0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 268 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 268 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 537 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 537 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 075 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 075 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 150 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 150 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 300 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 300 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 601 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 448 601 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 203 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 897 203 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 794 406 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 794 406 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 588 812 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 588 812 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 177 625 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 177 625 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 355 251 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 355 251 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 710 502 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 710 502 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 421 004 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 421 004 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 842 009 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 842 842 009 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 684 019 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 685 684 019 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 368 038 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 371 368 038 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 736 076 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 742 736 076 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 485 472 153 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 485 472 153 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 970 944 307 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 970 944 307 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 941 888 614 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 941 888 614 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 883 777 228 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 883 777 228 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 767 554 457 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 767 554 457 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 535 108 915 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 535 108 915 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 070 217 830 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 070 217 830 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 140 435 660 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 140 435 660 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 280 871 321 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 500 280 871 321 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 000 561 742 643 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 000 561 742 643 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 001 123 485 286 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 001 123 485 286 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 002 246 970 572 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 002 246 970 572 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 004 493 941 145 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 004 493 941 145 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 008 987 882 291 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 008 987 882 291 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 017 975 764 582 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 017 975 764 582 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 035 951 529 164 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 035 951 529 164 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 071 903 058 329 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 071 903 058 329 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 143 806 116 659 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 143 806 116 659 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 408 287 612 233 318 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 408 287 612 233 318 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 816 575 224 466 636 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 816 575 224 466 636 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 633 150 448 933 273 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 633 150 448 933 273 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 266 300 897 866 547 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 266 300 897 866 547 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 532 601 795 733 094 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 532 601 795 733 094 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 065 203 591 466 188 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 065 203 591 466 188 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 130 407 182 932 377 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 130 407 182 932 377 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 260 814 365 864 755 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 260 814 365 864 755 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 521 628 731 729 510 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 521 628 731 729 510 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 043 257 463 459 020 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 043 257 463 459 020 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 378 086 514 926 918 041 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 378 086 514 926 918 041 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 756 173 029 853 836 083 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 756 173 029 853 836 083 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 512 346 059 707 672 166 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 512 346 059 707 672 166 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 024 692 119 415 344 332 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 024 692 119 415 344 332 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 049 384 238 830 688 665 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 049 384 238 830 