0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 46;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 46 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 92;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 608 92 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 217 84;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 217 84 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 435 68;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 435 68 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 871 36;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 871 36 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 742 72;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 742 72 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 485 44;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 485 44 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 970 88;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 970 88 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 941 76;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 941 76 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 883 52;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 883 52 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 767 04;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 767 04 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 534 08;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 783 534 08 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 567 068 16;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 567 068 16 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 134 136 32;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 134 136 32 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 268 272 64;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 268 272 64 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 536 545 28;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 536 545 28 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 073 090 56;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 073 090 56 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 146 181 12;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 146 181 12 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 292 362 24;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 292 362 24 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 584 724 48;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 584 724 48 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 169 448 96;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 169 448 96 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 338 897 92;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 842 338 897 92 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 684 677 795 84;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 684 677 795 84 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 369 355 591 68;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 369 355 591 68 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 738 711 183 36;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 738 711 183 36 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 477 422 366 72;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 477 422 366 72 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 954 844 733 44;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 954 844 733 44 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 909 689 466 88;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 909 689 466 88 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 819 378 933 76;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 819 378 933 76 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 638 757 867 52;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 638 757 867 52 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 277 515 735 04;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 277 515 735 04 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 555 031 470 08;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 094 555 031 470 08 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 189 110 062 940 16;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 189 110 062 940 16 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 378 220 125 880 32;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 378 220 125 880 32 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 756 440 251 760 64;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 756 440 251 760 64 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 512 880 503 521 28;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 512 880 503 521 28 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 025 761 007 042 56;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 025 761 007 042 56 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 051 522 014 085 12;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 051 522 014 085 12 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 103 044 028 170 24;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 103 044 028 170 24 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 206 088 056 340 48;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 206 088 056 340 48 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 104 412 176 112 680 96;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 104 412 176 112 680 96 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 208 824 352 225 361 92;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 208 824 352 225 361 92 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 417 648 704 450 723 84;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 417 648 704 450 723 84 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 835 297 408 901 447 68;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 835 297 408 901 447 68 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 670 594 817 802 895 36;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 670 594 817 802 895 36 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 341 189 635 605 790 72;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 341 189 635 605 790 72 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 682 379 271 211 581 44;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 682 379 271 211 581 44 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 364 758 542 423 162 88;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 364 758 542 423 162 88 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 729 517 084 846 325 76;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 729 517 084 846 325 76 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 459 034 169 692 651 52;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 459 034 169 692 651 52 