0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 321 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 321 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 642 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 642 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 284 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 284 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 569 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 569 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 139 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 139 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 278 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 278 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 556 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 556 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 113 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 801 113 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 227 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 602 227 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 204 454 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 204 454 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 408 908 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 408 908 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 817 817 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 817 817 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 635 635 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 635 635 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 271 270 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 271 270 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 542 540 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 542 540 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 085 081 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 373 085 081 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 170 163 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 746 170 163 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 340 326 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 492 340 326 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 680 652 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 984 680 652 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 969 361 305 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 969 361 305 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 938 722 611 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 938 722 611 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 877 445 222 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 877 445 222 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 754 890 444 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 754 890 444 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 509 780 889 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 509 780 889 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 019 561 779 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 751 019 561 779 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 039 123 558 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 502 039 123 558 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 078 247 116 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 004 078 247 116 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 156 494 233 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 008 156 494 233 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 016 312 988 467 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 016 312 988 467 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 032 625 976 934 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 032 625 976 934 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 065 251 953 868 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 065 251 953 868 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 130 503 907 737 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 130 503 907 737 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 261 007 815 475 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 261 007 815 475 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 522 015 630 950 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 522 015 630 950 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 705 044 031 261 900 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 705 044 031 261 900 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 410 088 062 523 801 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 410 088 062 523 801 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 820 176 125 047 603 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 820 176 125 047 603 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 640 352 250 095 206 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 640 352 250 095 206 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 280 704 500 190 412 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 280 704 500 190 412 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 561 409 000 380 825 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 561 409 000 380 825 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 122 818 000 761 651 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 122 818 000 761 651 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 245 636 001 523 302 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 245 636 001 523 302 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 491 272 003 046 604 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 491 272 003 046 604 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 982 544 006 093 209 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 982 544 006 093 209 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 965 088 012 186 419 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 965 088 012 186 419 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 930 176 024 372 838 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 930 176 024 372 838 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 860 352 048 745 676 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 860 352 048 745 676 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 519 720 704 097 491 353 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 519 720 704 097 491 353 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 039 441 408 194 982 707 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 039 441 408 194 982 707 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 078 882 816 389 965 414 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 078 882 816 389 965 414 