0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 392;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 392 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 784;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 784 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 568;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 568 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 136;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 363 136 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 272;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 726 272 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 452 544;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 452 544 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 905 088;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 905 088 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 810 176;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 810 176 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 620 352;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 620 352 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 240 704;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 240 704 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 481 408;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 481 408 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 962 816;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 962 816 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 925 632;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 925 632 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 851 264;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 851 264 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 687 702 528;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 687 702 528 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 375 405 056;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 375 405 056 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 750 810 112;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 750 810 112 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 501 620 224;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 501 620 224 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 003 240 448;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 003 240 448 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 006 480 896;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 006 480 896 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 012 961 792;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 172 012 961 792 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 025 923 584;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 344 025 923 584 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 051 847 168;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 688 051 847 168 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 103 694 336;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 376 103 694 336 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 207 388 672;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 752 207 388 672 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 504 414 777 344;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 504 414 777 344 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 008 829 554 688;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 008 829 554 688 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 017 659 109 376;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 017 659 109 376 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 035 318 218 752;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 035 318 218 752 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 070 636 437 504;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 070 636 437 504 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 141 272 875 008;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 141 272 875 008 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 282 545 750 016;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 282 545 750 016 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 565 091 500 032;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 565 091 500 032 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 353 130 183 000 064;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 353 130 183 000 064 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 706 260 366 000 128;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 706 260 366 000 128 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 412 520 732 000 256;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 412 520 732 000 256 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 825 041 464 000 512;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 825 041 464 000 512 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 650 082 928 001 024;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 650 082 928 001 024 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 300 165 856 002 048;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 300 165 856 002 048 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 600 331 712 004 096;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 600 331 712 004 096 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 200 663 424 008 192;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 200 663 424 008 192 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 401 326 848 016 384;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 401 326 848 016 384 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 802 653 696 032 768;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 802 653 696 032 768 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 345 605 307 392 065 536;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 345 605 307 392 065 536 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 691 210 614 784 131 072;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 691 210 614 784 131 072 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 382 421 229 568 262 144;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 382 421 229 568 262 144 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 764 842 459 136 524 288;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 764 842 459 136 524 288 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 529 684 918 273 048 576;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 529 684 918 273 048 576 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 059 369 836 546 097 152;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 059 369 836 546 097 152 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 