0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 971 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 971 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 942 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 942 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 884 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 884 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 769 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 769 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 539 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 539 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 078 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 078 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 156 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 156 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 313 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 313 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 328 627 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 328 627 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 657 254 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 657 254 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 314 508 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 314 508 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 629 017 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 629 017 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 258 035 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 258 035 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 516 070 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 516 070 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 032 140 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 032 140 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 064 281 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 064 281 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 128 563 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 128 563 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 257 126 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 257 126 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 688 514 252 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 688 514 252 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 377 028 505 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 377 028 505 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 754 057 011 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 754 057 011 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 508 114 022 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 508 114 022 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 016 228 044 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 016 228 044 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 032 456 089 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 032 456 089 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 064 912 179 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 064 912 179 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 129 824 358 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 129 824 358 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 259 648 716 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 259 648 716 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 112 519 297 433 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 112 519 297 433 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 225 038 594 867 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 225 038 594 867 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 450 077 189 734 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 450 077 189 734 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 900 154 379 468 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 900 154 379 468 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 800 308 758 937 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 800 308 758 937 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 600 617 517 875 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 600 617 517 875 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 201 235 035 750 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 201 235 035 750 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 402 470 071 500 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 402 470 071 500 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 804 940 143 001 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 804 940 143 001 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 609 880 286 003 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 609 880 286 003 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 219 760 572 006 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 219 760 572 006 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 518 439 521 144 012 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 518 439 521 144 012 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 036 879 042 288 025 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 036 879 042 288 025 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 073 758 084 576 051 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 073 758 084 576 051 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 147 516 169 152 102 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 147 516 169 152 102 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 295 032 338 304 204 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 295 032 338 304 204 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 590 064 676 608 409 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 590 064 676 608 409 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 180 129 353 216 819 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 180 129 353 216 819 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 360 258 706 433 638 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 360 258 706 433 638 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 788 720 517 412 867 276 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 788 720 517 412 867 276 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 577 441 034 825 734 553 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 577 441 034 825 734 553 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 154 882 069 651 469 107 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 154 882 069 651 469 107 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 309 764 139 302 938 214 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 309 764 139 302 938 214 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 619 528 278 605 876 428 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 619 528 278 605 876 428 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 239 056 557 211 752 857 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 239 056 557 211 752 857 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 478 113 114 423 505 715 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 478 113 114 423 505 715 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 956 226 228 847 011 430 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 956 226 228 847 011 430 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 912 452 457 694 022 860 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 912 452 457 694 022 860 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 824 904 915 388 045 721 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 824 904 915 388 045 721 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 649 809 830 776 091 443 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 649 809 830 776 091 443 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 299 619 661 552 182 886 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 299 619 661 552 182 886 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 998 599 239 323 104 365 772 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 998 599 239 323 104 365 772 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 997 198 478 646 208 731 545 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 997 198 478 646 208 731 545 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 994 396 957 292 417 463 091 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 994 396 957 292 417 463 091 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 988 793 914 584 834 926 182 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 988 793 914 584 834 926 182 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 977 587 829 169 669 852 364 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 977 587 829 169 669 852 364 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 955 175 658 339 339 704 729 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 955 175 658 339 339 704 729 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 910 351 316 678 679 409 459 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 910 351 316 678 679 409 459 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 820 702 633 357 358 818 918 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 820 702 633 357 358 818 918 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 641 405 266 714 717 637 836 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 641 405 266 714 717 637 836 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 282 810 533 429 435 275 673 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 282 810 533 429 435 275 673 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 998 565 621 066 858 870 551 347 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 998 565 621 066 858 870 551 347 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 997 131 242 133 717 741 102 694 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 997 131 242 133 717 741 102 694 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 994 262 484 267 435 482 205 388 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 994 262 484 267 435 482 205 388 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 988 524 968 534 870 964 410 777 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 988 524 968 534 870 964 410 777 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 977 049 937 069 741 928 821 555 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 977 049 937 069 741 928 821 555 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 954 099 874 139 483 857 643 110 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 954 099 874 139 483 857 643 110 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 908 199 748 278 967 715 286 220 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 908 199 748 278 967 715 286 220 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 816 399 496 557 935 430 572 441 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 816 399 496 557 935 430 572 441 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 632 798 993 115 870 861 144 883 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 632 798 993 115 870 861 144 883 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 265 597 986 231 741 722 289 766 4;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 265 597 986 231 741 722 289 766 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 998 531 195 972 463 483 444 579 532 8;
  • 80) 0.588 378 906 249 999 999 999 999 999 998 531 195 972 463 483 444 579 532 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 997 062 391 944 926 966 889 159 065 6;
  • 81) 0.176 757 812 499 999 999 999 999 999 997 062 391 944 926 966 889 159 065 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 994 124 783 889 853 933 778 318 131 2;
  • 82) 0.353 515 624 999 999 999 999 999 999 994 124 783 889 853 933 778 318 131 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 988 249 567 779 707 867 556 636 262 4;
  • 83) 0.707 031 249 999 999 999 999 999 999 988 249 567 779 707 867 556 636 262 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 976 499 135 559 415 735 113 272 524 8;
  • 84) 0.414 062 499 999 999 999 999 999 999 976 499 135 559 415 735 113 272 524 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 952 998 271 118 831 470 226 545 049 6;
  • 85) 0.828 124 999 999 999 999 999 999 999 952 998 271 118 831 470 226 545 049 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 905 996 542 237 662 940 453 090 099 2;
  • 86) 0.656 249 999 999 999 999 999 999 999 905 996 542 237 662 940 453 090 099 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 811 993 084 475 325 880 906 180 198 4;
  • 87) 0.312 499 999 999 999 999 999 999 999 811 993 084 475 325 880 906 180 198 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 623 986 168 950 651 761 812 360 396 8;
  • 88) 0.624 999 999 999 999 999 999 999 999 623 986 168 950 651 761 812 360 396 8 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 247 972 337 901 303 523 624 720 793 6;
  • 89) 0.249 999 999 999 999 999 999 999 999 247 972 337 901 303 523 624 720 793 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 998 495 944 675 802 607 047 249 441 587 2;
  • 90) 0.499 999 999 999 999 999 999 999 998 495 944 675 802 607 047 249 441 587 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 996 991 889 351 605 214 094 498 883 174 4;
  • 91) 0.999 999 999 999 999 999 999 999 996 991 889 351 605 214 094 498 883 174 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 993 983 778 703 210 428 188 997 766 348 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 985 6 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