0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 319 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 319 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 639 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 639 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 278 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 278 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 556 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 556 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 113 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 113 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 227 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 227 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 454 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 454 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 908 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 908 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 817 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 817 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 203 635 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 203 635 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 407 270 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 407 270 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 814 540 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 814 540 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 629 081 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 629 081 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 258 163 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 258 163 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 516 326 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 516 326 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 032 652 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 032 652 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 065 305 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 065 305 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 130 611 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 130 611 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 261 222 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 261 222 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 968 522 444 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 968 522 444 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 937 044 889 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 937 044 889 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 874 089 779 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 874 089 779 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 748 179 558 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 748 179 558 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 496 359 116 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 496 359 116 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 992 718 233 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 992 718 233 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 985 436 467 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 985 436 467 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 970 872 934 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 970 872 934 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 941 745 868 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 941 745 868 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 883 491 737 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 883 491 737 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 031 766 983 475 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 031 766 983 475 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 063 533 966 950 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 063 533 966 950 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 127 067 933 900 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 127 067 933 900 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 254 135 867 801 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 254 135 867 801 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 508 271 735 603 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 508 271 735 603 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 016 543 471 206 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 016 543 471 206 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 410 033 086 942 412 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 410 033 086 942 412 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 820 066 173 884 825 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 820 066 173 884 825 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 640 132 347 769 651 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 640 132 347 769 651 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 280 264 695 539 302 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 280 264 695 539 302 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 560 529 391 078 604 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 560 529 391 078 604 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 121 058 782 157 209 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 121 058 782 157 209 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 242 117 564 314 419 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 242 117 564 314 419 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 484 235 128 628 838 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 484 235 128 628 838 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 968 470 257 257 676 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 968 470 257 257 676 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 936 940 514 515 353 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 936 940 514 515 353 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 873 881 029 030 707 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 873 881 029 030 707 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 747 762 058 061 414 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 747 762 058 061 414 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 519 495 524 116 122 828 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 519 495 524 116 122 828 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 038 991 048 232 245 657 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 038 991 048 232 245 657 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 077 982 096 464 491 315 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 077 982 096 464 491 315 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 155 964 192 928 982 630 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 155 964 192 928 982 630 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 311 928 385 857 965 260 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 311 928 385 857 965 260 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 623 856 771 715 930 521 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 623 856 771 715 930 521 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 247 713 543 431 861 043 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 247 713 543 431 861 043 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 495 427 086 863 722 086 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 495 427 086 863 722 086 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 990 854 173 727 444 172 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 990 854 173 727 444 172 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 981 708 347 454 888 345 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 981 708 347 454 888 345 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 251 963 416 694 909 776 691 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 251 963 416 694 909 776 691 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 503 926 833 389 819 553 382 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 503 926 833 389 819 553 382 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 007 853 666 779 639 106 764 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 007 853 666 779 639 106 764 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 015 707 333 559 278 213 529 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 015 707 333 559 278 213 529 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 031 414 667 118 556 427 059 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 031 414 667 118 556 427 059 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 062 829 334 237 112 854 118 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 062 829 334 237 112 854 118 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 125 658 668 474 225 708 236 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 125 658 668 474 225 708 236 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 251 317 336 948 451 416 473 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 251 317 336 948 451 416 473 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 502 634 673 896 902 832 947 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 502 634 673 896 902 832 947 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 001 005 269 347 793 805 665 894 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 001 005 269 347 793 805 665 894 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 002 010 538 695 587 611 331 788 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 002 010 538 695 587 611 331 788 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 004 021 077 391 175 222 663 577 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 004 021 077 391 175 222 663 577 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 008 042 154 782 350 445 327 155 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 008 042 154 782 350 445 327 155 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 016 084 309 564 700 890 654 310 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 016 084 309 564 700 890 654 310 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 032 168 619 129 401 781 308 620 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 032 168 619 129 401 781 308 620 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 064 337 238 258 803 562 617 241 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 064 337 238 258 803 562 617 241 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 128 674 476 517 607 125 234 483 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 128 674 476 517 607 125 234 483 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 257 348 953 035 214 250 468 966 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 257 348 953 035 214 250 468 966 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 514 697 906 070 428 500 937 932 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 514 697 906 070 428 500 937 932 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 001 029 395 812 140 857 001 875 865 6;
  • 78) 0.397 094 726 562 500 000 000 000 001 029 395 812 140 857 001 875 865 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 002 058 791 624 281 714 003 751 731 2;
  • 79) 0.794 189 453 125 000 000 000 000 002 058 791 624 281 714 003 751 731 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 004 117 583 248 563 428 007 503 462 4;
  • 80) 0.588 378 906 250 000 000 000 000 004 117 583 248 563 428 007 503 462 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 008 235 166 497 126 856 015 006 924 8;
  • 81) 0.176 757 812 500 000 000 000 000 008 235 166 497 126 856 015 006 924 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 016 470 332 994 253 712 030 013 849 6;
  • 82) 0.353 515 625 000 000 000 000 000 016 470 332 994 253 712 030 013 849 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 032 940 665 988 507 424 060 027 699 2;
  • 83) 0.707 031 250 000 000 000 000 000 032 940 665 988 507 424 060 027 699 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 065 881 331 977 014 848 120 055 398 4;
  • 84) 0.414 062 500 000 000 000 000 000 065 881 331 977 014 848 120 055 398 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 131 762 663 954 029 696 240 110 796 8;
  • 85) 0.828 125 000 000 000 000 000 000 131 762 663 954 029 696 240 110 796 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 263 525 327 908 059 392 480 221 593 6;
  • 86) 0.656 250 000 000 000 000 000 000 263 525 327 908 059 392 480 221 593 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 527 050 655 816 118 784 960 443 187 2;
  • 87) 0.312 500 000 000 000 000 000 000 527 050 655 816 118 784 960 443 187 2 × 2 = 0 + 0.625 000 000 000 000 000 000 001 054 101 311 632 237 569 920 886 374 4;
  • 88) 0.625 000 000 000 000 000 000 001 054 101 311 632 237 569 920 886 374 4 × 2 = 1 + 0.250 000 000 000 000 000 000 002 108 202 623 264 475 139 841 772 748 8;
  • 89) 0.250 000 000 000 000 000 000 002 108 202 623 264 475 139 841 772 748 8 × 2 = 0 + 0.500 000 000 000 000 000 000 004 216 405 246 528 950 279 683 545 497 6;
  • 90) 0.500 000 000 000 000 000 000 004 216 405 246 528 950 279 683 545 497 6 × 2 = 1 + 0.000 000 000 000 000 000 000 008 432 810 493 057 900 559 367 090 995 2;
  • 91) 0.000 000 000 000 000 000 000 008 432 810 493 057 900 559 367 090 995 2 × 2 = 0 + 0.000 000 000 000 000 000 000 016 865 620 986 115 801 118 734 181 990 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 8 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