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 157 765 632 779 930 828 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 157 765 632 779 930 828 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 315 531 265 559 861 657 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 315 531 265 559 861 657 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 631 062 531 119 723 315 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 631 062 531 119 723 315 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 262 125 062 239 446 630 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 262 125 062 239 446 630 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 524 250 124 478 893 260 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 524 250 124 478 893 260 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 813 048 500 248 957 786 521 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 813 048 500 248 957 786 521 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 626 097 000 497 915 573 043 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 626 097 000 497 915 573 043 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 252 194 000 995 831 146 086 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 252 194 000 995 831 146 086 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 504 388 001 991 662 292 172 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 504 388 001 991 662 292 172 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 008 776 003 983 324 584 345 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 008 776 003 983 324 584 345 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 017 552 007 966 649 168 691 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 017 552 007 966 649 168 691 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 035 104 015 933 298 337 382 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 035 104 015 933 298 337 382 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 070 208 031 866 596 674 764 8;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 070 208 031 866 596 674 764 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 140 416 063 733 193 349 529 6;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 140 416 063 733 193 349 529 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 280 832 127 466 386 699 059 2;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 280 832 127 466 386 699 059 2 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 561 664 254 932 773 398 118 4;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 561 664 254 932 773 398 118 4 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 001 123 328 509 865 546 796 236 8;
  • 68) 0.663 473 725 318 908 691 406 250 000 001 123 328 509 865 546 796 236 8 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 002 246 657 019 731 093 592 473 6;
  • 69) 0.326 947 450 637 817 382 812 500 000 002 246 657 019 731 093 592 473 6 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 004 493 314 039 462 187 184 947 2;
  • 70) 0.653 894 901 275 634 765 625 000 000 004 493 314 039 462 187 184 947 2 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 008 986 628 078 924 374 369 894 4;
  • 71) 0.307 789 802 551 269 531 250 000 000 008 986 628 078 924 374 369 894 4 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 017 973 256 157 848 748 739 788 8;
  • 72) 0.615 579 605 102 539 062 500 000 000 017 973 256 157 848 748 739 788 8 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 035 946 512 315 697 497 479 577 6;
  • 73) 0.231 159 210 205 078 125 000 000 000 035 946 512 315 697 497 479 577 6 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 071 893 024 631 394 994 959 155 2;
  • 74) 0.462 318 420 410 156 250 000 000 000 071 893 024 631 394 994 959 155 2 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 143 786 049 262 789 989 918 310 4;
  • 75) 0.924 636 840 820 312 500 000 000 000 143 786 049 262 789 989 918 310 4 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 287 572 098 525 579 979 836 620 8;
  • 76) 0.849 273 681 640 625 000 000 000 000 287 572 098 525 579 979 836 620 8 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 575 144 197 051 159 959 673 241 6;
  • 77) 0.698 547 363 281 250 000 000 000 000 575 144 197 051 159 959 673 241 6 × 2 = 1 + 0.397 094 726 562 500 000 000 000 001 150 288 394 102 319 919 346 483 2;
  • 78) 0.397 094 726 562 500 000 000 000 001 150 288 394 102 319 919 346 483 2 × 2 = 0 + 0.794 189 453 125 000 000 000 000 002 300 576 788 204 639 838 692 966 4;
  • 79) 0.794 189 453 125 000 000 000 000 002 300 576 788 204 639 838 692 966 4 × 2 = 1 + 0.588 378 906 250 000 000 000 000 004 601 153 576 409 279 677 385 932 8;
  • 80) 0.588 378 906 250 000 000 000 000 004 601 153 576 409 279 677 385 932 8 × 2 = 1 + 0.176 757 812 500 000 000 000 000 009 202 307 152 818 559 354 771 865 6;
  • 81) 0.176 757 812 500 000 000 000 000 009 202 307 152 818 559 354 771 865 6 × 2 = 0 + 0.353 515 625 000 000 000 000 000 018 404 614 305 637 118 709 543 731 2;
  • 82) 0.353 515 625 000 000 000 000 000 018 404 614 305 637 118 709 543 731 2 × 2 = 0 + 0.707 031 250 000 000 000 000 000 036 809 228 611 274 237 419 087 462 4;
  • 83) 0.707 031 250 000 000 000 000 000 036 809 228 611 274 237 419 087 462 4 × 2 = 1 + 0.414 062 500 000 000 000 000 000 073 618 457 222 548 474 838 174 924 8;
  • 84) 0.414 062 500 000 000 000 000 000 073 618 457 222 548 474 838 174 924 8 × 2 = 0 + 0.828 125 000 000 000 000 000 000 147 236 914 445 096 949 676 349 849 6;
  • 85) 0.828 125 000 000 000 000 000 000 147 236 914 445 096 949 676 349 849 6 × 2 = 1 + 0.656 250 000 000 000 000 000 000 294 473 828 890 193 899 352 699 699 2;
  • 86) 0.656 250 000 000 000 000 000 000 294 473 828 890 193 899 352 699 699 2 × 2 = 1 + 0.312 500 000 000 000 000 000 000 588 947 657 780 387 798 705 399 398 4;
  • 87) 0.312 500 000 000 000 000 000 000 588 947 657 780 387 798 705 399 398 4 × 2 = 0 + 0.625 000 000 000 000 000 000 001 177 895 315 560 775 597 410 798 796 8;
  • 88) 0.625 000 000 000 000 000 000 001 177 895 315 560 775 597 410 798 796 8 × 2 = 1 + 0.250 000 000 000 000 000 000 002 355 790 631 121 551 194 821 597 593 6;
  • 89) 0.250 000 000 000 000 000 000 002 355 790 631 121 551 194 821 597 593 6 × 2 = 0 + 0.500 000 000 000 000 000 000 004 711 581 262 243 102 389 643 195 187 2;
  • 90) 0.500 000 000 000 000 000 000 004 711 581 262 243 102 389 643 195 187 2 × 2 = 1 + 0.000 000 000 000 000 000 000 009 423 162 524 486 204 779 286 390 374 4;
  • 91) 0.000 000 000 000 000 000 000 009 423 162 524 486 204 779 286 390 374 4 × 2 = 0 + 0.000 000 000 000 000 000 000 018 846 325 048 972 409 558 572 780 748 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 160 6 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