688 665 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 098 768 477 661 377 331 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 098 768 477 661 377 331 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 197 536 955 322 754 662 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 197 536 955 322 754 662 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 395 073 910 645 509 324 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 395 073 910 645 509 324 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 952 790 147 821 291 018 649 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 952 790 147 821 291 018 649 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 905 580 295 642 582 037 299 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 905 580 295 642 582 037 299 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 811 160 591 285 164 074 598 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 811 160 591 285 164 074 598 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 622 321 182 570 328 149 196 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 622 321 182 570 328 149 196 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 244 642 365 140 656 298 393 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 244 642 365 140 656 298 393 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 489 284 730 281 312 596 787 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 489 284 730 281 312 596 787 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 978 569 460 562 625 193 574 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 978 569 460 562 625 193 574 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 957 138 921 125 250 387 148 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 957 138 921 125 250 387 148 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 914 277 842 250 500 774 297 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 914 277 842 250 500 774 297 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 828 555 684 501 001 548 595 2;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 828 555 684 501 001 548 595 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 657 111 369 002 003 097 190 4;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 657 111 369 002 003 097 190 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 314 222 738 004 006 194 380 8;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 314 222 738 004 006 194 380 8 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 998 628 445 476 008 012 388 761 6;
  • 67) 0.831 736 862 659 454 345 703 124 999 998 628 445 476 008 012 388 761 6 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 997 256 890 952 016 024 777 523 2;
  • 68) 0.663 473 725 318 908 691 406 249 999 997 256 890 952 016 024 777 523 2 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 994 513 781 904 032 049 555 046 4;
  • 69) 0.326 947 450 637 817 382 812 499 999 994 513 781 904 032 049 555 046 4 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 989 027 563 808 064 099 110 092 8;
  • 70) 0.653 894 901 275 634 765 624 999 999 989 027 563 808 064 099 110 092 8 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 978 055 127 616 128 198 220 185 6;
  • 71) 0.307 789 802 551 269 531 249 999 999 978 055 127 616 128 198 220 185 6 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 956 110 255 232 256 396 440 371 2;
  • 72) 0.615 579 605 102 539 062 499 999 999 956 110 255 232 256 396 440 371 2 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 912 220 510 464 512 792 880 742 4;
  • 73) 0.231 159 210 205 078 124 999 999 999 912 220 510 464 512 792 880 742 4 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 824 441 020 929 025 585 761 484 8;
  • 74) 0.462 318 420 410 156 249 999 999 999 824 441 020 929 025 585 761 484 8 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 648 882 041 858 051 171 522 969 6;
  • 75) 0.924 636 840 820 312 499 999 999 999 648 882 041 858 051 171 522 969 6 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 297 764 083 716 102 343 045 939 2;
  • 76) 0.849 273 681 640 624 999 999 999 999 297 764 083 716 102 343 045 939 2 × 2 = 1 + 0.698 547 363 281 249 999 999 999 998 595 528 167 432 204 686 091 878 4;
  • 77) 0.698 547 363 281 249 999 999 999 998 595 528 167 432 204 686 091 878 4 × 2 = 1 + 0.397 094 726 562 499 999 999 999 997 191 056 334 864 409 372 183 756 8;
  • 78) 0.397 094 726 562 499 999 999 999 997 191 056 334 864 409 372 183 756 8 × 2 = 0 + 0.794 189 453 124 999 999 999 999 994 382 112 669 728 818 744 367 513 6;
  • 79) 0.794 189 453 124 999 999 999 999 994 382 112 669 728 818 744 367 513 6 × 2 = 1 + 0.588 378 906 249 999 999 999 999 988 764 225 339 457 637 488 735 027 2;
  • 80) 0.588 378 906 249 999 999 999 999 988 764 225 339 457 637 488 735 027 2 × 2 = 1 + 0.176 757 812 499 999 999 999 999 977 528 450 678 915 274 977 470 054 4;
  • 81) 0.176 757 812 499 999 999 999 999 977 528 450 678 915 274 977 470 054 4 × 2 = 0 + 0.353 515 624 999 999 999 999 999 955 056 901 357 830 549 954 940 108 8;
  • 82) 0.353 515 624 999 999 999 999 999 955 056 901 357 830 549 954 940 108 8 × 2 = 0 + 0.707 031 249 999 999 999 999 999 910 113 802 715 661 099 909 880 217 6;
  • 83) 0.707 031 249 999 999 999 999 999 910 113 802 715 661 099 909 880 217 6 × 2 = 1 + 0.414 062 499 999 999 999 999 999 820 227 605 431 322 199 819 760 435 2;
  • 84) 0.414 062 499 999 999 999 999 999 820 227 605 431 322 199 819 760 435 2 × 2 = 0 + 0.828 124 999 999 999 999 999 999 640 455 210 862 644 399 639 520 870 4;
  • 85) 0.828 124 999 999 999 999 999 999 640 455 210 862 644 399 639 520 870 4 × 2 = 1 + 0.656 249 999 999 999 999 999 999 280 910 421 725 288 799 279 041 740 8;
  • 86) 0.656 249 999 999 999 999 999 999 280 910 421 725 288 799 279 041 740 8 × 2 = 1 + 0.312 499 999 999 999 999 999 998 561 820 843 450 577 598 558 083 481 6;
  • 87) 0.312 499 999 999 999 999 999 998 561 820 843 450 577 598 558 083 481 6 × 2 = 0 + 0.624 999 999 999 999 999 999 997 123 641 686 901 155 197 116 166 963 2;
  • 88) 0.624 999 999 999 999 999 999 997 123 641 686 901 155 197 116 166 963 2 × 2 = 1 + 0.249 999 999 999 999 999 999 994 247 283 373 802 310 394 232 333 926 4;
  • 89) 0.249 999 999 999 999 999 999 994 247 283 373 802 310 394 232 333 926 4 × 2 = 0 + 0.499 999 999 999 999 999 999 988 494 566 747 604 620 788 464 667 852 8;
  • 90) 0.499 999 999 999 999 999 999 988 494 566 747 604 620 788 464 667 852 8 × 2 = 0 + 0.999 999 999 999 999 999 999 976 989 133 495 209 241 576 929 335 705 6;
  • 91) 0.999 999 999 999 999 999 999 976 989 133 495 209 241 576 929 335 705 6 × 2 = 1 + 0.999 999 999 999 999 999 999 953 978 266 990 418 483 153 858 671 411 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 134 4 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