0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 75 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 75(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 75(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 75.

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 75 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 671 5;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 671 5 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 343;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 343 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 686;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 686 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 372;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 372 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 744;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 744 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 488;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 488 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 976;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 976 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 952;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 952 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 904;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 904 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 808;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 808 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 231 616;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 231 616 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 463 232;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 463 232 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 926 464;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 926 464 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 852 928;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 852 928 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 705 856;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 705 856 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 411 712;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 411 712 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 823 424;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 823 424 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 646 848;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 646 848 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 293 696;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 293 696 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 587 392;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 587 392 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 174 784;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 174 784 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 346 349 568;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 346 349 568 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 692 699 136;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 692 699 136 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 385 398 272;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 385 398 272 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 770 796 544;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 770 796 544 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 541 593 088;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 541 593 088 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 083 186 176;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 083 186 176 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 166 372 352;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 166 372 352 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 332 744 704;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 332 744 704 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 665 489 408;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 665 489 408 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 545 330 978 816;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 545 330 978 816 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 090 661 957 632;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 090 661 957 632 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 181 323 915 264;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 181 323 915 264 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 362 647 830 528;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 362 647 830 528 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 725 295 661 056;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 725 295 661 056 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 450 591 322 112;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 450 591 322 112 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 901 182 644 224;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 901 182 644 224 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 802 365 288 448;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 802 365 288 448 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 604 730 576 896;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 604 730 576 896 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 647 209 461 153 792;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 647 209 461 153 792 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 294 418 922 307 584;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 294 418 922 307 584 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 588 837 844 615 168;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 588 837 844 615 168 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 177 675 689 230 336;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 177 675 689 230 336 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 355 351 378 460 672;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 355 351 378 460 672 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 710 702 756 921 344;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 710 702 756 921 344 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 421 405 513 842 688;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 421 405 513 842 688 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 842 811 027 685 376;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 842 811 027 685 376 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 685 622 055 370 752;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 685 622 055 370 752 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 371 244 110 741 504;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 