0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 24 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 24(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 24(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 24.

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 24 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 48;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 48 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 96;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 96 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 92;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 225 92 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 84;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 451 84 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 903 68;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 903 68 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 807 36;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 807 36 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 614 72;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 614 72 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 229 44;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 229 44 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 458 88;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 458 88 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 917 76;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 917 76 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 835 52;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 835 52 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 671 04;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 787 671 04 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 575 342 08;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 575 342 08 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 150 684 16;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 150 684 16 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 301 368 32;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 301 368 32 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 602 736 64;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 602 736 64 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 205 473 28;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 205 473 28 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 410 946 56;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 410 946 56 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 821 893 12;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 821 893 12 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 643 786 24;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 643 786 24 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 287 572 48;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 923 287 572 48 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 575 144 96;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 846 575 144 96 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 693 150 289 92;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 693 150 289 92 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 386 300 579 84;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 386 300 579 84 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 772 601 159 68;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 772 601 159 68 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 545 202 319 36;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 545 202 319 36 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 090 404 638 72;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 090 404 638 72 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 180 809 277 44;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 180 809 277 44 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 361 618 554 88;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 361 618 554 88 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 723 237 109 76;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 723 237 109 76 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 446 474 219 52;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 446 474 219 52 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 892 948 439 04;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 892 948 439 04 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 785 896 878 08;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 197 785 896 878 08 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 395 571 793 756 16;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 395 571 793 756 16 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 791 143 587 512 32;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 791 143 587 512 32 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 582 287 175 024 64;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 582 287 175 024 64 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 164 574 350 049 28;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 164 574 350 049 28 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 329 148 700 098 56;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 329 148 700 098 56 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 658 297 400 197 12;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 658 297 400 197 12 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 316 594 800 394 24;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 316 594 800 394 24 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 633 189 600 788 48;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 633 189 600 788 48 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 213 266 379 201 576 96;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 213 266 379 201 576 96 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 426 532 758 403 153 92;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 426 532 758 403 153 92 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 853 065 516 806 307 84;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 853 065 516 806 307 84 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 706 131 033 612 615 68;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 706 131 033 612 615 68 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 412 262 067 225 231 36;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 412 262 067 225 231 36 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 824 524 134 450 462 72;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 824 524 134 450 462 72 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 649 048 268 900 925 44;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 649 048 268 900 925 44 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 298 096 537 801 850 88;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 298 096 537 801 850 88 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 596 193 075 603 701 76;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 