0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 297 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 297 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 594 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 594 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 188 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 188 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 377 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 377 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 755 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 755 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 510 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 510 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 020 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 020 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 041 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 041 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 596 083 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 596 083 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 192 166 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 192 166 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 384 332 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 384 332 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 768 665 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 768 665 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 537 331 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 537 331 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 074 662 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 074 662 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 149 324 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 149 324 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 298 649 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 298 649 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 597 299 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 597 299 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 489 194 598 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 489 194 598 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 978 389 196 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 978 389 196 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 956 778 393 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 956 778 393 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 913 556 787 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 913 556 787 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 827 113 574 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 827 113 574 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 654 227 148 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 654 227 148 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 308 454 297 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 308 454 297 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 616 908 595 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 616 908 595 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 233 817 190 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 233 817 190 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 467 634 380 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 467 634 380 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 935 268 761 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 935 268 761 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 870 537 523 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 870 537 523 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 019 741 075 046 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 019 741 075 046 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 039 482 150 092 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 039 482 150 092 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 078 964 300 185 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 078 964 300 185 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 157 928 600 371 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 157 928 600 371 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 315 857 200 742 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 315 857 200 742 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 631 714 401 484 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 631 714 401 484 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 263 428 802 969 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 263 428 802 969 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 526 857 605 939 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 526 857 605 939 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 637 053 715 211 878 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 637 053 715 211 878 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 274 107 430 423 756 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 274 107 430 423 756 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 548 214 860 847 513 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 548 214 860 847 513 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 096 429 721 695 027 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 096 429 721 695 027 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 192 859 443 390 054 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 192 859 443 390 054 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 385 718 886 780 108 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 385 718 886 780 108 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 771 437 773 560 217 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 771 437 773 560 217 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 542 875 547 120 435 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 542 875 547 120 435 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 085 751 094 240 870 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 085 751 094 240 870 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 171 502 188 481 740 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 171 502 188 481 740 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 516 343 004 376 963 481 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 516 343 004 376 963 481 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 032 686 008 753 926 963 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 032 686 008 753 926 963 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 065 372 017 507 853 926 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 065 372 017 507 853 926 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 130 744 035 015 707 852 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 130 744 035 015 707 852 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 261 488 070 031 415 705 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 261 488 070 031 415 705 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 522 976 140 062 831 411 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 522 976 140 062 831 411 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 045 952 280 125 662 822 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 045 952 280 125 662 822 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 091 904 560 251 325 644 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 091 904 560 251 325 644 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 183 809 120 502 651 289 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 183 809 120 502 651 289 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 367 618 241 005 302 579 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 367 618 241 005 302 579 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 248 735 236 482 010 605 158 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 248 735 236 482 010 605 158 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 497 470 472 964 021 210 316 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 497 470 472 964 021 210 316 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 994 940 945 928 042 420 633 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 994 940 945 928 042 420 633 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 989 881 891 856 084 841 267 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 989 881 891 856 084 841 267 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 979 763 783 712 169 682 534 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 979 763 783 712 169 682 534 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 959 527 567 424 339 365 068 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 959 527 567 424 339 365 068 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 919 055 134 848 678 730 137 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 919 055 134 848 678 730 137 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 838 110 269 697 357 460 275 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 838 110 269 697 357 460 275 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 676 220 539 394 714 920 550 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 676 220 539 394 714 920 550 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 352 441 078 789 429 841 100 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 352 441 078 789 429 841 100 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 998 704 882 157 578 859 682 201 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 998 704 882 157 578 859 682 201 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 997 409 764 315 157 719 364 403 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 997 409 764 315 157 719 364 403 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 994 819 528 630 315 438 728 806 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 994 819 528 630 315 438 728 806 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 989 639 057 260 630 877 457 612 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 989 639 057 260 630 877 457 612 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 979 278 114 521 261 754 915 225 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 979 278 114 521 261 754 915 225 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 958 556 229 042 523 509 830 451 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 958 556 229 042 523 509 830 451 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 917 112 458 085 047 019 660 902 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 917 112 458 085 047 019 660 902 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 834 224 916 170 094 039 321 804 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 834 224 916 170 094 039 321 804 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 668 449 832 340 188 078 643 609 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 668 449 832 340 188 078 643 609 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 336 899 664 680 376 157 287 219 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 336 899 664 680 376 157 287 219 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 998 673 799 329 360 752 314 574 438 4;
  • 79) 0.794 189 453 124 999 999 999 999 998 673 799 329 360 752 314 574 438 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 997 347 598 658 721 504 629 148 876 8;
  • 80) 0.588 378 906 249 999 999 999 999 997 347 598 658 721 504 629 148 876 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 994 695 197 317 443 009 258 297 753 6;
  • 81) 0.176 757 812 499 999 999 999 999 994 695 197 317 443 009 258 297 753 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 989 390 394 634 886 018 516 595 507 2;
  • 82) 0.353 515 624 999 999 999 999 999 989 390 394 634 886 018 516 595 507 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 978 780 789 269 772 037 033 191 014 4;
  • 83) 0.707 031 249 999 999 999 999 999 978 780 789 269 772 037 033 191 014 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 957 561 578 539 544 074 066 382 028 8;
  • 84) 0.414 062 499 999 999 999 999 999 957 561 578 539 544 074 066 382 028 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 915 123 157 079 088 148 132 764 057 6;
  • 85) 0.828 124 999 999 999 999 999 999 915 123 157 079 088 148 132 764 057 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 830 246 314 158 176 296 265 528 115 2;
  • 86) 0.656 249 999 999 999 999 999 999 830 246 314 158 176 296 265 528 115 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 660 492 628 316 352 592 531 056 230 4;
  • 87) 0.312 499 999 999 999 999 999 999 660 492 628 316 352 592 531 056 230 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 320 985 256 632 705 185 062 112 460 8;
  • 88) 0.624 999 999 999 999 999 999 999 320 985 256 632 705 185 062 112 460 8 × 2 = 1 + 0.249 999 999 999 999 999 999 998 641 970 513 265 410 370 124 224 921 6;
  • 89) 0.249 999 999 999 999 999 999 998 641 970 513 265 410 370 124 224 921 6 × 2 = 0 + 0.499 999 999 999 999 999 999 997 283 941 026 530 820 740 248 449 843 2;
  • 90) 0.499 999 999 999 999 999 999 997 283 941 026 530 820 740 248 449 843 2 × 2 = 0 + 0.999 999 999 999 999 999 999 994 567 882 053 061 641 480 496 899 686 4;
  • 91) 0.999 999 999 999 999 999 999 994 567 882 053 061 641 480 496 899 686 4 × 2 = 1 + 0.999 999 999 999 999 999 999 989 135 764 106 123 282 960 993 799 372 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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 001(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(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 1001


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 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


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 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 148 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 1001

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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