0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 293 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 293 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 586 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 586 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 172 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 172 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 345 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 345 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 691 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 691 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 382 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 382 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 764 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 764 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 529 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 529 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 059 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 059 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 190 118 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 190 118 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 380 236 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 380 236 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 760 473 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 760 473 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 520 947 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 520 947 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 041 894 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 041 894 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 083 788 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 083 788 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 167 577 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 167 577 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 335 155 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 335 155 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 670 310 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 670 310 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 340 620 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 340 620 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 954 681 241 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 954 681 241 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 909 362 483 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 909 362 483 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 818 724 966 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 818 724 966 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 637 449 932 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 637 449 932 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 274 899 865 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 274 899 865 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 549 799 731 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 549 799 731 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 099 599 462 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 099 599 462 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 199 198 924 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 199 198 924 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 398 397 849 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 398 397 849 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 008 796 795 699 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 008 796 795 699 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 017 593 591 398 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 017 593 591 398 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 035 187 182 796 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 035 187 182 796 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 070 374 365 593 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 070 374 365 593 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 140 748 731 187 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 140 748 731 187 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 281 497 462 374 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 281 497 462 374 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 562 994 924 748 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 562 994 924 748 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 125 989 849 497 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 125 989 849 497 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 251 979 698 995 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 251 979 698 995 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 636 503 959 397 990 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 636 503 959 397 990 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 273 007 918 795 980 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 273 007 918 795 980 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 546 015 837 591 961 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 546 015 837 591 961 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 092 031 675 183 923 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 092 031 675 183 923 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 184 063 350 367 846 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 184 063 350 367 846 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 368 126 700 735 692 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 368 126 700 735 692 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 736 253 401 471 385 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 736 253 401 471 385 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 472 506 802 942 771 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 472 506 802 942 771 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 945 013 605 885 542 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 945 013 605 885 542 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 757 890 027 211 771 084 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 757 890 027 211 771 084 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 515 780 054 423 542 169 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 515 780 054 423 542 169 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 031 560 108 847 084 339 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 031 560 108 847 084 339 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 063 120 217 694 168 678 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 063 120 217 694 168 678 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 126 240 435 388 337 356 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 126 240 435 388 337 356 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 252 480 870 776 674 713 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 252 480 870 776 674 713 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 504 961 741 553 349 427 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 504 961 741 553 349 427 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 009 923 483 106 698 854 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 009 923 483 106 698 854 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 019 846 966 213 397 708 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 019 846 966 213 397 708 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 039 693 932 426 795 417 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 039 693 932 426 795 417 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 079 387 864 853 590 835 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 079 387 864 853 590 835 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 248 158 775 729 707 181 670 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 248 158 775 729 707 181 670 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 496 317 551 459 414 363 340 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 496 317 551 459 414 363 340 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 992 635 102 918 828 726 681 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 992 635 102 918 828 726 681 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 985 270 205 837 657 453 363 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 985 270 205 837 657 453 363 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 970 540 411 675 314 906 726 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 970 540 411 675 314 906 726 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 941 080 823 350 629 813 452 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 941 080 823 350 629 813 452 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 882 161 646 701 259 626 905 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 882 161 646 701 259 626 905 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 764 323 293 402 519 253 811 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 764 323 293 402 519 253 811 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 528 646 586 805 038 507 622 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 528 646 586 805 038 507 622 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 057 293 173 610 077 015 244 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 057 293 173 610 077 015 244 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 998 114 586 347 220 154 030 489 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 998 114 586 347 220 154 030 489 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 996 229 172 694 440 308 060 979 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 996 229 172 694 440 308 060 979 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 992 458 345 388 880 616 121 958 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 992 458 345 388 880 616 121 958 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 984 916 690 777 761 232 243 916 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 984 916 690 777 761 232 243 916 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 969 833 381 555 522 464 487 833 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 969 833 381 555 522 464 487 833 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 939 666 763 111 044 928 975 667 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 939 666 763 111 044 928 975 667 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 879 333 526 222 089 857 951 334 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 879 333 526 222 089 857 951 334 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 758 667 052 444 179 715 902 668 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 758 667 052 444 179 715 902 668 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 517 334 104 888 359 431 805 337 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 517 334 104 888 359 431 805 337 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 034 668 209 776 718 863 610 675 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 034 668 209 776 718 863 610 675 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 998 069 336 419 553 437 727 221 350 4;
  • 79) 0.794 189 453 124 999 999 999 999 998 069 336 419 553 437 727 221 350 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 996 138 672 839 106 875 454 442 700 8;
  • 80) 0.588 378 906 249 999 999 999 999 996 138 672 839 106 875 454 442 700 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 992 277 345 678 213 750 908 885 401 6;
  • 81) 0.176 757 812 499 999 999 999 999 992 277 345 678 213 750 908 885 401 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 984 554 691 356 427 501 817 770 803 2;
  • 82) 0.353 515 624 999 999 999 999 999 984 554 691 356 427 501 817 770 803 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 969 109 382 712 855 003 635 541 606 4;
  • 83) 0.707 031 249 999 999 999 999 999 969 109 382 712 855 003 635 541 606 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 938 218 765 425 710 007 271 083 212 8;
  • 84) 0.414 062 499 999 999 999 999 999 938 218 765 425 710 007 271 083 212 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 876 437 530 851 420 014 542 166 425 6;
  • 85) 0.828 124 999 999 999 999 999 999 876 437 530 851 420 014 542 166 425 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 752 875 061 702 840 029 084 332 851 2;
  • 86) 0.656 249 999 999 999 999 999 999 752 875 061 702 840 029 084 332 851 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 505 750 123 405 680 058 168 665 702 4;
  • 87) 0.312 499 999 999 999 999 999 999 505 750 123 405 680 058 168 665 702 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 011 500 246 811 360 116 337 331 404 8;
  • 88) 0.624 999 999 999 999 999 999 999 011 500 246 811 360 116 337 331 404 8 × 2 = 1 + 0.249 999 999 999 999 999 999 998 023 000 493 622 720 232 674 662 809 6;
  • 89) 0.249 999 999 999 999 999 999 998 023 000 493 622 720 232 674 662 809 6 × 2 = 0 + 0.499 999 999 999 999 999 999 996 046 000 987 245 440 465 349 325 619 2;
  • 90) 0.499 999 999 999 999 999 999 996 046 000 987 245 440 465 349 325 619 2 × 2 = 0 + 0.999 999 999 999 999 999 999 992 092 001 974 490 880 930 698 651 238 4;
  • 91) 0.999 999 999 999 999 999 999 992 092 001 974 490 880 930 698 651 238 4 × 2 = 1 + 0.999 999 999 999 999 999 999 984 184 003 948 981 761 861 397 302 476 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 146 6 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