0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 42;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 42 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 346 84;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 346 84 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 693 68;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 693 68 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 387 36;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 387 36 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 774 72;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 774 72 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 549 44;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 549 44 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 098 88;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 098 88 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 197 76;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 197 76 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 395 52;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 395 52 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 791 04;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 791 04 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 582 08;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 582 08 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 467 164 16;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 467 164 16 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 934 328 32;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 934 328 32 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 868 656 64;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 868 656 64 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 737 313 28;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 737 313 28 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 474 626 56;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 474 626 56 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 949 253 12;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 949 253 12 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 898 506 24;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 898 506 24 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 797 012 48;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 797 012 48 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 594 024 96;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 594 024 96 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 175 188 049 92;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 175 188 049 92 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 350 376 099 84;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 350 376 099 84 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 700 752 199 68;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 700 752 199 68 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 401 504 399 36;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 401 504 399 36 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 803 008 798 72;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 803 008 798 72 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 606 017 597 44;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 606 017 597 44 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 212 035 194 88;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 212 035 194 88 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 424 070 389 76;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 424 070 389 76 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 848 140 779 52;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 848 140 779 52 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 273 696 281 559 04;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 273 696 281 559 04 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 547 392 563 118 08;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 547 392 563 118 08 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 094 785 126 236 16;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 094 785 126 236 16 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 189 570 252 472 32;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 189 570 252 472 32 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 379 140 504 944 64;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 379 140 504 944 64 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 758 281 009 889 28;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 758 281 009 889 28 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 516 562 019 778 56;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 516 562 019 778 56 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 331 033 124 039 557 12;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 331 033 124 039 557 12 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 662 066 248 079 114 24;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 662 066 248 079 114 24 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 324 132 496 158 228 48;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 324 132 496 158 228 48 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 648 264 992 316 456 96;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 648 264 992 316 456 96 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 296 529 984 632 913 92;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 296 529 984 632 913 92 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 593 059 969 265 827 84;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 593 059 969 265 827 84 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 186 119 938 531 655 68;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 186 119 938 531 655 68 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 372 239 877 063 311 36;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 372 239 877 063 311 36 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 744 479 754 126 622 72;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 744 479 754 126 622 72 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 488 959 508 253 245 44;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 488 959 508 253 245 44 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 977 919 016 506 490 88;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 977 919 016 506 490 88 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 955 838 033 012 981 76;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 955 838 033 012 981 76 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 911 676 066 025 963 52;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 