0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 665 84;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 665 84 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 331 68;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 331 68 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 663 36;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 663 36 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 326 72;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 326 72 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 653 44;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 653 44 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 306 88;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 306 88 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 613 76;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 613 76 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 227 52;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 227 52 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 455 04;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 306 455 04 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 612 910 08;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 612 910 08 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 225 820 16;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 225 820 16 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 451 640 32;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 451 640 32 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 903 280 64;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 903 280 64 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 806 561 28;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 806 561 28 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 613 122 56;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 613 122 56 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 226 245 12;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 226 245 12 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 452 490 24;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 452 490 24 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 904 980 48;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 020 904 980 48 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 809 960 96;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 041 809 960 96 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 619 921 92;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 083 619 921 92 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 167 239 843 84;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 167 239 843 84 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 334 479 687 68;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 334 479 687 68 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 668 959 375 36;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 668 959 375 36 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 337 918 750 72;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 337 918 750 72 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 675 837 501 44;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 675 837 501 44 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 351 675 002 88;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 351 675 002 88 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 658 703 350 005 76;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 658 703 350 005 76 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 317 406 700 011 52;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 317 406 700 011 52 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 634 813 400 023 04;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 634 813 400 023 04 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 269 626 800 046 08;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 269 626 800 046 08 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 539 253 600 092 16;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 539 253 600 092 16 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 078 507 200 184 32;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 078 507 200 184 32 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 157 014 400 368 64;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 157 014 400 368 64 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 314 028 800 737 28;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 314 028 800 737 28 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 628 057 601 474 56;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 628 057 601 474 56 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 256 115 202 949 12;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 256 115 202 949 12 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 512 230 405 898 24;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 512 230 405 898 24 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 024 460 811 796 48;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 024 460 811 796 48 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 322 048 921 623 592 96;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 322 048 921 623 592 96 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 644 097 843 247 185 92;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 644 097 843 247 185 92 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 288 195 686 494 371 84;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 288 195 686 494 371 84 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 576 391 372 988 743 68;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 576 391 372 988 743 68 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 152 782 745 977 487 36;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 152 782 745 977 487 36 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 305 565 491 954 974 72;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 305 565 491 954 974 72 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 611 130 983 909 949 44;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 611 130 983 909 949 44 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 222 261 967 819 898 88;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 222 261 967 819 898 88 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 444 523 935 639 797 76;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 444 523 935 639 797 76 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 404 889 047 871 279 595 52;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 404 889 047 871 279 595 52 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 809 778 095 742 559 191 04;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 