0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 68 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 68(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 68(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 68.

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 68 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 36;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 36 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 610 72;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 610 72 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 221 44;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 221 44 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 442 88;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 442 88 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 885 76;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 885 76 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 771 52;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 771 52 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 543 04;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 543 04 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 086 08;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 086 08 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 172 16;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 172 16 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 344 32;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 344 32 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 392 688 64;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 392 688 64 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 785 377 28;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 785 377 28 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 570 754 56;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 570 754 56 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 141 509 12;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 141 509 12 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 283 018 24;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 283 018 24 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 566 036 48;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 566 036 48 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 132 072 96;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 132 072 96 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 264 145 92;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 264 145 92 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 528 291 84;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 528 291 84 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 056 583 68;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 056 583 68 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 113 167 36;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 113 167 36 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 844 226 334 72;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 844 226 334 72 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 688 452 669 44;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 688 452 669 44 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 376 905 338 88;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 376 905 338 88 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 753 810 677 76;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 753 810 677 76 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 507 621 355 52;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 507 621 355 52 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 015 242 711 04;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 015 242 711 04 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 030 485 422 08;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 030 485 422 08 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 060 970 844 16;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 060 970 844 16 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 121 941 688 32;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 121 941 688 32 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 243 883 376 64;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 243 883 376 64 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 096 487 766 753 28;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 096 487 766 753 28 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 192 975 533 506 56;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 192 975 533 506 56 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 385 951 067 013 12;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 385 951 067 013 12 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 771 902 134 026 24;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 771 902 134 026 24 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 543 804 268 052 48;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 543 804 268 052 48 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 087 608 536 104 96;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 087 608 536 104 96 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 175 217 072 209 92;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 175 217 072 209 92 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 350 434 144 419 84;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 350 434 144 419 84 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 700 868 288 839 68;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 700 868 288 839 68 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 105 401 736 577 679 36;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 105 401 736 577 679 36 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 210 803 473 155 358 72;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 210 803 473 155 358 72 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 421 606 946 310 717 44;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 421 606 946 310 717 44 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 843 213 892 621 434 88;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 843 213 892 621 434 88 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 686 427 785 242 869 76;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 686 427 785 242 869 76 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 372 855 570 485 739 52;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 372 855 570 485 739 52 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 745 711 140 971 479 04;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 745 711 140 971 479 04 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 491 422 281 942 958 08;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 491 422 281 942 958 08 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 982 844 563 885 916 16;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 982 844 563 885 916 16 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 965 689 127 771 832 