0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 168;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 168 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 336;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 336 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 672;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 672 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 449 344;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 449 344 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 898 688;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 898 688 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 797 376;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 797 376 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 594 752;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 594 752 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 189 504;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 189 504 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 379 008;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 379 008 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 758 016;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 758 016 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 516 032;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 516 032 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 032 064;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 032 064 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 064 128;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 574 064 128 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 148 128 256;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 148 128 256 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 296 256 512;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 296 256 512 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 592 513 024;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 592 513 024 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 185 026 048;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 185 026 048 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 370 052 096;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 370 052 096 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 740 104 192;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 740 104 192 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 480 208 384;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 480 208 384 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 960 416 768;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 960 416 768 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 920 833 536;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 920 833 536 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 841 667 072;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 841 667 072 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 683 334 144;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 383 683 334 144 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 767 366 668 288;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 767 366 668 288 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 534 733 336 576;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 534 733 336 576 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 069 466 673 152;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 069 466 673 152 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 138 933 346 304;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 138 933 346 304 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 277 866 692 608;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 277 866 692 608 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 555 733 385 216;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 555 733 385 216 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 111 466 770 432;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 111 466 770 432 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 222 933 540 864;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 222 933 540 864 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 445 867 081 728;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 445 867 081 728 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 891 734 163 456;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 891 734 163 456 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 785 783 468 326 912;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 785 783 468 326 912 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 571 566 936 653 824;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 571 566 936 653 824 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 143 133 873 307 648;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 143 133 873 307 648 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 286 267 746 615 296;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 286 267 746 615 296 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 572 535 493 230 592;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 572 535 493 230 592 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 145 070 986 461 184;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 145 070 986 461 184 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 290 141 972 922 368;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 290 141 972 922 368 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 580 283 945 844 736;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 580 283 945 844 736 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 425 160 567 891 689 472;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 425 160 567 891 689 472 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 850 321 135 783 378 944;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 850 321 135 783 378 944 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 700 642 271 566 757 888;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 700 642 271 566 757 888 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 401 284 543 133 515 776;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 401 284 543 133 515 776 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 802 569 086 267 031 552;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 802 569 086 267 031 552 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 605 138 172 534 063 104;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 605 138 172 534 063 104 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 210 276 345 068 126 208;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 210 276 345 068 126 208 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 420 552 690 136 252 416;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 420 552 690 136 252 416 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 841 105 380 272 504 832;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 841 105 380 272 504 832 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 682 210 760 545 009 664;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 682 210 760 545 009 664 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 563 364 421 521 090 019 328;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 563 364 421 521 090 019 328 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 126 728 843 042 180 038 656;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 126 728 843 042 180 038 656 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 253 457 686 084 360 077 312;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 253 457 686 084 360 077 312 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 506 915 372 168 720 154 624;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 506 915 372 168 720 154 624 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 013 830 744 337 440 309 248;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 013 830 744 337 440 309 248 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 027 661 488 674 880 618 496;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 027 661 488 674 880 618 496 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 055 322 977 349 761 236 992;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 055 322 977 349 761 236 992 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 110 645 954 699 522 473 984;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 110 645 954 699 522 473 984 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 221 291 909 399 044 947 968;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 221 291 909 399 044 947 968 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 442 583 818 798 089 895 936;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 442 583 818 798 089 895 936 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 885 167 637 596 179 791 872;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 885 167 637 596 179 791 872 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 001 770 335 275 192 359 583 744;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 001 770 335 275 192 359 583 744 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 003 540 670 550 384 719 167 488;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 003 540 670 550 384 719 167 488 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 007 081 341 100 769 438 334 976;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 007 081 341 100 769 438 334 976 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 014 162 682 201 538 876 669 952;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 014 162 682 201 538 876 669 952 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 028 325 364 403 077 753 339 904;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 028 325 364 403 077 753 339 904 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 056 650 728 806 155 506 679 808;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 056 650 728 806 155 506 679 808 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 113 301 457 612 311 013 359 616;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 113 301 457 612 311 013 359 616 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 226 602 915 224 622 026 719 232;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 226 602 915 224 622 026 719 232 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 453 205 830 449 244 053 438 464;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 453 205 830 449 244 053 438 464 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 906 411 660 898 488 106 876 928;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 906 411 660 898 488 106 876 928 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 001 812 823 321 796 976 213 753 856;
  • 75) 0.924 636 840 820 312 500 000 000 000 001 812 823 321 796 976 213 753 856 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 003 625 646 643 593 952 427 507 712;
  • 76) 0.849 273 681 640 625 000 000 000 000 003 625 646 643 593 952 427 507 712 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 007 251 293 287 187 904 855 015 424;
  • 77) 0.698 547 363 281 250 000 000 000 000 007 251 293 287 187 904 855 015 424 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 014 502 586 574 375 809 710 030 848;
  • 78) 0.397 094 726 562 500 000 000 000 000 014 502 586 574 375 809 710 030 848 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 029 005 173 148 751 619 420 061 696;
  • 79) 0.794 189 453 125 000 000 000 000 000 029 005 173 148 751 619 420 061 696 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 058 010 346 297 503 238 840 123 392;
  • 80) 0.588 378 906 250 000 000 000 000 000 058 010 346 297 503 238 840 123 392 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 116 020 692 595 006 477 680 246 784;
  • 81) 0.176 757 812 500 000 000 000 000 000 116 020 692 595 006 477 680 246 784 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 232 041 385 190 012 955 360 493 568;
  • 82) 0.353 515 625 000 000 000 000 000 000 232 041 385 190 012 955 360 493 568 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 464 082 770 380 025 910 720 987 136;
  • 83) 0.707 031 250 000 000 000 000 000 000 464 082 770 380 025 910 720 987 136 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 928 165 540 760 051 821 441 974 272;
  • 84) 0.414 062 500 000 000 000 000 000 000 928 165 540 760 051 821 441 974 272 × 2 = 0 + 0.828 125 000 000 000 000 000 000 001 856 331 081 520 103 642 883 948 544;
  • 85) 0.828 125 000 000 000 000 000 000 001 856 331 081 520 103 642 883 948 544 × 2 = 1 + 0.656 250 000 000 000 000 000 000 003 712 662 163 040 207 285 767 897 088;
  • 86) 0.656 250 000 000 000 000 000 000 003 712 662 163 040 207 285 767 897 088 × 2 = 1 + 0.312 500 000 000 000 000 000 000 007 425 324 326 080 414 571 535 794 176;
  • 87) 0.312 500 000 000 000 000 000 000 007 425 324 326 080 414 571 535 794 176 × 2 = 0 + 0.625 000 000 000 000 000 000 000 014 850 648 652 160 829 143 071 588 352;
  • 88) 0.625 000 000 000 000 000 000 000 014 850 648 652 160 829 143 071 588 352 × 2 = 1 + 0.250 000 000 000 000 000 000 000 029 701 297 304 321 658 286 143 176 704;
  • 89) 0.250 000 000 000 000 000 000 000 029 701 297 304 321 658 286 143 176 704 × 2 = 0 + 0.500 000 000 000 000 000 000 000 059 402 594 608 643 316 572 286 353 408;
  • 90) 0.500 000 000 000 000 000 000 000 059 402 594 608 643 316 572 286 353 408 × 2 = 1 + 0.000 000 000 000 000 000 000 000 118 805 189 217 286 633 144 572 706 816;
  • 91) 0.000 000 000 000 000 000 000 000 118 805 189 217 286 633 144 572 706 816 × 2 = 0 + 0.000 000 000 000 000 000 000 000 237 610 378 434 573 266 289 145 413 632;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 084 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