0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 317 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 317 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 635 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 635 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 271 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 271 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 542 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 542 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 084 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 084 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 169 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 169 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 339 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 339 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 678 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 678 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 356 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 356 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 202 713 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 202 713 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 405 427 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 405 427 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 810 854 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 810 854 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 621 708 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 621 708 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 243 417 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 243 417 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 486 835 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 486 835 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 973 670 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 973 670 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 947 340 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 947 340 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 894 681 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 894 681 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 983 789 363 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 983 789 363 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 967 578 726 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 967 578 726 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 935 157 452 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 935 157 452 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 870 314 905 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 870 314 905 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 740 629 811 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 740 629 811 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 481 259 622 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 481 259 622 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 962 519 244 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 962 519 244 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 925 038 489 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 925 038 489 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 850 076 979 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 850 076 979 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 700 153 958 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 700 153 958 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 400 307 916 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 400 307 916 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 030 800 615 833 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 030 800 615 833 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 061 601 231 667 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 061 601 231 667 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 123 202 463 334 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 123 202 463 334 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 246 404 926 668 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 246 404 926 668 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 492 809 853 337 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 492 809 853 337 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 985 619 706 675 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 985 619 706 675 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 971 239 413 350 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 971 239 413 350 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 942 478 826 700 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 942 478 826 700 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 639 884 957 653 401 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 639 884 957 653 401 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 279 769 915 306 803 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 279 769 915 306 803 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 559 539 830 613 606 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 559 539 830 613 606 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 119 079 661 227 212 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 119 079 661 227 212 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 238 159 322 454 425 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 238 159 322 454 425 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 476 318 644 908 851 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 476 318 644 908 851 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 952 637 289 817 702 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 952 637 289 817 702 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 905 274 579 635 404 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 905 274 579 635 404 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 810 549 159 270 809 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 810 549 159 270 809 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 621 098 318 541 619 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 621 098 318 541 619 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 519 242 196 637 083 238 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 519 242 196 637 083 238 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 038 484 393 274 166 476 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 038 484 393 274 166 476 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 076 968 786 548 332 953 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 076 968 786 548 332 953 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 153 937 573 096 665 907 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 153 937 573 096 665 907 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 307 875 146 193 331 814 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 307 875 146 193 331 814 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 615 750 292 386 663 628 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 615 750 292 386 663 628 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 231 500 584 773 327 257 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 231 500 584 773 327 257 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 463 001 169 546 654 515 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 463 001 169 546 654 515 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 926 002 339 093 309 030 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 926 002 339 093 309 030 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 852 004 678 186 618 060 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 852 004 678 186 618 060 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 251 704 009 356 373 236 121 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 251 704 009 356 373 236 121 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 503 408 018 712 746 472 243 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 503 408 018 712 746 472 243 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 006 816 037 425 492 944 486 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 006 816 037 425 492 944 486 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 013 632 074 850 985 888 972 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 013 632 074 850 985 888 972 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 027 264 149 701 971 777 945 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 027 264 149 701 971 777 945 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 054 528 299 403 943 555 891 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 054 528 299 403 943 555 891 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 109 056 598 807 887 111 782 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 109 056 598 807 887 111 782 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 218 113 197 615 774 223 564 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 218 113 197 615 774 223 564 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 436 226 395 231 548 447 129 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 436 226 395 231 548 447 129 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 872 452 790 463 096 894 259 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 872 452 790 463 096 894 259 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 001 744 905 580 926 193 788 518 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 001 744 905 580 926 193 788 518 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 003 489 811 161 852 387 577 036 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 003 489 811 161 852 387 577 036 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 006 979 622 323 704 775 154 073 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 006 979 622 323 704 775 154 073 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 013 959 244 647 409 550 308 147 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 013 959 244 647 409 550 308 147 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 027 918 489 294 819 100 616 294 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 027 918 489 294 819 100 616 294 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 055 836 978 589 638 201 232 588 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 055 836 978 589 638 201 232 588 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 111 673 957 179 276 402 465 177 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 111 673 957 179 276 402 465 177 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 223 347 914 358 552 804 930 355 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 223 347 914 358 552 804 930 355 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 446 695 828 717 105 609 860 710 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 446 695 828 717 105 609 860 710 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 893 391 657 434 211 219 721 420 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 893 391 657 434 211 219 721 420 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 001 786 783 314 868 422 439 442 841 6;
  • 79) 0.794 189 453 125 000 000 000 000 001 786 783 314 868 422 439 442 841 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 003 573 566 629 736 844 878 885 683 2;
  • 80) 0.588 378 906 250 000 000 000 000 003 573 566 629 736 844 878 885 683 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 007 147 133 259 473 689 757 771 366 4;
  • 81) 0.176 757 812 500 000 000 000 000 007 147 133 259 473 689 757 771 366 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 014 294 266 518 947 379 515 542 732 8;
  • 82) 0.353 515 625 000 000 000 000 000 014 294 266 518 947 379 515 542 732 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 028 588 533 037 894 759 031 085 465 6;
  • 83) 0.707 031 250 000 000 000 000 000 028 588 533 037 894 759 031 085 465 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 057 177 066 075 789 518 062 170 931 2;
  • 84) 0.414 062 500 000 000 000 000 000 057 177 066 075 789 518 062 170 931 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 114 354 132 151 579 036 124 341 862 4;
  • 85) 0.828 125 000 000 000 000 000 000 114 354 132 151 579 036 124 341 862 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 228 708 264 303 158 072 248 683 724 8;
  • 86) 0.656 250 000 000 000 000 000 000 228 708 264 303 158 072 248 683 724 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 457 416 528 606 316 144 497 367 449 6;
  • 87) 0.312 500 000 000 000 000 000 000 457 416 528 606 316 144 497 367 449 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 914 833 057 212 632 288 994 734 899 2;
  • 88) 0.625 000 000 000 000 000 000 000 914 833 057 212 632 288 994 734 899 2 × 2 = 1 + 0.250 000 000 000 000 000 000 001 829 666 114 425 264 577 989 469 798 4;
  • 89) 0.250 000 000 000 000 000 000 001 829 666 114 425 264 577 989 469 798 4 × 2 = 0 + 0.500 000 000 000 000 000 000 003 659 332 228 850 529 155 978 939 596 8;
  • 90) 0.500 000 000 000 000 000 000 003 659 332 228 850 529 155 978 939 596 8 × 2 = 1 + 0.000 000 000 000 000 000 000 007 318 664 457 701 058 311 957 879 193 6;
  • 91) 0.000 000 000 000 000 000 000 007 318 664 457 701 058 311 957 879 193 6 × 2 = 0 + 0.000 000 000 000 000 000 000 014 637 328 915 402 116 623 915 758 387 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 158 9 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