809 778 095 742 559 191 04 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 619 556 191 485 118 382 08;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 619 556 191 485 118 382 08 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 239 112 382 970 236 764 16;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 239 112 382 970 236 764 16 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 478 224 765 940 473 528 32;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 478 224 765 940 473 528 32 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 956 449 531 880 947 056 64;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 956 449 531 880 947 056 64 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 912 899 063 761 894 113 28;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 912 899 063 761 894 113 28 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 825 798 127 523 788 226 56;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 825 798 127 523 788 226 56 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 651 596 255 047 576 453 12;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 651 596 255 047 576 453 12 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 303 192 510 095 152 906 24;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 303 192 510 095 152 906 24 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 030 606 385 020 190 305 812 48;
  • 59) 0.417 311 472 119 763 493 537 902 832 030 606 385 020 190 305 812 48 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 061 212 770 040 380 611 624 96;
  • 60) 0.834 622 944 239 526 987 075 805 664 061 212 770 040 380 611 624 96 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 122 425 540 080 761 223 249 92;
  • 61) 0.669 245 888 479 053 974 151 611 328 122 425 540 080 761 223 249 92 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 244 851 080 161 522 446 499 84;
  • 62) 0.338 491 776 958 107 948 303 222 656 244 851 080 161 522 446 499 84 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 489 702 160 323 044 892 999 68;
  • 63) 0.676 983 553 916 215 896 606 445 312 489 702 160 323 044 892 999 68 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 979 404 320 646 089 785 999 36;
  • 64) 0.353 967 107 832 431 793 212 890 624 979 404 320 646 089 785 999 36 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 958 808 641 292 179 571 998 72;
  • 65) 0.707 934 215 664 863 586 425 781 249 958 808 641 292 179 571 998 72 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 917 617 282 584 359 143 997 44;
  • 66) 0.415 868 431 329 727 172 851 562 499 917 617 282 584 359 143 997 44 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 835 234 565 168 718 287 994 88;
  • 67) 0.831 736 862 659 454 345 703 124 999 835 234 565 168 718 287 994 88 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 670 469 130 337 436 575 989 76;
  • 68) 0.663 473 725 318 908 691 406 249 999 670 469 130 337 436 575 989 76 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 340 938 260 674 873 151 979 52;
  • 69) 0.326 947 450 637 817 382 812 499 999 340 938 260 674 873 151 979 52 × 2 = 0 + 0.653 894 901 275 634 765 624 999 998 681 876 521 349 746 303 959 04;
  • 70) 0.653 894 901 275 634 765 624 999 998 681 876 521 349 746 303 959 04 × 2 = 1 + 0.307 789 802 551 269 531 249 999 997 363 753 042 699 492 607 918 08;
  • 71) 0.307 789 802 551 269 531 249 999 997 363 753 042 699 492 607 918 08 × 2 = 0 + 0.615 579 605 102 539 062 499 999 994 727 506 085 398 985 215 836 16;
  • 72) 0.615 579 605 102 539 062 499 999 994 727 506 085 398 985 215 836 16 × 2 = 1 + 0.231 159 210 205 078 124 999 999 989 455 012 170 797 970 431 672 32;
  • 73) 0.231 159 210 205 078 124 999 999 989 455 012 170 797 970 431 672 32 × 2 = 0 + 0.462 318 420 410 156 249 999 999 978 910 024 341 595 940 863 344 64;
  • 74) 0.462 318 420 410 156 249 999 999 978 910 024 341 595 940 863 344 64 × 2 = 0 + 0.924 636 840 820 312 499 999 999 957 820 048 683 191 881 726 689 28;
  • 75) 0.924 636 840 820 312 499 999 999 957 820 048 683 191 881 726 689 28 × 2 = 1 + 0.849 273 681 640 624 999 999 999 915 640 097 366 383 763 453 378 56;
  • 76) 0.849 273 681 640 624 999 999 999 915 640 097 366 383 763 453 378 56 × 2 = 1 + 0.698 547 363 281 249 999 999 999 831 280 194 732 767 526 906 757 12;
  • 77) 0.698 547 363 281 249 999 999 999 831 280 194 732 767 526 906 757 12 × 2 = 1 + 0.397 094 726 562 499 999 999 999 662 560 389 465 535 053 813 514 24;
  • 78) 0.397 094 726 562 499 999 999 999 662 560 389 465 535 053 813 514 24 × 2 = 0 + 0.794 189 453 124 999 999 999 999 325 120 778 931 070 107 627 028 48;
  • 79) 0.794 189 453 124 999 999 999 999 325 120 778 931 070 107 627 028 48 × 2 = 1 + 0.588 378 906 249 999 999 999 998 650 241 557 862 140 215 254 056 96;
  • 80) 0.588 378 906 249 999 999 999 998 650 241 557 862 140 215 254 056 96 × 2 = 1 + 0.176 757 812 499 999 999 999 997 300 483 115 724 280 430 508 113 92;
  • 81) 0.176 757 812 499 999 999 999 997 300 483 115 724 280 430 508 113 92 × 2 = 0 + 0.353 515 624 999 999 999 999 994 600 966 231 448 560 861 016 227 84;
  • 82) 0.353 515 624 999 999 999 999 994 600 966 231 448 560 861 016 227 84 × 2 = 0 + 0.707 031 249 999 999 999 999 989 201 932 462 897 121 722 032 455 68;
  • 83) 0.707 031 249 999 999 999 999 989 201 932 462 897 121 722 032 455 68 × 2 = 1 + 0.414 062 499 999 999 999 999 978 403 864 925 794 243 444 064 911 36;
  • 84) 0.414 062 499 999 999 999 999 978 403 864 925 794 243 444 064 911 36 × 2 = 0 + 0.828 124 999 999 999 999 999 956 807 729 851 588 486 888 129 822 72;
  • 85) 0.828 124 999 999 999 999 999 956 807 729 851 588 486 888 129 822 72 × 2 = 1 + 0.656 249 999 999 999 999 999 913 615 459 703 176 973 776 259 645 44;
  • 86) 0.656 249 999 999 999 999 999 913 615 459 703 176 973 776 259 645 44 × 2 = 1 + 0.312 499 999 999 999 999 999 827 230 919 406 353 947 552 519 290 88;
  • 87) 0.312 499 999 999 999 999 999 827 230 919 406 353 947 552 519 290 88 × 2 = 0 + 0.624 999 999 999 999 999 999 654 461 838 812 707 895 105 038 581 76;
  • 88) 0.624 999 999 999 999 999 999 654 461 838 812 707 895 105 038 581 76 × 2 = 1 + 0.249 999 999 999 999 999 999 308 923 677 625 415 790 210 077 163 52;
  • 89) 0.249 999 999 999 999 999 999 308 923 677 625 415 790 210 077 163 52 × 2 = 0 + 0.499 999 999 999 999 999 998 617 847 355 250 831 580 420 154 327 04;
  • 90) 0.499 999 999 999 999 999 998 617 847 355 250 831 580 420 154 327 04 × 2 = 0 + 0.999 999 999 999 999 999 997 235 694 710 501 663 160 840 308 654 08;
  • 91) 0.999 999 999 999 999 999 997 235 694 710 501 663 160 840 308 654 08 × 2 = 1 + 0.999 999 999 999 999 999 994 471 389 421 003 326 321 680 617 308 16;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 332 92 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