0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 955 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 955 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 911 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 911 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 822 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 822 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 644 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 644 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 289 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 289 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 579 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 579 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 158 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 158 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 162 316 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 162 316 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 324 633 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 324 633 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 649 267 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 649 267 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 298 534 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 298 534 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 597 068 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 597 068 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 194 137 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 194 137 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 388 275 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 388 275 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 776 550 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 776 550 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 553 100 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 553 100 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 106 201 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 106 201 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 342 212 403 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 342 212 403 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 684 424 806 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 684 424 806 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 368 849 612 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 368 849 612 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 737 699 225 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 737 699 225 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 475 398 451 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 475 398 451 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 950 796 902 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 950 796 902 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 901 593 804 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 901 593 804 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 803 187 609 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 803 187 609 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 606 375 219 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 606 375 219 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 212 750 438 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 212 750 438 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 110 425 500 876 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 110 425 500 876 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 220 851 001 753 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 220 851 001 753 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 441 702 003 507 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 441 702 003 507 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 883 404 007 014 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 883 404 007 014 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 766 808 014 028 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 766 808 014 028 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 533 616 028 057 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 533 616 028 057 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 067 232 056 115 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 067 232 056 115 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 134 464 112 230 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 134 464 112 230 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 268 928 224 460 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 268 928 224 460 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 128 537 856 448 921 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 128 537 856 448 921 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 257 075 712 897 843 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 257 075 712 897 843 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 514 151 425 795 686 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 514 151 425 795 686 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 028 302 851 591 372 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 028 302 851 591 372 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 056 605 703 182 745 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 056 605 703 182 745 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 113 211 406 365 491 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 113 211 406 365 491 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 226 422 812 730 982 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 226 422 812 730 982 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 452 845 625 461 964 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 452 845 625 461 964 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 696 905 691 250 923 929 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 696 905 691 250 923 929 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 393 811 382 501 847 859 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 393 811 382 501 847 859 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 787 622 765 003 695 718 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 787 622 765 003 695 718 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 575 245 530 007 391 436 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 575 245 530 007 391 436 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 150 491 060 014 782 873 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 150 491 060 014 782 873 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 300 982 120 029 565 747 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 300 982 120 029 565 747 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 601 964 240 059 131 494 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 601 964 240 059 131 494 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 203 928 480 118 262 988 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 203 928 480 118 262 988 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 407 856 960 236 525 977 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 407 856 960 236 525 977 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 815 713 920 473 051 955 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 815 713 920 473 051 955 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 631 427 840 946 103 910 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 631 427 840 946 103 910 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 262 855 681 892 207 820 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 262 855 681 892 207 820 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 998 525 711 363 784 415 641 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 998 525 711 363 784 415 641 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 997 051 422 727 568 831 283 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 997 051 422 727 568 831 283 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 994 102 845 455 137 662 566 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 994 102 845 455 137 662 566 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 988 205 690 910 275 325 132 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 988 205 690 910 275 325 132 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 976 411 381 820 550 650 265 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 976 411 381 820 550 650 265 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 952 822 763 641 101 300 531 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 952 822 763 641 101 300 531 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 905 645 527 282 202 601 062 4;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 905 645 527 282 202 601 062 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 811 291 054 564 405 202 124 8;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 811 291 054 564 405 202 124 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 622 582 109 128 810 404 249 6;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 622 582 109 128 810 404 249 6 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 245 164 218 257 620 808 499 2;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 245 164 218 257 620 808 499 2 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 998 490 328 436 515 241 616 998 4;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 998 490 328 436 515 241 616 998 4 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 996 980 656 873 030 483 233 996 8;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 996 980 656 873 030 483 233 996 8 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 993 961 313 746 060 966 467 993 6;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 993 961 313 746 060 966 467 993 6 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 987 922 627 492 121 932 935 987 2;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 987 922 627 492 121 932 935 987 2 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 975 845 254 984 243 865 871 974 4;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 975 845 254 984 243 865 871 974 4 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 951 690 509 968 487 731 743 948 8;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 951 690 509 968 487 731 743 948 8 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 903 381 019 936 975 463 487 897 6;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 903 381 019 936 975 463 487 897 6 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 806 762 039 873 950 926 975 795 2;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 806 762 039 873 950 926 975 795 2 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 613 524 079 747 901 853 951 590 4;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 613 524 079 747 901 853 951 590 4 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 227 048 159 495 803 707 903 180 8;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 227 048 159 495 803 707 903 180 8 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 998 454 096 318 991 607 415 806 361 6;
  • 78) 0.397 094 726 562 499 999 999 999 999 998 454 096 318 991 607 415 806 361 6 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 996 908 192 637 983 214 831 612 723 2;
  • 79) 0.794 189 453 124 999 999 999 999 999 996 908 192 637 983 214 831 612 723 2 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 993 816 385 275 966 429 663 225 446 4;
  • 80) 0.588 378 906 249 999 999 999 999 999 993 816 385 275 966 429 663 225 446 4 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 987 632 770 551 932 859 326 450 892 8;
  • 81) 0.176 757 812 499 999 999 999 999 999 987 632 770 551 932 859 326 450 892 8 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 975 265 541 103 865 718 652 901 785 6;
  • 82) 0.353 515 624 999 999 999 999 999 999 975 265 541 103 865 718 652 901 785 6 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 950 531 082 207 731 437 305 803 571 2;
  • 83) 0.707 031 249 999 999 999 999 999 999 950 531 082 207 731 437 305 803 571 2 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 901 062 164 415 462 874 611 607 142 4;
  • 84) 0.414 062 499 999 999 999 999 999 999 901 062 164 415 462 874 611 607 142 4 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 802 124 328 830 925 749 223 214 284 8;
  • 85) 0.828 124 999 999 999 999 999 999 999 802 124 328 830 925 749 223 214 284 8 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 604 248 657 661 851 498 446 428 569 6;
  • 86) 0.656 249 999 999 999 999 999 999 999 604 248 657 661 851 498 446 428 569 6 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 208 497 315 323 702 996 892 857 139 2;
  • 87) 0.312 499 999 999 999 999 999 999 999 208 497 315 323 702 996 892 857 139 2 × 2 = 0 + 0.624 999 999 999 999 999 999 999 998 416 994 630 647 405 993 785 714 278 4;
  • 88) 0.624 999 999 999 999 999 999 999 998 416 994 630 647 405 993 785 714 278 4 × 2 = 1 + 0.249 999 999 999 999 999 999 999 996 833 989 261 294 811 987 571 428 556 8;
  • 89) 0.249 999 999 999 999 999 999 999 996 833 989 261 294 811 987 571 428 556 8 × 2 = 0 + 0.499 999 999 999 999 999 999 999 993 667 978 522 589 623 975 142 857 113 6;
  • 90) 0.499 999 999 999 999 999 999 999 993 667 978 522 589 623 975 142 857 113 6 × 2 = 0 + 0.999 999 999 999 999 999 999 999 987 335 957 045 179 247 950 285 714 227 2;
  • 91) 0.999 999 999 999 999 999 999 999 987 335 957 045 179 247 950 285 714 227 2 × 2 = 1 + 0.999 999 999 999 999 999 999 999 974 671 914 090 358 495 900 571 428 454 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 977 8 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