0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 988 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 988 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 976 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 976 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 953 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 953 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 907 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 907 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 814 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 814 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 628 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 628 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 257 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 257 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 166 515 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 166 515 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 333 030 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 333 030 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 666 060 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 666 060 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 332 121 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 332 121 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 664 243 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 664 243 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 328 486 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 328 486 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 656 972 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 656 972 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 313 945 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 313 945 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 627 891 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 627 891 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 255 782 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 255 782 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 346 511 564 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 346 511 564 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 693 023 129 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 693 023 129 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 386 046 259 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 386 046 259 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 772 092 518 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 772 092 518 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 544 185 036 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 544 185 036 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 088 370 073 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 088 370 073 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 176 740 147 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 176 740 147 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 353 480 294 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 353 480 294 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 706 960 588 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 706 960 588 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 413 921 177 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 413 921 177 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 114 827 842 355 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 114 827 842 355 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 229 655 684 710 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 229 655 684 710 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 459 311 369 420 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 459 311 369 420 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 918 622 738 841 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 918 622 738 841 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 837 245 477 683 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 837 245 477 683 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 674 490 955 366 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 674 490 955 366 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 348 981 910 732 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 348 981 910 732 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 697 963 821 465 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 697 963 821 465 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 395 927 642 931 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 395 927 642 931 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 791 855 285 862 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 791 855 285 862 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 261 583 710 571 724 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 261 583 710 571 724 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 523 167 421 143 449 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 523 167 421 143 449 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 046 334 842 286 899 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 046 334 842 286 899 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 092 669 684 573 798 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 092 669 684 573 798 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 185 339 369 147 596 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 185 339 369 147 596 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 370 678 738 295 193 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 370 678 738 295 193 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 741 357 476 590 387 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 741 357 476 590 387 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 482 714 953 180 774 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 482 714 953 180 774 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 965 429 906 361 548 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 965 429 906 361 548 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 930 859 812 723 097 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 930 859 812 723 097 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 579 861 719 625 446 195 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 579 861 719 625 446 195 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 159 723 439 250 892 390 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 159 723 439 250 892 390 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 319 446 878 501 784 780 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 319 446 878 501 784 780 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 638 893 757 003 569 561 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 638 893 757 003 569 561 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 277 787 514 007 139 123 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 277 787 514 007 139 123 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 555 575 028 014 278 246 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 555 575 028 014 278 246 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 111 150 056 028 556 492 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 111 150 056 028 556 492 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 222 300 112 057 112 985 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 222 300 112 057 112 985 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 444 600 224 114 225 971 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 444 600 224 114 225 971 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 889 200 448 228 451 942 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 889 200 448 228 451 942 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 001 778 400 896 456 903 884 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 001 778 400 896 456 903 884 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 003 556 801 792 913 807 769 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 003 556 801 792 913 807 769 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 007 113 603 585 827 615 539 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 007 113 603 585 827 615 539 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 014 227 207 171 655 231 078 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 014 227 207 171 655 231 078 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 028 454 414 343 310 462 156 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 028 454 414 343 310 462 156 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 056 908 828 686 620 924 313 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 056 908 828 686 620 924 313 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 113 817 657 373 241 848 627 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 113 817 657 373 241 848 627 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 227 635 314 746 483 697 254 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 227 635 314 746 483 697 254 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 455 270 629 492 967 394 508 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 455 270 629 492 967 394 508 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 910 541 258 985 934 789 017 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 910 541 258 985 934 789 017 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 001 821 082 517 971 869 578 035 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 001 821 082 517 971 869 578 035 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 003 642 165 035 943 739 156 070 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 003 642 165 035 943 739 156 070 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 007 284 330 071 887 478 312 140 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 007 284 330 071 887 478 312 140 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 014 568 660 143 774 956 624 281 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 014 568 660 143 774 956 624 281 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 029 137 320 287 549 913 248 563 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 029 137 320 287 549 913 248 563 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 058 274 640 575 099 826 497 126 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 058 274 640 575 099 826 497 126 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 116 549 281 150 199 652 994 252 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 116 549 281 150 199 652 994 252 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 233 098 562 300 399 305 988 505 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 233 098 562 300 399 305 988 505 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 466 197 124 600 798 611 977 011 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 466 197 124 600 798 611 977 011 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 932 394 249 201 597 223 954 022 4;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 932 394 249 201 597 223 954 022 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 001 864 788 498 403 194 447 908 044 8;
  • 79) 0.794 189 453 125 000 000 000 000 000 001 864 788 498 403 194 447 908 044 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 003 729 576 996 806 388 895 816 089 6;
  • 80) 0.588 378 906 250 000 000 000 000 000 003 729 576 996 806 388 895 816 089 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 007 459 153 993 612 777 791 632 179 2;
  • 81) 0.176 757 812 500 000 000 000 000 000 007 459 153 993 612 777 791 632 179 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 014 918 307 987 225 555 583 264 358 4;
  • 82) 0.353 515 625 000 000 000 000 000 000 014 918 307 987 225 555 583 264 358 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 029 836 615 974 451 111 166 528 716 8;
  • 83) 0.707 031 250 000 000 000 000 000 000 029 836 615 974 451 111 166 528 716 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 059 673 231 948 902 222 333 057 433 6;
  • 84) 0.414 062 500 000 000 000 000 000 000 059 673 231 948 902 222 333 057 433 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 119 346 463 897 804 444 666 114 867 2;
  • 85) 0.828 125 000 000 000 000 000 000 000 119 346 463 897 804 444 666 114 867 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 238 692 927 795 608 889 332 229 734 4;
  • 86) 0.656 250 000 000 000 000 000 000 000 238 692 927 795 608 889 332 229 734 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 477 385 855 591 217 778 664 459 468 8;
  • 87) 0.312 500 000 000 000 000 000 000 000 477 385 855 591 217 778 664 459 468 8 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 954 771 711 182 435 557 328 918 937 6;
  • 88) 0.625 000 000 000 000 000 000 000 000 954 771 711 182 435 557 328 918 937 6 × 2 = 1 + 0.250 000 000 000 000 000 000 000 001 909 543 422 364 871 114 657 837 875 2;
  • 89) 0.250 000 000 000 000 000 000 000 001 909 543 422 364 871 114 657 837 875 2 × 2 = 0 + 0.500 000 000 000 000 000 000 000 003 819 086 844 729 742 229 315 675 750 4;
  • 90) 0.500 000 000 000 000 000 000 000 003 819 086 844 729 742 229 315 675 750 4 × 2 = 1 + 0.000 000 000 000 000 000 000 000 007 638 173 689 459 484 458 631 351 500 8;
  • 91) 0.000 000 000 000 000 000 000 000 007 638 173 689 459 484 458 631 351 500 8 × 2 = 0 + 0.000 000 000 000 000 000 000 000 015 276 347 378 918 968 917 262 703 001 6;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 994 2 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