894 645 151 924 224 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 501 477 789 290 303 848 448;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 501 477 789 290 303 848 448 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 002 955 578 580 607 696 896;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 002 955 578 580 607 696 896 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 005 911 157 161 215 393 792;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 005 911 157 161 215 393 792 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 011 822 314 322 430 787 584;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 011 822 314 322 430 787 584 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 023 644 628 644 861 575 168;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 023 644 628 644 861 575 168 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 047 289 257 289 723 150 336;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 047 289 257 289 723 150 336 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 094 578 514 579 446 300 672;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 094 578 514 579 446 300 672 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 189 157 029 158 892 601 344;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 189 157 029 158 892 601 344 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 378 314 058 317 785 202 688;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 378 314 058 317 785 202 688 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 756 628 116 635 570 405 376;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 756 628 116 635 570 405 376 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 001 513 256 233 271 140 810 752;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 001 513 256 233 271 140 810 752 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 003 026 512 466 542 281 621 504;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 003 026 512 466 542 281 621 504 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 006 053 024 933 084 563 243 008;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 006 053 024 933 084 563 243 008 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 012 106 049 866 169 126 486 016;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 012 106 049 866 169 126 486 016 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 024 212 099 732 338 252 972 032;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 024 212 099 732 338 252 972 032 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 048 424 199 464 676 505 944 064;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 048 424 199 464 676 505 944 064 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 096 848 398 929 353 011 888 128;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 096 848 398 929 353 011 888 128 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 193 696 797 858 706 023 776 256;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 193 696 797 858 706 023 776 256 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 387 393 595 717 412 047 552 512;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 387 393 595 717 412 047 552 512 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 774 787 191 434 824 095 105 024;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 774 787 191 434 824 095 105 024 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 001 549 574 382 869 648 190 210 048;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 001 549 574 382 869 648 190 210 048 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 003 099 148 765 739 296 380 420 096;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 003 099 148 765 739 296 380 420 096 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 006 198 297 531 478 592 760 840 192;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 006 198 297 531 478 592 760 840 192 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 012 396 595 062 957 185 521 680 384;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 012 396 595 062 957 185 521 680 384 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 024 793 190 125 914 371 043 360 768;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 024 793 190 125 914 371 043 360 768 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 049 586 380 251 828 742 086 721 536;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 049 586 380 251 828 742 086 721 536 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 099 172 760 503 657 484 173 443 072;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 099 172 760 503 657 484 173 443 072 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 198 345 521 007 314 968 346 886 144;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 198 345 521 007 314 968 346 886 144 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 396 691 042 014 629 936 693 772 288;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 396 691 042 014 629 936 693 772 288 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 793 382 084 029 259 873 387 544 576;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 793 382 084 029 259 873 387 544 576 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 001 586 764 168 058 519 746 775 089 152;
  • 84) 0.414 062 500 000 000 000 000 000 000 001 586 764 168 058 519 746 775 089 152 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 003 173 528 336 117 039 493 550 178 304;
  • 85) 0.828 125 000 000 000 000 000 000 000 003 173 528 336 117 039 493 550 178 304 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 006 347 056 672 234 078 987 100 356 608;
  • 86) 0.656 250 000 000 000 000 000 000 000 006 347 056 672 234 078 987 100 356 608 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 012 694 113 344 468 157 974 200 713 216;
  • 87) 0.312 500 000 000 000 000 000 000 000 012 694 113 344 468 157 974 200 713 216 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 025 388 226 688 936 315 948 401 426 432;
  • 88) 0.625 000 000 000 000 000 000 000 000 025 388 226 688 936 315 948 401 426 432 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 050 776 453 377 872 631 896 802 852 864;
  • 89) 0.250 000 000 000 000 000 000 000 000 050 776 453 377 872 631 896 802 852 864 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 101 552 906 755 745 263 793 605 705 728;
  • 90) 0.500 000 000 000 000 000 000 000 000 101 552 906 755 745 263 793 605 705 728 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 203 105 813 511 490 527 587 211 411 456;
  • 91) 0.000 000 000 000 000 000 000 000 000 203 105 813 511 490 527 587 211 411 456 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 406 211 627 022 981 055 174 422 822 912;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 194 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