0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 42 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 42(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 42(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 42.

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 42 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 84;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 304 84 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 609 68;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 609 68 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 219 36;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 219 36 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 438 72;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 438 72 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 877 44;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 877 44 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 754 88;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 754 88 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 509 76;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 509 76 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 019 52;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 019 52 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 039 04;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 039 04 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 078 08;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 078 08 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 392 156 16;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 392 156 16 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 784 312 32;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 784 312 32 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 568 624 64;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 568 624 64 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 137 249 28;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 137 249 28 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 274 498 56;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 274 498 56 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 548 997 12;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 548 997 12 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 097 994 24;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 097 994 24 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 195 988 48;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 195 988 48 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 391 976 96;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 391 976 96 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 783 953 92;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 783 953 92 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 567 907 84;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 921 567 907 84 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 843 135 815 68;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 843 135 815 68 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 686 271 631 36;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 686 271 631 36 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 372 543 262 72;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 372 543 262 72 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 745 086 525 44;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 745 086 525 44 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 490 173 050 88;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 490 173 050 88 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 980 346 101 76;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 980 346 101 76 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 960 692 203 52;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 960 692 203 52 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 921 384 407 04;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 921 384 407 04 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 842 768 814 08;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 842 768 814 08 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 685 537 628 16;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 047 685 537 628 16 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 095 371 075 256 32;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 095 371 075 256 32 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 190 742 150 512 64;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 190 742 150 512 64 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 381 484 301 025 28;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 381 484 301 025 28 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 762 968 602 050 56;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 762 968 602 050 56 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 525 937 204 101 12;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 525 937 204 101 12 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 051 874 408 202 24;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 051 874 408 202 24 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 103 748 816 404 48;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 103 748 816 404 48 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 207 497 632 808 96;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 207 497 632 808 96 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 414 995 265 617 92;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 414 995 265 617 92 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 104 829 990 531 235 84;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 104 829 990 531 235 84 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 209 659 981 062 471 68;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 209 659 981 062 471 68 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 419 319 962 124 943 36;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 419 319 962 124 943 36 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 838 639 924 249 886 72;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 838 639 924 249 886 72 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 677 279 848 499 773 44;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 677 279 848 499 773 44 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 354 559 696 999 546 88;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 354 559 696 999 546 88 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 709 119 393 999 093 76;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 709 119 393 999 093 76 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 418 238 787 998 187 52;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 418 238 787 998 187 52 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 836 477 575 996 375 04;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 836 477 575 996 375 04 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 672 955 151 992 750 