596 193 075 603 701 76 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 141 192 386 151 207 403 52;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 141 192 386 151 207 403 52 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 282 384 772 302 414 807 04;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 282 384 772 302 414 807 04 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 564 769 544 604 829 614 08;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 564 769 544 604 829 614 08 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 129 539 089 209 659 228 16;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 129 539 089 209 659 228 16 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 259 078 178 419 318 456 32;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 259 078 178 419 318 456 32 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 518 156 356 838 636 912 64;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 518 156 356 838 636 912 64 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 036 312 713 677 273 825 28;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 036 312 713 677 273 825 28 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 072 625 427 354 547 650 56;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 072 625 427 354 547 650 56 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 145 250 854 709 095 301 12;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 145 250 854 709 095 301 12 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 290 501 709 418 190 602 24;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 290 501 709 418 190 602 24 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 581 003 418 836 381 204 48;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 581 003 418 836 381 204 48 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 001 162 006 837 672 762 408 96;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 001 162 006 837 672 762 408 96 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 002 324 013 675 345 524 817 92;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 002 324 013 675 345 524 817 92 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 004 648 027 350 691 049 635 84;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 004 648 027 350 691 049 635 84 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 009 296 054 701 382 099 271 68;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 009 296 054 701 382 099 271 68 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 018 592 109 402 764 198 543 36;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 018 592 109 402 764 198 543 36 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 037 184 218 805 528 397 086 72;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 037 184 218 805 528 397 086 72 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 074 368 437 611 056 794 173 44;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 074 368 437 611 056 794 173 44 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 148 736 875 222 113 588 346 88;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 148 736 875 222 113 588 346 88 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 297 473 750 444 227 176 693 76;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 297 473 750 444 227 176 693 76 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 594 947 500 888 454 353 387 52;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 594 947 500 888 454 353 387 52 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 001 189 895 001 776 908 706 775 04;
  • 73) 0.231 159 210 205 078 125 000 000 000 001 189 895 001 776 908 706 775 04 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 002 379 790 003 553 817 413 550 08;
  • 74) 0.462 318 420 410 156 250 000 000 000 002 379 790 003 553 817 413 550 08 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 004 759 580 007 107 634 827 100 16;
  • 75) 0.924 636 840 820 312 500 000 000 000 004 759 580 007 107 634 827 100 16 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 009 519 160 014 215 269 654 200 32;
  • 76) 0.849 273 681 640 625 000 000 000 000 009 519 160 014 215 269 654 200 32 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 019 038 320 028 430 539 308 400 64;
  • 77) 0.698 547 363 281 250 000 000 000 000 019 038 320 028 430 539 308 400 64 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 038 076 640 056 861 078 616 801 28;
  • 78) 0.397 094 726 562 500 000 000 000 000 038 076 640 056 861 078 616 801 28 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 076 153 280 113 722 157 233 602 56;
  • 79) 0.794 189 453 125 000 000 000 000 000 076 153 280 113 722 157 233 602 56 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 152 306 560 227 444 314 467 205 12;
  • 80) 0.588 378 906 250 000 000 000 000 000 152 306 560 227 444 314 467 205 12 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 304 613 120 454 888 628 934 410 24;
  • 81) 0.176 757 812 500 000 000 000 000 000 304 613 120 454 888 628 934 410 24 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 609 226 240 909 777 257 868 820 48;
  • 82) 0.353 515 625 000 000 000 000 000 000 609 226 240 909 777 257 868 820 48 × 2 = 0 + 0.707 031 250 000 000 000 000 000 001 218 452 481 819 554 515 737 640 96;
  • 83) 0.707 031 250 000 000 000 000 000 001 218 452 481 819 554 515 737 640 96 × 2 = 1 + 0.414 062 500 000 000 000 000 000 002 436 904 963 639 109 031 475 281 92;
  • 84) 0.414 062 500 000 000 000 000 000 002 436 904 963 639 109 031 475 281 92 × 2 = 0 + 0.828 125 000 000 000 000 000 000 004 873 809 927 278 218 062 950 563 84;
  • 85) 0.828 125 000 000 000 000 000 000 004 873 809 927 278 218 062 950 563 84 × 2 = 1 + 0.656 250 000 000 000 000 000 000 009 747 619 854 556 436 125 901 127 68;
  • 86) 0.656 250 000 000 000 000 000 000 009 747 619 854 556 436 125 901 127 68 × 2 = 1 + 0.312 500 000 000 000 000 000 000 019 495 239 709 112 872 251 802 255 36;
  • 87) 0.312 500 000 000 000 000 000 000 019 495 239 709 112 872 251 802 255 36 × 2 = 0 + 0.625 000 000 000 000 000 000 000 038 990 479 418 225 744 503 604 510 72;
  • 88) 0.625 000 000 000 000 000 000 000 038 990 479 418 225 744 503 604 510 72 × 2 = 1 + 0.250 000 000 000 000 000 000 000 077 980 958 836 451 489 007 209 021 44;
  • 89) 0.250 000 000 000 000 000 000 000 077 980 958 836 451 489 007 209 021 44 × 2 = 0 + 0.500 000 000 000 000 000 000 000 155 961 917 672 902 978 014 418 042 88;
  • 90) 0.500 000 000 000 000 000 000 000 155 961 917 672 902 978 014 418 042 88 × 2 = 1 + 0.000 000 000 000 000 000 000 000 311 923 835 345 805 956 028 836 085 76;
  • 91) 0.000 000 000 000 000 000 000 000 311 923 835 345 805 956 028 836 085 76 × 2 = 0 + 0.000 000 000 000 000 000 000 000 623 847 670 691 611 912 057 672 171 52;

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 24(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 24(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 24(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 24 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