32;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 965 689 127 771 832 32 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 139 931 378 255 543 664 64;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 139 931 378 255 543 664 64 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 279 862 756 511 087 329 28;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 279 862 756 511 087 329 28 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 559 725 513 022 174 658 56;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 559 725 513 022 174 658 56 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 119 451 026 044 349 317 12;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 119 451 026 044 349 317 12 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 238 902 052 088 698 634 24;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 238 902 052 088 698 634 24 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 477 804 104 177 397 268 48;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 477 804 104 177 397 268 48 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 955 608 208 354 794 536 96;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 955 608 208 354 794 536 96 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 911 216 416 709 589 073 92;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 911 216 416 709 589 073 92 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 822 432 833 419 178 147 84;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 822 432 833 419 178 147 84 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 644 865 666 838 356 295 68;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 644 865 666 838 356 295 68 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 289 731 333 676 712 591 36;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 289 731 333 676 712 591 36 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 998 579 462 667 353 425 182 72;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 998 579 462 667 353 425 182 72 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 997 158 925 334 706 850 365 44;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 997 158 925 334 706 850 365 44 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 994 317 850 669 413 700 730 88;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 994 317 850 669 413 700 730 88 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 988 635 701 338 827 401 461 76;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 988 635 701 338 827 401 461 76 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 977 271 402 677 654 802 923 52;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 977 271 402 677 654 802 923 52 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 954 542 805 355 309 605 847 04;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 954 542 805 355 309 605 847 04 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 909 085 610 710 619 211 694 08;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 909 085 610 710 619 211 694 08 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 818 171 221 421 238 423 388 16;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 818 171 221 421 238 423 388 16 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 636 342 442 842 476 846 776 32;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 636 342 442 842 476 846 776 32 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 272 684 885 684 953 693 552 64;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 272 684 885 684 953 693 552 64 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 998 545 369 771 369 907 387 105 28;
  • 73) 0.231 159 210 205 078 124 999 999 999 998 545 369 771 369 907 387 105 28 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 997 090 739 542 739 814 774 210 56;
  • 74) 0.462 318 420 410 156 249 999 999 999 997 090 739 542 739 814 774 210 56 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 994 181 479 085 479 629 548 421 12;
  • 75) 0.924 636 840 820 312 499 999 999 999 994 181 479 085 479 629 548 421 12 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 988 362 958 170 959 259 096 842 24;
  • 76) 0.849 273 681 640 624 999 999 999 999 988 362 958 170 959 259 096 842 24 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 976 725 916 341 918 518 193 684 48;
  • 77) 0.698 547 363 281 249 999 999 999 999 976 725 916 341 918 518 193 684 48 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 953 451 832 683 837 036 387 368 96;
  • 78) 0.397 094 726 562 499 999 999 999 999 953 451 832 683 837 036 387 368 96 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 906 903 665 367 674 072 774 737 92;
  • 79) 0.794 189 453 124 999 999 999 999 999 906 903 665 367 674 072 774 737 92 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 813 807 330 735 348 145 549 475 84;
  • 80) 0.588 378 906 249 999 999 999 999 999 813 807 330 735 348 145 549 475 84 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 627 614 661 470 696 291 098 951 68;
  • 81) 0.176 757 812 499 999 999 999 999 999 627 614 661 470 696 291 098 951 68 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 255 229 322 941 392 582 197 903 36;
  • 82) 0.353 515 624 999 999 999 999 999 999 255 229 322 941 392 582 197 903 36 × 2 = 0 + 0.707 031 249 999 999 999 999 999 998 510 458 645 882 785 164 395 806 72;
  • 83) 0.707 031 249 999 999 999 999 999 998 510 458 645 882 785 164 395 806 72 × 2 = 1 + 0.414 062 499 999 999 999 999 999 997 020 917 291 765 570 328 791 613 44;
  • 84) 0.414 062 499 999 999 999 999 999 997 020 917 291 765 570 328 791 613 44 × 2 = 0 + 0.828 124 999 999 999 999 999 999 994 041 834 583 531 140 657 583 226 88;
  • 85) 0.828 124 999 999 999 999 999 999 994 041 834 583 531 140 657 583 226 88 × 2 = 1 + 0.656 249 999 999 999 999 999 999 988 083 669 167 062 281 315 166 453 76;
  • 86) 0.656 249 999 999 999 999 999 999 988 083 669 167 062 281 315 166 453 76 × 2 = 1 + 0.312 499 999 999 999 999 999 999 976 167 338 334 124 562 630 332 907 52;
  • 87) 0.312 499 999 999 999 999 999 999 976 167 338 334 124 562 630 332 907 52 × 2 = 0 + 0.624 999 999 999 999 999 999 999 952 334 676 668 249 125 260 665 815 04;
  • 88) 0.624 999 999 999 999 999 999 999 952 334 676 668 249 125 260 665 815 04 × 2 = 1 + 0.249 999 999 999 999 999 999 999 904 669 353 336 498 250 521 331 630 08;
  • 89) 0.249 999 999 999 999 999 999 999 904 669 353 336 498 250 521 331 630 08 × 2 = 0 + 0.499 999 999 999 999 999 999 999 809 338 706 672 996 501 042 663 260 16;
  • 90) 0.499 999 999 999 999 999 999 999 809 338 706 672 996 501 042 663 260 16 × 2 = 0 + 0.999 999 999 999 999 999 999 999 618 677 413 345 993 002 085 326 520 32;
  • 91) 0.999 999 999 999 999 999 999 999 618 677 413 345 993 002 085 326 520 32 × 2 = 1 + 0.999 999 999 999 999 999 999 999 237 354 826 691 986 004 170 653 040 64;

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 68(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 68(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 68(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 68 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