371 244 110 741 504 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 742 488 221 483 008;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 742 488 221 483 008 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 245 484 976 442 966 016;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 245 484 976 442 966 016 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 490 969 952 885 932 032;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 490 969 952 885 932 032 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 981 939 905 771 864 064;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 981 939 905 771 864 064 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 963 879 811 543 728 128;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 963 879 811 543 728 128 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 927 759 623 087 456 256;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 927 759 623 087 456 256 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 855 519 246 174 912 512;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 855 519 246 174 912 512 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 711 038 492 349 825 024;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 711 038 492 349 825 024 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 422 076 984 699 650 048;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 422 076 984 699 650 048 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 844 153 969 399 300 096;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 844 153 969 399 300 096 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 688 307 938 798 600 192;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 688 307 938 798 600 192 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 251 376 615 877 597 200 384;
  • 62) 0.338 491 776 958 107 948 303 222 656 251 376 615 877 597 200 384 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 502 753 231 755 194 400 768;
  • 63) 0.676 983 553 916 215 896 606 445 312 502 753 231 755 194 400 768 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 005 506 463 510 388 801 536;
  • 64) 0.353 967 107 832 431 793 212 890 625 005 506 463 510 388 801 536 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 011 012 927 020 777 603 072;
  • 65) 0.707 934 215 664 863 586 425 781 250 011 012 927 020 777 603 072 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 022 025 854 041 555 206 144;
  • 66) 0.415 868 431 329 727 172 851 562 500 022 025 854 041 555 206 144 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 044 051 708 083 110 412 288;
  • 67) 0.831 736 862 659 454 345 703 125 000 044 051 708 083 110 412 288 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 088 103 416 166 220 824 576;
  • 68) 0.663 473 725 318 908 691 406 250 000 088 103 416 166 220 824 576 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 176 206 832 332 441 649 152;
  • 69) 0.326 947 450 637 817 382 812 500 000 176 206 832 332 441 649 152 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 352 413 664 664 883 298 304;
  • 70) 0.653 894 901 275 634 765 625 000 000 352 413 664 664 883 298 304 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 704 827 329 329 766 596 608;
  • 71) 0.307 789 802 551 269 531 250 000 000 704 827 329 329 766 596 608 × 2 = 0 + 0.615 579 605 102 539 062 500 000 001 409 654 658 659 533 193 216;
  • 72) 0.615 579 605 102 539 062 500 000 001 409 654 658 659 533 193 216 × 2 = 1 + 0.231 159 210 205 078 125 000 000 002 819 309 317 319 066 386 432;
  • 73) 0.231 159 210 205 078 125 000 000 002 819 309 317 319 066 386 432 × 2 = 0 + 0.462 318 420 410 156 250 000 000 005 638 618 634 638 132 772 864;
  • 74) 0.462 318 420 410 156 250 000 000 005 638 618 634 638 132 772 864 × 2 = 0 + 0.924 636 840 820 312 500 000 000 011 277 237 269 276 265 545 728;
  • 75) 0.924 636 840 820 312 500 000 000 011 277 237 269 276 265 545 728 × 2 = 1 + 0.849 273 681 640 625 000 000 000 022 554 474 538 552 531 091 456;
  • 76) 0.849 273 681 640 625 000 000 000 022 554 474 538 552 531 091 456 × 2 = 1 + 0.698 547 363 281 250 000 000 000 045 108 949 077 105 062 182 912;
  • 77) 0.698 547 363 281 250 000 000 000 045 108 949 077 105 062 182 912 × 2 = 1 + 0.397 094 726 562 500 000 000 000 090 217 898 154 210 124 365 824;
  • 78) 0.397 094 726 562 500 000 000 000 090 217 898 154 210 124 365 824 × 2 = 0 + 0.794 189 453 125 000 000 000 000 180 435 796 308 420 248 731 648;
  • 79) 0.794 189 453 125 000 000 000 000 180 435 796 308 420 248 731 648 × 2 = 1 + 0.588 378 906 250 000 000 000 000 360 871 592 616 840 497 463 296;
  • 80) 0.588 378 906 250 000 000 000 000 360 871 592 616 840 497 463 296 × 2 = 1 + 0.176 757 812 500 000 000 000 000 721 743 185 233 680 994 926 592;
  • 81) 0.176 757 812 500 000 000 000 000 721 743 185 233 680 994 926 592 × 2 = 0 + 0.353 515 625 000 000 000 000 001 443 486 370 467 361 989 853 184;
  • 82) 0.353 515 625 000 000 000 000 001 443 486 370 467 361 989 853 184 × 2 = 0 + 0.707 031 250 000 000 000 000 002 886 972 740 934 723 979 706 368;
  • 83) 0.707 031 250 000 000 000 000 002 886 972 740 934 723 979 706 368 × 2 = 1 + 0.414 062 500 000 000 000 000 005 773 945 481 869 447 959 412 736;
  • 84) 0.414 062 500 000 000 000 000 005 773 945 481 869 447 959 412 736 × 2 = 0 + 0.828 125 000 000 000 000 000 011 547 890 963 738 895 918 825 472;
  • 85) 0.828 125 000 000 000 000 000 011 547 890 963 738 895 918 825 472 × 2 = 1 + 0.656 250 000 000 000 000 000 023 095 781 927 477 791 837 650 944;
  • 86) 0.656 250 000 000 000 000 000 023 095 781 927 477 791 837 650 944 × 2 = 1 + 0.312 500 000 000 000 000 000 046 191 563 854 955 583 675 301 888;
  • 87) 0.312 500 000 000 000 000 000 046 191 563 854 955 583 675 301 888 × 2 = 0 + 0.625 000 000 000 000 000 000 092 383 127 709 911 167 350 603 776;
  • 88) 0.625 000 000 000 000 000 000 092 383 127 709 911 167 350 603 776 × 2 = 1 + 0.250 000 000 000 000 000 000 184 766 255 419 822 334 701 207 552;
  • 89) 0.250 000 000 000 000 000 000 184 766 255 419 822 334 701 207 552 × 2 = 0 + 0.500 000 000 000 000 000 000 369 532 510 839 644 669 402 415 104;
  • 90) 0.500 000 000 000 000 000 000 369 532 510 839 644 669 402 415 104 × 2 = 1 + 0.000 000 000 000 000 000 000 739 065 021 679 289 338 804 830 208;
  • 91) 0.000 000 000 000 000 000 000 739 065 021 679 289 338 804 830 208 × 2 = 0 + 0.000 000 000 000 000 000 001 478 130 043 358 578 677 609 660 416;

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 75(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 75(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 75(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 75 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