911 676 066 025 963 52 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 823 352 132 051 927 04;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 823 352 132 051 927 04 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 247 646 704 264 103 854 08;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 247 646 704 264 103 854 08 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 495 293 408 528 207 708 16;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 495 293 408 528 207 708 16 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 990 586 817 056 415 416 32;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 990 586 817 056 415 416 32 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 981 173 634 112 830 832 64;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 981 173 634 112 830 832 64 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 962 347 268 225 661 665 28;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 962 347 268 225 661 665 28 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 924 694 536 451 323 330 56;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 924 694 536 451 323 330 56 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 849 389 072 902 646 661 12;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 849 389 072 902 646 661 12 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 698 778 145 805 293 322 24;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 698 778 145 805 293 322 24 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 397 556 291 610 586 644 48;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 397 556 291 610 586 644 48 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 795 112 583 221 173 288 96;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 795 112 583 221 173 288 96 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 253 590 225 166 442 346 577 92;
  • 62) 0.338 491 776 958 107 948 303 222 656 253 590 225 166 442 346 577 92 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 507 180 450 332 884 693 155 84;
  • 63) 0.676 983 553 916 215 896 606 445 312 507 180 450 332 884 693 155 84 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 014 360 900 665 769 386 311 68;
  • 64) 0.353 967 107 832 431 793 212 890 625 014 360 900 665 769 386 311 68 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 028 721 801 331 538 772 623 36;
  • 65) 0.707 934 215 664 863 586 425 781 250 028 721 801 331 538 772 623 36 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 057 443 602 663 077 545 246 72;
  • 66) 0.415 868 431 329 727 172 851 562 500 057 443 602 663 077 545 246 72 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 114 887 205 326 155 090 493 44;
  • 67) 0.831 736 862 659 454 345 703 125 000 114 887 205 326 155 090 493 44 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 229 774 410 652 310 180 986 88;
  • 68) 0.663 473 725 318 908 691 406 250 000 229 774 410 652 310 180 986 88 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 459 548 821 304 620 361 973 76;
  • 69) 0.326 947 450 637 817 382 812 500 000 459 548 821 304 620 361 973 76 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 919 097 642 609 240 723 947 52;
  • 70) 0.653 894 901 275 634 765 625 000 000 919 097 642 609 240 723 947 52 × 2 = 1 + 0.307 789 802 551 269 531 250 000 001 838 195 285 218 481 447 895 04;
  • 71) 0.307 789 802 551 269 531 250 000 001 838 195 285 218 481 447 895 04 × 2 = 0 + 0.615 579 605 102 539 062 500 000 003 676 390 570 436 962 895 790 08;
  • 72) 0.615 579 605 102 539 062 500 000 003 676 390 570 436 962 895 790 08 × 2 = 1 + 0.231 159 210 205 078 125 000 000 007 352 781 140 873 925 791 580 16;
  • 73) 0.231 159 210 205 078 125 000 000 007 352 781 140 873 925 791 580 16 × 2 = 0 + 0.462 318 420 410 156 250 000 000 014 705 562 281 747 851 583 160 32;
  • 74) 0.462 318 420 410 156 250 000 000 014 705 562 281 747 851 583 160 32 × 2 = 0 + 0.924 636 840 820 312 500 000 000 029 411 124 563 495 703 166 320 64;
  • 75) 0.924 636 840 820 312 500 000 000 029 411 124 563 495 703 166 320 64 × 2 = 1 + 0.849 273 681 640 625 000 000 000 058 822 249 126 991 406 332 641 28;
  • 76) 0.849 273 681 640 625 000 000 000 058 822 249 126 991 406 332 641 28 × 2 = 1 + 0.698 547 363 281 250 000 000 000 117 644 498 253 982 812 665 282 56;
  • 77) 0.698 547 363 281 250 000 000 000 117 644 498 253 982 812 665 282 56 × 2 = 1 + 0.397 094 726 562 500 000 000 000 235 288 996 507 965 625 330 565 12;
  • 78) 0.397 094 726 562 500 000 000 000 235 288 996 507 965 625 330 565 12 × 2 = 0 + 0.794 189 453 125 000 000 000 000 470 577 993 015 931 250 661 130 24;
  • 79) 0.794 189 453 125 000 000 000 000 470 577 993 015 931 250 661 130 24 × 2 = 1 + 0.588 378 906 250 000 000 000 000 941 155 986 031 862 501 322 260 48;
  • 80) 0.588 378 906 250 000 000 000 000 941 155 986 031 862 501 322 260 48 × 2 = 1 + 0.176 757 812 500 000 000 000 001 882 311 972 063 725 002 644 520 96;
  • 81) 0.176 757 812 500 000 000 000 001 882 311 972 063 725 002 644 520 96 × 2 = 0 + 0.353 515 625 000 000 000 000 003 764 623 944 127 450 005 289 041 92;
  • 82) 0.353 515 625 000 000 000 000 003 764 623 944 127 450 005 289 041 92 × 2 = 0 + 0.707 031 250 000 000 000 000 007 529 247 888 254 900 010 578 083 84;
  • 83) 0.707 031 250 000 000 000 000 007 529 247 888 254 900 010 578 083 84 × 2 = 1 + 0.414 062 500 000 000 000 000 015 058 495 776 509 800 021 156 167 68;
  • 84) 0.414 062 500 000 000 000 000 015 058 495 776 509 800 021 156 167 68 × 2 = 0 + 0.828 125 000 000 000 000 000 030 116 991 553 019 600 042 312 335 36;
  • 85) 0.828 125 000 000 000 000 000 030 116 991 553 019 600 042 312 335 36 × 2 = 1 + 0.656 250 000 000 000 000 000 060 233 983 106 039 200 084 624 670 72;
  • 86) 0.656 250 000 000 000 000 000 060 233 983 106 039 200 084 624 670 72 × 2 = 1 + 0.312 500 000 000 000 000 000 120 467 966 212 078 400 169 249 341 44;
  • 87) 0.312 500 000 000 000 000 000 120 467 966 212 078 400 169 249 341 44 × 2 = 0 + 0.625 000 000 000 000 000 000 240 935 932 424 156 800 338 498 682 88;
  • 88) 0.625 000 000 000 000 000 000 240 935 932 424 156 800 338 498 682 88 × 2 = 1 + 0.250 000 000 000 000 000 000 481 871 864 848 313 600 676 997 365 76;
  • 89) 0.250 000 000 000 000 000 000 481 871 864 848 313 600 676 997 365 76 × 2 = 0 + 0.500 000 000 000 000 000 000 963 743 729 696 627 201 353 994 731 52;
  • 90) 0.500 000 000 000 000 000 000 963 743 729 696 627 201 353 994 731 52 × 2 = 1 + 0.000 000 000 000 000 000 001 927 487 459 393 254 402 707 989 463 04;
  • 91) 0.000 000 000 000 000 000 001 927 487 459 393 254 402 707 989 463 04 × 2 = 0 + 0.000 000 000 000 000 000 003 854 974 918 786 508 805 415 978 926 08;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 71 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