0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 982 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 982 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 964 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 964 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 929 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 929 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 859 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 859 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 718 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 718 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 436 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 436 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 873 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 873 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 747 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 747 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 331 494 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 331 494 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 662 988 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 662 988 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 325 977 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 325 977 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 651 955 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 651 955 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 303 910 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 303 910 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 607 820 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 607 820 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 215 641 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 215 641 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 431 283 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 431 283 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 862 566 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 862 566 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 725 132 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 725 132 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 691 450 265 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 691 450 265 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 382 900 531 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 382 900 531 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 765 801 062 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 765 801 062 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 531 602 124 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 531 602 124 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 063 204 249 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 063 204 249 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 126 408 499 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 126 408 499 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 252 816 998 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 252 816 998 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 505 633 996 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 505 633 996 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 011 267 993 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 011 267 993 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 114 022 535 987 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 114 022 535 987 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 228 045 071 974 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 228 045 071 974 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 456 090 143 948 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 456 090 143 948 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 912 180 287 897 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 912 180 287 897 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 824 360 575 795 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 824 360 575 795 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 648 721 151 590 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 648 721 151 590 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 297 442 303 180 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 297 442 303 180 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 594 884 606 361 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 594 884 606 361 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 189 769 212 723 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 189 769 212 723 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 379 538 425 446 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 379 538 425 446 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 759 076 850 892 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 759 076 850 892 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 521 518 153 701 785 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 521 518 153 701 785 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 043 036 307 403 571 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 043 036 307 403 571 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 086 072 614 807 142 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 086 072 614 807 142 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 172 145 229 614 284 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 172 145 229 614 284 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 344 290 459 228 569 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 344 290 459 228 569 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 688 580 918 457 139 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 688 580 918 457 139 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 377 161 836 914 278 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 377 161 836 914 278 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 754 323 673 828 556 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 754 323 673 828 556 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 508 647 347 657 113 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 508 647 347 657 113 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 579 017 294 695 314 227 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 579 017 294 695 314 227 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 158 034 589 390 628 454 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 158 034 589 390 628 454 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 316 069 178 781 256 908 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 316 069 178 781 256 908 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 632 138 357 562 513 817 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 632 138 357 562 513 817 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 264 276 715 125 027 635 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 264 276 715 125 027 635 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 528 553 430 250 055 270 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 528 553 430 250 055 270 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 057 106 860 500 110 540 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 057 106 860 500 110 540 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 114 213 721 000 221 081 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 114 213 721 000 221 081 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 228 427 442 000 442 163 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 228 427 442 000 442 163 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 456 854 884 000 884 326 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 456 854 884 000 884 326 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 913 709 768 001 768 652 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 913 709 768 001 768 652 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 001 827 419 536 003 537 305 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 001 827 419 536 003 537 305 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 003 654 839 072 007 074 611 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 003 654 839 072 007 074 611 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 007 309 678 144 014 149 222 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 007 309 678 144 014 149 222 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 014 619 356 288 028 298 444 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 014 619 356 288 028 298 444 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 029 238 712 576 056 596 889 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 029 238 712 576 056 596 889 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 058 477 425 152 113 193 779 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 058 477 425 152 113 193 779 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 116 954 850 304 226 387 558 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 116 954 850 304 226 387 558 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 233 909 700 608 452 775 116 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 233 909 700 608 452 775 116 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 467 819 401 216 905 550 233 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 467 819 401 216 905 550 233 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 935 638 802 433 811 100 467 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 935 638 802 433 811 100 467 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 001 871 277 604 867 622 200 934 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 001 871 277 604 867 622 200 934 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 003 742 555 209 735 244 401 868 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 003 742 555 209 735 244 401 868 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 007 485 110 419 470 488 803 737 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 007 485 110 419 470 488 803 737 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 014 970 220 838 940 977 607 475 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 014 970 220 838 940 977 607 475 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 029 940 441 677 881 955 214 950 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 029 940 441 677 881 955 214 950 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 059 880 883 355 763 910 429 900 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 059 880 883 355 763 910 429 900 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 119 761 766 711 527 820 859 801 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 119 761 766 711 527 820 859 801 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 239 523 533 423 055 641 719 603 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 239 523 533 423 055 641 719 603 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 479 047 066 846 111 283 439 206 4;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 479 047 066 846 111 283 439 206 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 958 094 133 692 222 566 878 412 8;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 958 094 133 692 222 566 878 412 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 001 916 188 267 384 445 133 756 825 6;
  • 80) 0.588 378 906 250 000 000 000 000 000 001 916 188 267 384 445 133 756 825 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 003 832 376 534 768 890 267 513 651 2;
  • 81) 0.176 757 812 500 000 000 000 000 000 003 832 376 534 768 890 267 513 651 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 007 664 753 069 537 780 535 027 302 4;
  • 82) 0.353 515 625 000 000 000 000 000 000 007 664 753 069 537 780 535 027 302 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 015 329 506 139 075 561 070 054 604 8;
  • 83) 0.707 031 250 000 000 000 000 000 000 015 329 506 139 075 561 070 054 604 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 030 659 012 278 151 122 140 109 209 6;
  • 84) 0.414 062 500 000 000 000 000 000 000 030 659 012 278 151 122 140 109 209 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 061 318 024 556 302 244 280 218 419 2;
  • 85) 0.828 125 000 000 000 000 000 000 000 061 318 024 556 302 244 280 218 419 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 122 636 049 112 604 488 560 436 838 4;
  • 86) 0.656 250 000 000 000 000 000 000 000 122 636 049 112 604 488 560 436 838 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 245 272 098 225 208 977 120 873 676 8;
  • 87) 0.312 500 000 000 000 000 000 000 000 245 272 098 225 208 977 120 873 676 8 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 490 544 196 450 417 954 241 747 353 6;
  • 88) 0.625 000 000 000 000 000 000 000 000 490 544 196 450 417 954 241 747 353 6 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 981 088 392 900 835 908 483 494 707 2;
  • 89) 0.250 000 000 000 000 000 000 000 000 981 088 392 900 835 908 483 494 707 2 × 2 = 0 + 0.500 000 000 000 000 000 000 000 001 962 176 785 801 671 816 966 989 414 4;
  • 90) 0.500 000 000 000 000 000 000 000 001 962 176 785 801 671 816 966 989 414 4 × 2 = 1 + 0.000 000 000 000 000 000 000 000 003 924 353 571 603 343 633 933 978 828 8;
  • 91) 0.000 000 000 000 000 000 000 000 003 924 353 571 603 343 633 933 978 828 8 × 2 = 0 + 0.000 000 000 000 000 000 000 000 007 848 707 143 206 687 267 867 957 657 6;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2 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