0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 033;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 033 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 066;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 066 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 132;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 132 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 264;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 264 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 528;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 896 528 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 793 056;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 793 056 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 586 112;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 586 112 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 172 224;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 172 224 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 344 448;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 344 448 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 688 896;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 688 896 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 377 792;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 377 792 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 755 584;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 755 584 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 511 168;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 511 168 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 022 336;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 022 336 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 044 672;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 044 672 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 588 089 344;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 588 089 344 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 176 178 688;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 176 178 688 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 352 357 376;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 352 357 376 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 704 714 752;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 704 714 752 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 409 429 504;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 409 429 504 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 818 859 008;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 818 859 008 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 637 718 016;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 637 718 016 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 275 436 032;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 275 436 032 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 550 872 064;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 550 872 064 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 101 744 128;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 101 744 128 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 530 203 488 256;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 530 203 488 256 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 060 406 976 512;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 060 406 976 512 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 120 813 953 024;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 120 813 953 024 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 241 627 906 048;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 241 627 906 048 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 483 255 812 096;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 483 255 812 096 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 966 511 624 192;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 966 511 624 192 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 933 023 248 384;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 933 023 248 384 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 866 046 496 768;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 866 046 496 768 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 732 092 993 536;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 732 092 993 536 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 464 185 987 072;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 783 464 185 987 072 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 928 371 974 144;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 566 928 371 974 144 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 133 856 743 948 288;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 133 856 743 948 288 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 267 713 487 896 576;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 267 713 487 896 576 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 535 426 975 793 152;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 535 426 975 793 152 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 070 853 951 586 304;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 070 853 951 586 304 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 141 707 903 172 608;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 141 707 903 172 608 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 283 415 806 345 216;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 283 415 806 345 216 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 566 831 612 690 432;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 566 831 612 690 432 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 133 663 225 380 864;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 133 663 225 380 864 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 698 267 326 450 761 728;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 698 267 326 450 761 728 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 396 534 652 901 523 456;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 396 534 652 901 523 456 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 793 069 305 803 046 912;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 793 069 305 803 046 912 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 586 138 611 606 093 824;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 586 138 611 606 093 824 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 172 277 223 212 187 648;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 172 277 223 212 187 648 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 344 554 446 424 375 296;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 344 554 446 424 375 296 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 689 108 892 848 750 592;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 689 108 892 848 750 592 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 378 217 785 697 501 184;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 378 217 785 697 501 184 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 756 435 571 395 002 368;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 756 435 571 395 002 368 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 512 871 142 790 004 736;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 512 871 142 790 004 736 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 251 025 742 285 580 009 472;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 251 025 742 285 580 009 472 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 502 051 484 571 160 018 944;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 502 051 484 571 160 018 944 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 004 102 969 142 320 037 888;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 004 102 969 142 320 037 888 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 008 205 938 284 640 075 776;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 008 205 938 284 640 075 776 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 016 411 876 569 280 151 552;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 016 411 876 569 280 151 552 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 032 823 753 138 560 303 104;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 032 823 753 138 560 303 104 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 065 647 506 277 120 606 208;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 065 647 506 277 120 606 208 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 131 295 012 554 241 212 416;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 131 295 012 554 241 212 416 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 262 590 025 108 482 424 832;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 262 590 025 108 482 424 832 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 525 180 050 216 964 849 664;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 525 180 050 216 964 849 664 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 001 050 360 100 433 929 699 328;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 001 050 360 100 433 929 699 328 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 002 100 720 200 867 859 398 656;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 002 100 720 200 867 859 398 656 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 004 201 440 401 735 718 797 312;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 004 201 440 401 735 718 797 312 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 008 402 880 803 471 437 594 624;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 008 402 880 803 471 437 594 624 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 016 805 761 606 942 875 189 248;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 016 805 761 606 942 875 189 248 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 033 611 523 213 885 750 378 496;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 033 611 523 213 885 750 378 496 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 067 223 046 427 771 500 756 992;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 067 223 046 427 771 500 756 992 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 134 446 092 855 543 001 513 984;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 134 446 092 855 543 001 513 984 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 268 892 185 711 086 003 027 968;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 268 892 185 711 086 003 027 968 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 537 784 371 422 172 006 055 936;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 537 784 371 422 172 006 055 936 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 001 075 568 742 844 344 012 111 872;
  • 76) 0.849 273 681 640 625 000 000 000 000 001 075 568 742 844 344 012 111 872 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 002 151 137 485 688 688 024 223 744;
  • 77) 0.698 547 363 281 250 000 000 000 000 002 151 137 485 688 688 024 223 744 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 004 302 274 971 377 376 048 447 488;
  • 78) 0.397 094 726 562 500 000 000 000 000 004 302 274 971 377 376 048 447 488 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 008 604 549 942 754 752 096 894 976;
  • 79) 0.794 189 453 125 000 000 000 000 000 008 604 549 942 754 752 096 894 976 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 017 209 099 885 509 504 193 789 952;
  • 80) 0.588 378 906 250 000 000 000 000 000 017 209 099 885 509 504 193 789 952 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 034 418 199 771 019 008 387 579 904;
  • 81) 0.176 757 812 500 000 000 000 000 000 034 418 199 771 019 008 387 579 904 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 068 836 399 542 038 016 775 159 808;
  • 82) 0.353 515 625 000 000 000 000 000 000 068 836 399 542 038 016 775 159 808 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 137 672 799 084 076 033 550 319 616;
  • 83) 0.707 031 250 000 000 000 000 000 000 137 672 799 084 076 033 550 319 616 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 275 345 598 168 152 067 100 639 232;
  • 84) 0.414 062 500 000 000 000 000 000 000 275 345 598 168 152 067 100 639 232 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 550 691 196 336 304 134 201 278 464;
  • 85) 0.828 125 000 000 000 000 000 000 000 550 691 196 336 304 134 201 278 464 × 2 = 1 + 0.656 250 000 000 000 000 000 000 001 101 382 392 672 608 268 402 556 928;
  • 86) 0.656 250 000 000 000 000 000 000 001 101 382 392 672 608 268 402 556 928 × 2 = 1 + 0.312 500 000 000 000 000 000 000 002 202 764 785 345 216 536 805 113 856;
  • 87) 0.312 500 000 000 000 000 000 000 002 202 764 785 345 216 536 805 113 856 × 2 = 0 + 0.625 000 000 000 000 000 000 000 004 405 529 570 690 433 073 610 227 712;
  • 88) 0.625 000 000 000 000 000 000 000 004 405 529 570 690 433 073 610 227 712 × 2 = 1 + 0.250 000 000 000 000 000 000 000 008 811 059 141 380 866 147 220 455 424;
  • 89) 0.250 000 000 000 000 000 000 000 008 811 059 141 380 866 147 220 455 424 × 2 = 0 + 0.500 000 000 000 000 000 000 000 017 622 118 282 761 732 294 440 910 848;
  • 90) 0.500 000 000 000 000 000 000 000 017 622 118 282 761 732 294 440 910 848 × 2 = 1 + 0.000 000 000 000 000 000 000 000 035 244 236 565 523 464 588 881 821 696;
  • 91) 0.000 000 000 000 000 000 000 000 035 244 236 565 523 464 588 881 821 696 × 2 = 0 + 0.000 000 000 000 000 000 000 000 070 488 473 131 046 929 177 763 643 392;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 016 5 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