08;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 672 955 151 992 750 08 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 139 345 910 303 985 500 16;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 139 345 910 303 985 500 16 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 278 691 820 607 971 000 32;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 278 691 820 607 971 000 32 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 557 383 641 215 942 000 64;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 557 383 641 215 942 000 64 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 114 767 282 431 884 001 28;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 114 767 282 431 884 001 28 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 229 534 564 863 768 002 56;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 229 534 564 863 768 002 56 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 459 069 129 727 536 005 12;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 459 069 129 727 536 005 12 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 918 138 259 455 072 010 24;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 918 138 259 455 072 010 24 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 836 276 518 910 144 020 48;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 836 276 518 910 144 020 48 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 672 553 037 820 288 040 96;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 672 553 037 820 288 040 96 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 345 106 075 640 576 081 92;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 345 106 075 640 576 081 92 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 998 690 212 151 281 152 163 84;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 998 690 212 151 281 152 163 84 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 997 380 424 302 562 304 327 68;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 997 380 424 302 562 304 327 68 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 994 760 848 605 124 608 655 36;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 994 760 848 605 124 608 655 36 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 989 521 697 210 249 217 310 72;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 989 521 697 210 249 217 310 72 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 979 043 394 420 498 434 621 44;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 979 043 394 420 498 434 621 44 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 958 086 788 840 996 869 242 88;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 958 086 788 840 996 869 242 88 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 916 173 577 681 993 738 485 76;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 916 173 577 681 993 738 485 76 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 832 347 155 363 987 476 971 52;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 832 347 155 363 987 476 971 52 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 664 694 310 727 974 953 943 04;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 664 694 310 727 974 953 943 04 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 329 388 621 455 949 907 886 08;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 329 388 621 455 949 907 886 08 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 998 658 777 242 911 899 815 772 16;
  • 72) 0.615 579 605 102 539 062 499 999 999 998 658 777 242 911 899 815 772 16 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 997 317 554 485 823 799 631 544 32;
  • 73) 0.231 159 210 205 078 124 999 999 999 997 317 554 485 823 799 631 544 32 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 994 635 108 971 647 599 263 088 64;
  • 74) 0.462 318 420 410 156 249 999 999 999 994 635 108 971 647 599 263 088 64 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 989 270 217 943 295 198 526 177 28;
  • 75) 0.924 636 840 820 312 499 999 999 999 989 270 217 943 295 198 526 177 28 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 978 540 435 886 590 397 052 354 56;
  • 76) 0.849 273 681 640 624 999 999 999 999 978 540 435 886 590 397 052 354 56 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 957 080 871 773 180 794 104 709 12;
  • 77) 0.698 547 363 281 249 999 999 999 999 957 080 871 773 180 794 104 709 12 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 914 161 743 546 361 588 209 418 24;
  • 78) 0.397 094 726 562 499 999 999 999 999 914 161 743 546 361 588 209 418 24 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 828 323 487 092 723 176 418 836 48;
  • 79) 0.794 189 453 124 999 999 999 999 999 828 323 487 092 723 176 418 836 48 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 656 646 974 185 446 352 837 672 96;
  • 80) 0.588 378 906 249 999 999 999 999 999 656 646 974 185 446 352 837 672 96 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 313 293 948 370 892 705 675 345 92;
  • 81) 0.176 757 812 499 999 999 999 999 999 313 293 948 370 892 705 675 345 92 × 2 = 0 + 0.353 515 624 999 999 999 999 999 998 626 587 896 741 785 411 350 691 84;
  • 82) 0.353 515 624 999 999 999 999 999 998 626 587 896 741 785 411 350 691 84 × 2 = 0 + 0.707 031 249 999 999 999 999 999 997 253 175 793 483 570 822 701 383 68;
  • 83) 0.707 031 249 999 999 999 999 999 997 253 175 793 483 570 822 701 383 68 × 2 = 1 + 0.414 062 499 999 999 999 999 999 994 506 351 586 967 141 645 402 767 36;
  • 84) 0.414 062 499 999 999 999 999 999 994 506 351 586 967 141 645 402 767 36 × 2 = 0 + 0.828 124 999 999 999 999 999 999 989 012 703 173 934 283 290 805 534 72;
  • 85) 0.828 124 999 999 999 999 999 999 989 012 703 173 934 283 290 805 534 72 × 2 = 1 + 0.656 249 999 999 999 999 999 999 978 025 406 347 868 566 581 611 069 44;
  • 86) 0.656 249 999 999 999 999 999 999 978 025 406 347 868 566 581 611 069 44 × 2 = 1 + 0.312 499 999 999 999 999 999 999 956 050 812 695 737 133 163 222 138 88;
  • 87) 0.312 499 999 999 999 999 999 999 956 050 812 695 737 133 163 222 138 88 × 2 = 0 + 0.624 999 999 999 999 999 999 999 912 101 625 391 474 266 326 444 277 76;
  • 88) 0.624 999 999 999 999 999 999 999 912 101 625 391 474 266 326 444 277 76 × 2 = 1 + 0.249 999 999 999 999 999 999 999 824 203 250 782 948 532 652 888 555 52;
  • 89) 0.249 999 999 999 999 999 999 999 824 203 250 782 948 532 652 888 555 52 × 2 = 0 + 0.499 999 999 999 999 999 999 999 648 406 501 565 897 065 305 777 111 04;
  • 90) 0.499 999 999 999 999 999 999 999 648 406 501 565 897 065 305 777 111 04 × 2 = 0 + 0.999 999 999 999 999 999 999 999 296 813 003 131 794 130 611 554 222 08;
  • 91) 0.999 999 999 999 999 999 999 999 296 813 003 131 794 130 611 554 222 08 × 2 = 1 + 0.999 999 999 999 999 999 999 998 593 626 006 263 588 261 223 108 444 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 42(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 42(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 42(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 42 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