118 739 673 092 194 304;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 118 739 673 092 194 304 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 237 479 346 184 388 608;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 237 479 346 184 388 608 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 474 958 692 368 777 216;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 474 958 692 368 777 216 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 949 917 384 737 554 432;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 949 917 384 737 554 432 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 899 834 769 475 108 864;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 899 834 769 475 108 864 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 907 799 669 538 950 217 728;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 907 799 669 538 950 217 728 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 815 599 339 077 900 435 456;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 815 599 339 077 900 435 456 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 631 198 678 155 800 870 912;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 631 198 678 155 800 870 912 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 262 397 356 311 601 741 824;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 262 397 356 311 601 741 824 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 524 794 712 623 203 483 648;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 524 794 712 623 203 483 648 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 049 589 425 246 406 967 296;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 049 589 425 246 406 967 296 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 099 178 850 492 813 934 592;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 099 178 850 492 813 934 592 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 198 357 700 985 627 869 184;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 198 357 700 985 627 869 184 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 396 715 401 971 255 738 368;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 396 715 401 971 255 738 368 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 793 430 803 942 511 476 736;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 793 430 803 942 511 476 736 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 001 586 861 607 885 022 953 472;
  • 66) 0.415 868 431 329 727 172 851 562 500 001 586 861 607 885 022 953 472 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 003 173 723 215 770 045 906 944;
  • 67) 0.831 736 862 659 454 345 703 125 000 003 173 723 215 770 045 906 944 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 006 347 446 431 540 091 813 888;
  • 68) 0.663 473 725 318 908 691 406 250 000 006 347 446 431 540 091 813 888 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 012 694 892 863 080 183 627 776;
  • 69) 0.326 947 450 637 817 382 812 500 000 012 694 892 863 080 183 627 776 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 025 389 785 726 160 367 255 552;
  • 70) 0.653 894 901 275 634 765 625 000 000 025 389 785 726 160 367 255 552 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 050 779 571 452 320 734 511 104;
  • 71) 0.307 789 802 551 269 531 250 000 000 050 779 571 452 320 734 511 104 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 101 559 142 904 641 469 022 208;
  • 72) 0.615 579 605 102 539 062 500 000 000 101 559 142 904 641 469 022 208 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 203 118 285 809 282 938 044 416;
  • 73) 0.231 159 210 205 078 125 000 000 000 203 118 285 809 282 938 044 416 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 406 236 571 618 565 876 088 832;
  • 74) 0.462 318 420 410 156 250 000 000 000 406 236 571 618 565 876 088 832 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 812 473 143 237 131 752 177 664;
  • 75) 0.924 636 840 820 312 500 000 000 000 812 473 143 237 131 752 177 664 × 2 = 1 + 0.849 273 681 640 625 000 000 000 001 624 946 286 474 263 504 355 328;
  • 76) 0.849 273 681 640 625 000 000 000 001 624 946 286 474 263 504 355 328 × 2 = 1 + 0.698 547 363 281 250 000 000 000 003 249 892 572 948 527 008 710 656;
  • 77) 0.698 547 363 281 250 000 000 000 003 249 892 572 948 527 008 710 656 × 2 = 1 + 0.397 094 726 562 500 000 000 000 006 499 785 145 897 054 017 421 312;
  • 78) 0.397 094 726 562 500 000 000 000 006 499 785 145 897 054 017 421 312 × 2 = 0 + 0.794 189 453 125 000 000 000 000 012 999 570 291 794 108 034 842 624;
  • 79) 0.794 189 453 125 000 000 000 000 012 999 570 291 794 108 034 842 624 × 2 = 1 + 0.588 378 906 250 000 000 000 000 025 999 140 583 588 216 069 685 248;
  • 80) 0.588 378 906 250 000 000 000 000 025 999 140 583 588 216 069 685 248 × 2 = 1 + 0.176 757 812 500 000 000 000 000 051 998 281 167 176 432 139 370 496;
  • 81) 0.176 757 812 500 000 000 000 000 051 998 281 167 176 432 139 370 496 × 2 = 0 + 0.353 515 625 000 000 000 000 000 103 996 562 334 352 864 278 740 992;
  • 82) 0.353 515 625 000 000 000 000 000 103 996 562 334 352 864 278 740 992 × 2 = 0 + 0.707 031 250 000 000 000 000 000 207 993 124 668 705 728 557 481 984;
  • 83) 0.707 031 250 000 000 000 000 000 207 993 124 668 705 728 557 481 984 × 2 = 1 + 0.414 062 500 000 000 000 000 000 415 986 249 337 411 457 114 963 968;
  • 84) 0.414 062 500 000 000 000 000 000 415 986 249 337 411 457 114 963 968 × 2 = 0 + 0.828 125 000 000 000 000 000 000 831 972 498 674 822 914 229 927 936;
  • 85) 0.828 125 000 000 000 000 000 000 831 972 498 674 822 914 229 927 936 × 2 = 1 + 0.656 250 000 000 000 000 000 001 663 944 997 349 645 828 459 855 872;
  • 86) 0.656 250 000 000 000 000 000 001 663 944 997 349 645 828 459 855 872 × 2 = 1 + 0.312 500 000 000 000 000 000 003 327 889 994 699 291 656 919 711 744;
  • 87) 0.312 500 000 000 000 000 000 003 327 889 994 699 291 656 919 711 744 × 2 = 0 + 0.625 000 000 000 000 000 000 006 655 779 989 398 583 313 839 423 488;
  • 88) 0.625 000 000 000 000 000 000 006 655 779 989 398 583 313 839 423 488 × 2 = 1 + 0.250 000 000 000 000 000 000 013 311 559 978 797 166 627 678 846 976;
  • 89) 0.250 000 000 000 000 000 000 013 311 559 978 797 166 627 678 846 976 × 2 = 0 + 0.500 000 000 000 000 000 000 026 623 119 957 594 333 255 357 693 952;
  • 90) 0.500 000 000 000 000 000 000 026 623 119 957 594 333 255 357 693 952 × 2 = 1 + 0.000 000 000 000 000 000 000 053 246 239 915 188 666 510 715 387 904;
  • 91) 0.000 000 000 000 000 000 000 053 246 239 915 188 666 510 715 387 904 × 2 = 0 + 0.000 000 000 000 000 000 000 106 492 479 830 377 333 021 430 775 808;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 196 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