0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 91 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 91(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 91(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 91.

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 91 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 671 82;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 671 82 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 343 64;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 343 64 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 687 28;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 687 28 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 374 56;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 374 56 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 749 12;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 749 12 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 498 24;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 498 24 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 996 48;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 996 48 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 992 96;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 992 96 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 985 92;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 985 92 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 971 84;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 971 84 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 231 943 68;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 231 943 68 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 463 887 36;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 463 887 36 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 927 774 72;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 927 774 72 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 855 549 44;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 855 549 44 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 711 098 88;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 711 098 88 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 422 197 76;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 422 197 76 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 844 395 52;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 844 395 52 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 688 791 04;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 688 791 04 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 377 582 08;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 377 582 08 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 755 164 16;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 755 164 16 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 510 328 32;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 510 328 32 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 020 656 64;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 020 656 64 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 694 041 313 28;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 694 041 313 28 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 388 082 626 56;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 388 082 626 56 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 776 165 253 12;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 776 165 253 12 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 552 330 506 24;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 552 330 506 24 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 104 661 012 48;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 104 661 012 48 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 209 322 024 96;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 209 322 024 96 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 418 644 049 92;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 418 644 049 92 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 837 288 099 84;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 837 288 099 84 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 545 674 576 199 68;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 545 674 576 199 68 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 091 349 152 399 36;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 091 349 152 399 36 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 182 698 304 798 72;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 182 698 304 798 72 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 365 396 609 597 44;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 365 396 609 597 44 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 730 793 219 194 88;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 730 793 219 194 88 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 461 586 438 389 76;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 461 586 438 389 76 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 923 172 876 779 52;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 923 172 876 779 52 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 846 345 753 559 04;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 846 345 753 559 04 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 692 691 507 118 08;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 692 691 507 118 08 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 647 385 383 014 236 16;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 647 385 383 014 236 16 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 294 770 766 028 472 32;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 294 770 766 028 472 32 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 589 541 532 056 944 64;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 589 541 532 056 944 64 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 179 083 064 113 889 28;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 179 083 064 113 889 28 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 358 166 128 227 778 56;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 358 166 128 227 778 56 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 716 332 256 455 557 12;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 716 332 256 455 557 12 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 432 664 512 911 114 24;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 432 664 512 911 114 24 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 865 329 025 822 228 48;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 865 329 025 822 228 48 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 730 658 051 644 456 96;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 730 658 051 644 456 96 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 461 316 103 288 913 