0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 118;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 118 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 236;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 236 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 472;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 472 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 944;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 944 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 888;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 888 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 795 776;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 795 776 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 591 552;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 591 552 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 183 104;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 183 104 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 366 208;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 366 208 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 732 416;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 732 416 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 464 832;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 464 832 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 929 664;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 929 664 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 859 328;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 859 328 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 718 656;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 718 656 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 295 437 312;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 295 437 312 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 590 874 624;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 590 874 624 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 181 749 248;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 181 749 248 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 363 498 496;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 363 498 496 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 726 996 992;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 726 996 992 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 453 993 984;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 453 993 984 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 907 987 968;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 907 987 968 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 815 975 936;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 815 975 936 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 631 951 872;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 631 951 872 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 263 903 744;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 263 903 744 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 766 527 807 488;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 766 527 807 488 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 533 055 614 976;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 533 055 614 976 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 066 111 229 952;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 066 111 229 952 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 132 222 459 904;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 132 222 459 904 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 264 444 919 808;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 264 444 919 808 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 528 889 839 616;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 528 889 839 616 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 057 779 679 232;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 057 779 679 232 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 115 559 358 464;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 115 559 358 464 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 231 118 716 928;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 231 118 716 928 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 462 237 433 856;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 462 237 433 856 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 924 474 867 712;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 924 474 867 712 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 569 848 949 735 424;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 569 848 949 735 424 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 139 697 899 470 848;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 139 697 899 470 848 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 279 395 798 941 696;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 279 395 798 941 696 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 558 791 597 883 392;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 558 791 597 883 392 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 117 583 195 766 784;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 117 583 195 766 784 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 235 166 391 533 568;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 235 166 391 533 568 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 470 332 783 067 136;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 470 332 783 067 136 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 940 665 566 134 272;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 940 665 566 134 272 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 881 331 132 268 544;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 881 331 132 268 544 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 699 762 662 264 537 088;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 699 762 662 264 537 088 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 399 525 324 529 074 176;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 399 525 324 529 074 176 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 799 050 649 058 148 352;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 799 050 649 058 148 352 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 598 101 298 116 296 704;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 598 101 298 116 296 704 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 196 202 596 232 593 408;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 196 202 596 232 593 408 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 392 405 192 465 186 816;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 392 405 192 465 186 816 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 784 810 384 930 373 632;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 784 810 384 930 373 632 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 569 620 769 860 747 264;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 569 620 769 860 747 264 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 563 139 241 539 721 494 528;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 563 139 241 539 721 494 528 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 126 278 483 079 442 989 056;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 126 278 483 079 442 989 056 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 252 556 966 158 885 978 112;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 252 556 966 158 885 978 112 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 505 113 932 317 771 956 224;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 505 113 932 317 771 956 224 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 010 227 864 635 543 912 448;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 010 227 864 635 543 912 448 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 020 455 729 271 087 824 896;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 020 455 729 271 087 824 896 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 040 911 458 542 175 649 792;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 040 911 458 542 175 649 792 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 081 822 917 084 351 299 584;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 081 822 917 084 351 299 584 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 163 645 834 168 702 599 168;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 163 645 834 168 702 599 168 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 327 291 668 337 405 198 336;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 327 291 668 337 405 198 336 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 654 583 336 674 810 396 672;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 654 583 336 674 810 396 672 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 001 309 166 673 349 620 793 344;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 001 309 166 673 349 620 793 344 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 002 618 333 346 699 241 586 688;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 002 618 333 346 699 241 586 688 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 005 236 666 693 398 483 173 376;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 005 236 666 693 398 483 173 376 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 010 473 333 386 796 966 346 752;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 010 473 333 386 796 966 346 752 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 020 946 666 773 593 932 693 504;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 020 946 666 773 593 932 693 504 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 041 893 333 547 187 865 387 008;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 041 893 333 547 187 865 387 008 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 083 786 667 094 375 730 774 016;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 083 786 667 094 375 730 774 016 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 167 573 334 188 751 461 548 032;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 167 573 334 188 751 461 548 032 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 335 146 668 377 502 923 096 064;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 335 146 668 377 502 923 096 064 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 670 293 336 755 005 846 192 128;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 670 293 336 755 005 846 192 128 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 001 340 586 673 510 011 692 384 256;
  • 75) 0.924 636 840 820 312 500 000 000 000 001 340 586 673 510 011 692 384 256 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 002 681 173 347 020 023 384 768 512;
  • 76) 0.849 273 681 640 625 000 000 000 000 002 681 173 347 020 023 384 768 512 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 005 362 346 694 040 046 769 537 024;
  • 77) 0.698 547 363 281 250 000 000 000 000 005 362 346 694 040 046 769 537 024 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 010 724 693 388 080 093 539 074 048;
  • 78) 0.397 094 726 562 500 000 000 000 000 010 724 693 388 080 093 539 074 048 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 021 449 386 776 160 187 078 148 096;
  • 79) 0.794 189 453 125 000 000 000 000 000 021 449 386 776 160 187 078 148 096 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 042 898 773 552 320 374 156 296 192;
  • 80) 0.588 378 906 250 000 000 000 000 000 042 898 773 552 320 374 156 296 192 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 085 797 547 104 640 748 312 592 384;
  • 81) 0.176 757 812 500 000 000 000 000 000 085 797 547 104 640 748 312 592 384 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 171 595 094 209 281 496 625 184 768;
  • 82) 0.353 515 625 000 000 000 000 000 000 171 595 094 209 281 496 625 184 768 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 343 190 188 418 562 993 250 369 536;
  • 83) 0.707 031 250 000 000 000 000 000 000 343 190 188 418 562 993 250 369 536 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 686 380 376 837 125 986 500 739 072;
  • 84) 0.414 062 500 000 000 000 000 000 000 686 380 376 837 125 986 500 739 072 × 2 = 0 + 0.828 125 000 000 000 000 000 000 001 372 760 753 674 251 973 001 478 144;
  • 85) 0.828 125 000 000 000 000 000 000 001 372 760 753 674 251 973 001 478 144 × 2 = 1 + 0.656 250 000 000 000 000 000 000 002 745 521 507 348 503 946 002 956 288;
  • 86) 0.656 250 000 000 000 000 000 000 002 745 521 507 348 503 946 002 956 288 × 2 = 1 + 0.312 500 000 000 000 000 000 000 005 491 043 014 697 007 892 005 912 576;
  • 87) 0.312 500 000 000 000 000 000 000 005 491 043 014 697 007 892 005 912 576 × 2 = 0 + 0.625 000 000 000 000 000 000 000 010 982 086 029 394 015 784 011 825 152;
  • 88) 0.625 000 000 000 000 000 000 000 010 982 086 029 394 015 784 011 825 152 × 2 = 1 + 0.250 000 000 000 000 000 000 000 021 964 172 058 788 031 568 023 650 304;
  • 89) 0.250 000 000 000 000 000 000 000 021 964 172 058 788 031 568 023 650 304 × 2 = 0 + 0.500 000 000 000 000 000 000 000 043 928 344 117 576 063 136 047 300 608;
  • 90) 0.500 000 000 000 000 000 000 000 043 928 344 117 576 063 136 047 300 608 × 2 = 1 + 0.000 000 000 000 000 000 000 000 087 856 688 235 152 126 272 094 601 216;
  • 91) 0.000 000 000 000 000 000 000 000 087 856 688 235 152 126 272 094 601 216 × 2 = 0 + 0.000 000 000 000 000 000 000 000 175 713 376 470 304 252 544 189 202 432;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 059 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