0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 3;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 307 3 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 614 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 614 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 229 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 229 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 458 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 458 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 916 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 916 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 833 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 833 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 667 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 667 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 334 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 334 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 668 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 668 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 337 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 197 337 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 675 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 394 675 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 789 350 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 789 350 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 578 700 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 578 700 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 157 401 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 157 401 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 314 803 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 314 803 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 629 606 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 629 606 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 259 212 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 259 212 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 518 425 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 518 425 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 036 851 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 981 036 851 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 073 702 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 962 073 702 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 147 404 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 924 147 404 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 848 294 809 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 848 294 809 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 696 589 619 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 696 589 619 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 393 179 238 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 393 179 238 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 786 358 476 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 786 358 476 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 572 716 953 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 572 716 953 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 145 433 907 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 145 433 907 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 290 867 814 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 290 867 814 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 581 735 628 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 581 735 628 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 163 471 257 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 025 163 471 257 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 326 942 515 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 050 326 942 515 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 100 653 885 030 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 100 653 885 030 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 201 307 770 060 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 201 307 770 060 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 402 615 540 121 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 402 615 540 121 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 805 231 080 243 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 805 231 080 243 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 610 462 160 486 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 610 462 160 486 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 220 924 320 972 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 220 924 320 972 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 441 848 641 945 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 441 848 641 945 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 883 697 283 891 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 883 697 283 891 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 767 394 567 782 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 767 394 567 782 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 107 534 789 135 564 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 107 534 789 135 564 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 215 069 578 271 129 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 215 069 578 271 129 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 430 139 156 542 259 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 430 139 156 542 259 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 860 278 313 084 518 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 860 278 313 084 518 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 720 556 626 169 036 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 720 556 626 169 036 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 441 113 252 338 073 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 441 113 252 338 073 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 882 226 504 676 147 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 882 226 504 676 147 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 764 453 009 352 294 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 764 453 009 352 294 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 528 906 018 704 588 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 528 906 018 704 588 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 071 057 812 037 409 177 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 071 057 812 037 409 177 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 142 115 624 074 818 355 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 142 115 624 074 818 355 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 284 231 248 149 636 710 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 284 231 248 149 636 710 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 568 462 496 299 273 420 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 568 462 496 299 273 420 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 136 924 992 598 546 841 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 136 924 992 598 546 841 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 273 849 985 197 093 683 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 273 849 985 197 093 683 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 547 699 970 394 187 366 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 547 699 970 394 187 366 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 095 399 940 788 374 732 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 095 399 940 788 374 732 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 190 799 881 576 749 465 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 190 799 881 576 749 465 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 381 599 763 153 498 931 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 381 599 763 153 498 931 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 763 199 526 306 997 862 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 763 199 526 306 997 862 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 001 526 399 052 613 995 724 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 001 526 399 052 613 995 724 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 003 052 798 105 227 991 449 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 003 052 798 105 227 991 449 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 006 105 596 210 455 982 899 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 006 105 596 210 455 982 899 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 012 211 192 420 911 965 798 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 012 211 192 420 911 965 798 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 024 422 384 841 823 931 596 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 024 422 384 841 823 931 596 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 048 844 769 683 647 863 193 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 048 844 769 683 647 863 193 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 097 689 539 367 295 726 387 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 097 689 539 367 295 726 387 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 195 379 078 734 591 452 774 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 195 379 078 734 591 452 774 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 390 758 157 469 182 905 548 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 390 758 157 469 182 905 548 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 781 516 314 938 365 811 097 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 781 516 314 938 365 811 097 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 001 563 032 629 876 731 622 195 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 001 563 032 629 876 731 622 195 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 003 126 065 259 753 463 244 390 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 003 126 065 259 753 463 244 390 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 006 252 130 519 506 926 488 780 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 006 252 130 519 506 926 488 780 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 012 504 261 039 013 852 977 561 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 012 504 261 039 013 852 977 561 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 025 008 522 078 027 705 955 123 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 025 008 522 078 027 705 955 123 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 050 017 044 156 055 411 910 246 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 050 017 044 156 055 411 910 246 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 100 034 088 312 110 823 820 492 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 100 034 088 312 110 823 820 492 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 200 068 176 624 221 647 640 985 6;
  • 79) 0.794 189 453 125 000 000 000 000 000 200 068 176 624 221 647 640 985 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 400 136 353 248 443 295 281 971 2;
  • 80) 0.588 378 906 250 000 000 000 000 000 400 136 353 248 443 295 281 971 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 800 272 706 496 886 590 563 942 4;
  • 81) 0.176 757 812 500 000 000 000 000 000 800 272 706 496 886 590 563 942 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 001 600 545 412 993 773 181 127 884 8;
  • 82) 0.353 515 625 000 000 000 000 000 001 600 545 412 993 773 181 127 884 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 003 201 090 825 987 546 362 255 769 6;
  • 83) 0.707 031 250 000 000 000 000 000 003 201 090 825 987 546 362 255 769 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 006 402 181 651 975 092 724 511 539 2;
  • 84) 0.414 062 500 000 000 000 000 000 006 402 181 651 975 092 724 511 539 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 012 804 363 303 950 185 449 023 078 4;
  • 85) 0.828 125 000 000 000 000 000 000 012 804 363 303 950 185 449 023 078 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 025 608 726 607 900 370 898 046 156 8;
  • 86) 0.656 250 000 000 000 000 000 000 025 608 726 607 900 370 898 046 156 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 051 217 453 215 800 741 796 092 313 6;
  • 87) 0.312 500 000 000 000 000 000 000 051 217 453 215 800 741 796 092 313 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 102 434 906 431 601 483 592 184 627 2;
  • 88) 0.625 000 000 000 000 000 000 000 102 434 906 431 601 483 592 184 627 2 × 2 = 1 + 0.250 000 000 000 000 000 000 000 204 869 812 863 202 967 184 369 254 4;
  • 89) 0.250 000 000 000 000 000 000 000 204 869 812 863 202 967 184 369 254 4 × 2 = 0 + 0.500 000 000 000 000 000 000 000 409 739 625 726 405 934 368 738 508 8;
  • 90) 0.500 000 000 000 000 000 000 000 409 739 625 726 405 934 368 738 508 8 × 2 = 1 + 0.000 000 000 000 000 000 000 000 819 479 251 452 811 868 737 477 017 6;
  • 91) 0.000 000 000 000 000 000 000 000 819 479 251 452 811 868 737 477 017 6 × 2 = 0 + 0.000 000 000 000 000 000 000 001 638 958 502 905 623 737 474 954 035 2;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 65 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100