92;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 461 316 103 288 913 92 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 922 632 206 577 827 84;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 922 632 206 577 827 84 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 245 845 264 413 155 655 68;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 245 845 264 413 155 655 68 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 491 690 528 826 311 311 36;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 491 690 528 826 311 311 36 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 983 381 057 652 622 622 72;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 983 381 057 652 622 622 72 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 966 762 115 305 245 245 44;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 966 762 115 305 245 245 44 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 933 524 230 610 490 490 88;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 933 524 230 610 490 490 88 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 867 048 461 220 980 981 76;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 867 048 461 220 980 981 76 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 734 096 922 441 961 963 52;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 734 096 922 441 961 963 52 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 468 193 844 883 923 927 04;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 468 193 844 883 923 927 04 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 936 387 689 767 847 854 08;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 936 387 689 767 847 854 08 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 872 775 379 535 695 708 16;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 872 775 379 535 695 708 16 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 251 745 550 759 071 391 416 32;
  • 62) 0.338 491 776 958 107 948 303 222 656 251 745 550 759 071 391 416 32 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 503 491 101 518 142 782 832 64;
  • 63) 0.676 983 553 916 215 896 606 445 312 503 491 101 518 142 782 832 64 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 006 982 203 036 285 565 665 28;
  • 64) 0.353 967 107 832 431 793 212 890 625 006 982 203 036 285 565 665 28 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 013 964 406 072 571 131 330 56;
  • 65) 0.707 934 215 664 863 586 425 781 250 013 964 406 072 571 131 330 56 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 027 928 812 145 142 262 661 12;
  • 66) 0.415 868 431 329 727 172 851 562 500 027 928 812 145 142 262 661 12 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 055 857 624 290 284 525 322 24;
  • 67) 0.831 736 862 659 454 345 703 125 000 055 857 624 290 284 525 322 24 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 111 715 248 580 569 050 644 48;
  • 68) 0.663 473 725 318 908 691 406 250 000 111 715 248 580 569 050 644 48 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 223 430 497 161 138 101 288 96;
  • 69) 0.326 947 450 637 817 382 812 500 000 223 430 497 161 138 101 288 96 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 446 860 994 322 276 202 577 92;
  • 70) 0.653 894 901 275 634 765 625 000 000 446 860 994 322 276 202 577 92 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 893 721 988 644 552 405 155 84;
  • 71) 0.307 789 802 551 269 531 250 000 000 893 721 988 644 552 405 155 84 × 2 = 0 + 0.615 579 605 102 539 062 500 000 001 787 443 977 289 104 810 311 68;
  • 72) 0.615 579 605 102 539 062 500 000 001 787 443 977 289 104 810 311 68 × 2 = 1 + 0.231 159 210 205 078 125 000 000 003 574 887 954 578 209 620 623 36;
  • 73) 0.231 159 210 205 078 125 000 000 003 574 887 954 578 209 620 623 36 × 2 = 0 + 0.462 318 420 410 156 250 000 000 007 149 775 909 156 419 241 246 72;
  • 74) 0.462 318 420 410 156 250 000 000 007 149 775 909 156 419 241 246 72 × 2 = 0 + 0.924 636 840 820 312 500 000 000 014 299 551 818 312 838 482 493 44;
  • 75) 0.924 636 840 820 312 500 000 000 014 299 551 818 312 838 482 493 44 × 2 = 1 + 0.849 273 681 640 625 000 000 000 028 599 103 636 625 676 964 986 88;
  • 76) 0.849 273 681 640 625 000 000 000 028 599 103 636 625 676 964 986 88 × 2 = 1 + 0.698 547 363 281 250 000 000 000 057 198 207 273 251 353 929 973 76;
  • 77) 0.698 547 363 281 250 000 000 000 057 198 207 273 251 353 929 973 76 × 2 = 1 + 0.397 094 726 562 500 000 000 000 114 396 414 546 502 707 859 947 52;
  • 78) 0.397 094 726 562 500 000 000 000 114 396 414 546 502 707 859 947 52 × 2 = 0 + 0.794 189 453 125 000 000 000 000 228 792 829 093 005 415 719 895 04;
  • 79) 0.794 189 453 125 000 000 000 000 228 792 829 093 005 415 719 895 04 × 2 = 1 + 0.588 378 906 250 000 000 000 000 457 585 658 186 010 831 439 790 08;
  • 80) 0.588 378 906 250 000 000 000 000 457 585 658 186 010 831 439 790 08 × 2 = 1 + 0.176 757 812 500 000 000 000 000 915 171 316 372 021 662 879 580 16;
  • 81) 0.176 757 812 500 000 000 000 000 915 171 316 372 021 662 879 580 16 × 2 = 0 + 0.353 515 625 000 000 000 000 001 830 342 632 744 043 325 759 160 32;
  • 82) 0.353 515 625 000 000 000 000 001 830 342 632 744 043 325 759 160 32 × 2 = 0 + 0.707 031 250 000 000 000 000 003 660 685 265 488 086 651 518 320 64;
  • 83) 0.707 031 250 000 000 000 000 003 660 685 265 488 086 651 518 320 64 × 2 = 1 + 0.414 062 500 000 000 000 000 007 321 370 530 976 173 303 036 641 28;
  • 84) 0.414 062 500 000 000 000 000 007 321 370 530 976 173 303 036 641 28 × 2 = 0 + 0.828 125 000 000 000 000 000 014 642 741 061 952 346 606 073 282 56;
  • 85) 0.828 125 000 000 000 000 000 014 642 741 061 952 346 606 073 282 56 × 2 = 1 + 0.656 250 000 000 000 000 000 029 285 482 123 904 693 212 146 565 12;
  • 86) 0.656 250 000 000 000 000 000 029 285 482 123 904 693 212 146 565 12 × 2 = 1 + 0.312 500 000 000 000 000 000 058 570 964 247 809 386 424 293 130 24;
  • 87) 0.312 500 000 000 000 000 000 058 570 964 247 809 386 424 293 130 24 × 2 = 0 + 0.625 000 000 000 000 000 000 117 141 928 495 618 772 848 586 260 48;
  • 88) 0.625 000 000 000 000 000 000 117 141 928 495 618 772 848 586 260 48 × 2 = 1 + 0.250 000 000 000 000 000 000 234 283 856 991 237 545 697 172 520 96;
  • 89) 0.250 000 000 000 000 000 000 234 283 856 991 237 545 697 172 520 96 × 2 = 0 + 0.500 000 000 000 000 000 000 468 567 713 982 475 091 394 345 041 92;
  • 90) 0.500 000 000 000 000 000 000 468 567 713 982 475 091 394 345 041 92 × 2 = 1 + 0.000 000 000 000 000 000 000 937 135 427 964 950 182 788 690 083 84;
  • 91) 0.000 000 000 000 000 000 000 937 135 427 964 950 182 788 690 083 84 × 2 = 0 + 0.000 000 000 000 000 000 001 874 270 855 929 900 365 577 380 167 68;

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 91(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 91(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 91(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 91 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