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 138 918 068 339 385 303 04;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 138 918 068 339 385 303 04 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 277 836 136 678 770 606 08;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 277 836 136 678 770 606 08 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 555 672 273 357 541 212 16;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 555 672 273 357 541 212 16 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 111 344 546 715 082 424 32;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 111 344 546 715 082 424 32 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 222 689 093 430 164 848 64;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 222 689 093 430 164 848 64 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 445 378 186 860 329 697 28;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 445 378 186 860 329 697 28 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 890 756 373 720 659 394 56;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 890 756 373 720 659 394 56 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 781 512 747 441 318 789 12;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 781 512 747 441 318 789 12 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 563 025 494 882 637 578 24;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 563 025 494 882 637 578 24 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 126 050 989 765 275 156 48;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 126 050 989 765 275 156 48 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 998 252 101 979 530 550 312 96;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 998 252 101 979 530 550 312 96 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 996 504 203 959 061 100 625 92;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 996 504 203 959 061 100 625 92 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 993 008 407 918 122 201 251 84;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 993 008 407 918 122 201 251 84 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 986 016 815 836 244 402 503 68;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 986 016 815 836 244 402 503 68 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 972 033 631 672 488 805 007 36;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 972 033 631 672 488 805 007 36 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 944 067 263 344 977 610 014 72;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 944 067 263 344 977 610 014 72 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 888 134 526 689 955 220 029 44;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 888 134 526 689 955 220 029 44 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 776 269 053 379 910 440 058 88;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 776 269 053 379 910 440 058 88 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 552 538 106 759 820 880 117 76;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 552 538 106 759 820 880 117 76 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 105 076 213 519 641 760 235 52;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 105 076 213 519 641 760 235 52 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 998 210 152 427 039 283 520 471 04;
  • 72) 0.615 579 605 102 539 062 499 999 999 998 210 152 427 039 283 520 471 04 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 996 420 304 854 078 567 040 942 08;
  • 73) 0.231 159 210 205 078 124 999 999 999 996 420 304 854 078 567 040 942 08 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 992 840 609 708 157 134 081 884 16;
  • 74) 0.462 318 420 410 156 249 999 999 999 992 840 609 708 157 134 081 884 16 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 985 681 219 416 314 268 163 768 32;
  • 75) 0.924 636 840 820 312 499 999 999 999 985 681 219 416 314 268 163 768 32 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 971 362 438 832 628 536 327 536 64;
  • 76) 0.849 273 681 640 624 999 999 999 999 971 362 438 832 628 536 327 536 64 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 942 724 877 665 257 072 655 073 28;
  • 77) 0.698 547 363 281 249 999 999 999 999 942 724 877 665 257 072 655 073 28 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 885 449 755 330 514 145 310 146 56;
  • 78) 0.397 094 726 562 499 999 999 999 999 885 449 755 330 514 145 310 146 56 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 770 899 510 661 028 290 620 293 12;
  • 79) 0.794 189 453 124 999 999 999 999 999 770 899 510 661 028 290 620 293 12 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 541 799 021 322 056 581 240 586 24;
  • 80) 0.588 378 906 249 999 999 999 999 999 541 799 021 322 056 581 240 586 24 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 083 598 042 644 113 162 481 172 48;
  • 81) 0.176 757 812 499 999 999 999 999 999 083 598 042 644 113 162 481 172 48 × 2 = 0 + 0.353 515 624 999 999 999 999 999 998 167 196 085 288 226 324 962 344 96;
  • 82) 0.353 515 624 999 999 999 999 999 998 167 196 085 288 226 324 962 344 96 × 2 = 0 + 0.707 031 249 999 999 999 999 999 996 334 392 170 576 452 649 924 689 92;
  • 83) 0.707 031 249 999 999 999 999 999 996 334 392 170 576 452 649 924 689 92 × 2 = 1 + 0.414 062 499 999 999 999 999 999 992 668 784 341 152 905 299 849 379 84;
  • 84) 0.414 062 499 999 999 999 999 999 992 668 784 341 152 905 299 849 379 84 × 2 = 0 + 0.828 124 999 999 999 999 999 999 985 337 568 682 305 810 599 698 759 68;
  • 85) 0.828 124 999 999 999 999 999 999 985 337 568 682 305 810 599 698 759 68 × 2 = 1 + 0.656 249 999 999 999 999 999 999 970 675 137 364 611 621 199 397 519 36;
  • 86) 0.656 249 999 999 999 999 999 999 970 675 137 364 611 621 199 397 519 36 × 2 = 1 + 0.312 499 999 999 999 999 999 999 941 350 274 729 223 242 398 795 038 72;
  • 87) 0.312 499 999 999 999 999 999 999 941 350 274 729 223 242 398 795 038 72 × 2 = 0 + 0.624 999 999 999 999 999 999 999 882 700 549 458 446 484 797 590 077 44;
  • 88) 0.624 999 999 999 999 999 999 999 882 700 549 458 446 484 797 590 077 44 × 2 = 1 + 0.249 999 999 999 999 999 999 999 765 401 098 916 892 969 595 180 154 88;
  • 89) 0.249 999 999 999 999 999 999 999 765 401 098 916 892 969 595 180 154 88 × 2 = 0 + 0.499 999 999 999 999 999 999 999 530 802 197 833 785 939 190 360 309 76;
  • 90) 0.499 999 999 999 999 999 999 999 530 802 197 833 785 939 190 360 309 76 × 2 = 0 + 0.999 999 999 999 999 999 999 999 061 604 395 667 571 878 380 720 619 52;
  • 91) 0.999 999 999 999 999 999 999 999 061 604 395 667 571 878 380 720 619 52 × 2 = 1 + 0.999 999 999 999 999 999 999 998 123 208 791 335 143 756 761 441 239 04;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 23 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